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Teaching creative thinking Appleby College

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If correlation doesn’t imply causation, then what does? It is thinking Appleby, a commonplace of a written Chigwell School scientific discussion that correlation does not imply causation. Business Week recently ran an spoof article pointing out some amusing examples of the dangers of inferring causation from correlation. For example, the article points out that Facebook’s growth has been strongly correlated with the yield on Greek government bonds: (credit) Despite this strong correlation, it would not be wise to conclude that the teaching Appleby College, success of a college essay Kendall Facebook has somehow caused the current (2009-2012) Greek debt crisis, nor that the Greek debt crisis has caused the adoption of Appleby College Facebook!

Of course, while it’s all very well to piously state that correlation doesn’t imply causation, it does leave us with a conundrum: under what conditions, exactly, can we use experimental data to deduce a causal relationship between two or more variables? The standard scientific answer to how to a written School this question is that (with some caveats) we can infer causality from a well designed randomized controlled experiment. Unfortunately, while this answer is satisfying in principle and sometimes useful in practice, it’s often impractical or impossible to do a randomized controlled experiment. And so we’re left with the question of whether there are other procedures we can use to infer causality from teaching creative thinking Appleby experimental data. And, given that we can find more general procedures for inferring causal relationships, what does causality mean, anyway, for how we reason about a system?

It might seem that the answers to such fundamental questions would have been settled long ago. Help Writing A College Kendall College? In fact, they turn out to be surprisingly subtle questions. Over the past few decades, a group of scientists have developed a theory of causal inference intended to address these and teaching creative, other related questions. This theory can be thought of as an algebra or language for reasoning about cause and effect. Many elements of the theory have been laid out in a famous book by one of the Griffith University (Navitas), main contributors to the theory, Judea Pearl. Although the teaching, theory of causal inference is not yet fully formed, and is still undergoing development, what has already been accomplished is interesting and worth understanding. In this post I will describe one small but important part of the theory of causal inference, a causal calculus developed by how to argument Saint High Pearl. This causal calculus is Appleby College, a set of three simple but powerful algebraic rules which can be used to make inferences about essays Milano causal relationships.

In particular, I’ll explain how the causal calculus can sometimes (but not always!) be used to infer causation from a set of data, even when a randomized controlled experiment is teaching creative thinking Appleby College, not possible. Also in the post, I’ll describe some of the a college College, limits of the causal calculus, and creative thinking Appleby, some of my own speculations and questions. The post is writing, a little technically detailed at points. However, the first three sections of the post are non-technical, and I hope will be of teaching creative thinking broad interest. Throughout the post I’ve included occasional “Problems for the author”, where I describe problems I’d like to solve, or things I’d like to understand better. College Admission Lake? Feel free to ignore these if you find them distracting, but I hope they’ll give you some sense of what I find interesting about the subject. Incidentally, I’m sure many of these problems have already been solved by others; I’m not claiming that these are all open research problems, although perhaps some are. They’re simply things I’d like to understand better. Creative Appleby College? Also in the post I’ve included some exercises for the reader, and how to essays Appleby, some slightly harder problems for the reader.

You may find it informative to work through these exercises and problems. Before diving in, one final caveat: I am not an expert on causal inference, nor on statistics. The reason I wrote this post was to help me internalize the ideas of the causal calculus. Occasionally, one finds a presentation of teaching creative Appleby College a technical subject which is beautifully clear and illuminating, a presentation where the author has seen right through the subject, and help Kendall College, is able to convey that crystalized understanding to others. That’s a great aspirational goal, but I don’t yet have that understanding of causal inference, and these notes don’t meet that standard. Nonetheless, I hope others will find my notes useful, and that experts will speak up to Appleby College correct any errors or misapprehensions on my part. Let me start by explaining two example problems to illustrate some of the expository, difficulties we run into when making inferences about causality. College? The first is known as Simpson’s paradox. To explain Simpson’s paradox I’ll use a concrete example based on the passage of the Civil Rights Act in the United States in 1964.

In the US House of Representatives, 61 percent of Democrats voted for the Civil Rights Act, while a much higher percentage, 80 percent, of Republicans voted for the Act. You might think that we could conclude from essays this that being Republican, rather than Democrat, was an thinking important factor in causing someone to vote for the Civil Rights Act. However, the picture changes if we include an additional factor in the analysis, namely, whether a legislator came from a Northern or Southern state. If we include that extra factor, the essays Appleby, situation completely reverses, in both the creative thinking, North and the South. Here’s how it breaks down: North: Democrat (94 percent), Republican (85 percent)

South: Democrat (7 percent), Republican (0 percent) Yes, you read that right: in Istituto Milano both the thinking Appleby College, North and the South, a larger fraction of Democrats than Republicans voted for the Act, despite the fact that overall a larger fraction of Republicans than Democrats voted for Istituto, the Act. You might wonder how this can possibly be true. I’ll quickly state the thinking College, raw voting numbers, so you can check that the arithmetic works out, and then I’ll explain why it’s true. You can skip the numbers if you trust my arithmetic. North: Democrat (145/154, 94 percent), Republican (138/162, 85 percent) South: Democrat (7/94, 7 percent), Republican (0/10, 0 percent)

Overall: Democrat (152/248, 61 percent), Republican (138/172, 80 percent) One way of understanding what’s going on write argument essay Anthony’s High, is to note that a far greater proportion of Democrat (as opposed to Republican) legislators were from the thinking Appleby College, South. In fact, at write essays Appleby the time the House had 94 Democrats, and creative thinking Appleby, only 10 Republicans. Because of this enormous difference, the very low fraction (7 percent) of southern Democrats voting for the Act dragged down the Democrats’ overall percentage much more than did the even lower fraction (0 percent) of southern Republicans who voted for the Act. (The numbers above are for the House of Congress.

The numbers were different in the Senate, but the write Appleby, same overall phenomenon occurred. I’ve taken the numbers from Wikipedia’s article about Simpson’s paradox, and teaching creative thinking College, there are more details there.) If we take a naive causal point of view, this result looks like a paradox. As I said above, the overall voting pattern seems to suggest that being Republican, rather than Democrat, was an important causal factor in how to write essays College voting for creative College, the Civil Rights Act. Yet if we look at the individual statistics in essay School both the North and the South, then we’d come to the exact opposite conclusion. To state the Appleby, same result more abstractly, Simpson’s paradox is the fact that the correlation between two variables can actually be reversed when additional factors are considered. How To Write Essay Saint? So two variables which appear correlated can become anticorrelated when another factor is taken into account. You might wonder if results like those we saw in voting on the Civil Rights Act are simply an unusual fluke. But, in fact, this is not that uncommon. Wikipedia’s page on Simpson’s paradox lists many important and similar real-world examples ranging from understanding whether there is gender-bias in university admissions to which treatment works best for kidney stones. In each case, understanding the causal relationships turns out to be much more complex than one might at first think.

I’ll now go through a second example of creative thinking College Simpson’s paradox, the kidney stone treatment example just mentioned, because it helps drive home just how bad our intuitions about statistics and causality are. Imagine you suffer from kidney stones, and your Doctor offers you two choices: treatment A or treatment B. Your Doctor tells you that the argument Saint, two treatments have been tested in a trial, and creative thinking Appleby, treatment A was effective for a higher percentage of write article (Navitas) patients than treatment B. If you’re like most people, at this point you’d say “Well, okay, I’ll go with treatment A”. Here’s the gotcha. Keep in creative College mind that this really happened . Suppose you divide patients in write a written Chigwell the trial up into those with large kidney stones, and those with small kidney stones. Then even though treatment A was effective for a higher overall percentage of patients than treatment B, treatment B was effective for a higher percentage of patients in creative both groups , i.e., for both large and small kidney stones. So your Doctor could just as honestly have said “Well, you have large [or small] kidney stones, and how to Chigwell, treatment B worked for a higher percentage of patients with large [or small] kidney stones than treatment A”. If your Doctor had made either one of these statements, then if you’re like most people you’d have decided to go with treatment B, i.e., the exact opposite teaching creative Appleby treatment.

The kidney stone example relies, of course, on the same kind of arithmetic as in the Civil Rights Act voting, and it’s worth stopping to figure out for yourself how the claims I made above could possibly be true. Admission Forest? If you’re having trouble, you can click through to the Wikipedia page, which has all the details of the teaching Appleby College, numbers. Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Help Writing? Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of thinking Appleby College statistical evidence are obviously not just a little wrong, but utterly, horribly wrong. Perhaps I’m alone in having terrible intuition about how to interpret statistics. But frankly I wouldn’t be surprised if most people share my confusion.

I often wonder how many people with real decision-making power – politicians, judges, and so on – are making decisions based on statistical studies, and yet they don’t understand even basic things like Simpson’s paradox. Or, to how to expository essays Appleby put it another way, they have not the first clue about statistics. Partial evidence may be worse than no evidence if it leads to an illusion of knowledge, and so to overconfidence and certainty where none is justified. It’s better to know that you don’t know. Correlation, causation, smoking, and creative thinking College, lung cancer. As a second example of the difficulties in establishing causality, consider the relationship between cigarette smoking and lung cancer. Writing Essays In College Marangoni? In 1964 the United States’ Surgeon General issued a report claiming that cigarette smoking causes lung cancer. Teaching Thinking Appleby College? Unfortunately, according to Pearl the help a college essay Kendall, evidence in the report was based primarily on correlations between cigarette smoking and lung cancer.

As a result the teaching thinking, report came under attack not just by tobacco companies, but also by some of the world’s most prominent statisticians, including the great Ronald Fisher. A Written Essay Chigwell? They claimed that there could be a hidden factor – maybe some kind of genetic factor – which caused both lung cancer and teaching thinking College, people to want to smoke (i.e., nicotine craving). If that was true, then while smoking and lung cancer would be correlated, the decision to smoke or not smoke would have no impact on writing, whether you got lung cancer. Now, you might scoff at this notion. But derision isn’t a principled argument. And, as the example of Simpson’s paradox showed, determining causality on the basis of creative thinking Appleby College correlations is tricky, at best, and can potentially lead to contradictory conclusions.

It’d be much better to have a principled way of using data to conclude that the relationship between smoking and help Kendall College, lung cancer is not just a correlation, but rather that there truly is a causal relationship. One way of demonstrating this kind of causal connection is to do a randomized, controlled experiment. Thinking? We suppose there is some experimenter who has the help a college essay Kendall, power to teaching creative thinking Appleby intervene with a person, literally forcing them to either smoke (or not) according to the whim of the experimenter. The experimenter takes a large group of people, and a seo article University, randomly divides them into two halves. One half are forced to smoke, while the other half are forced not to teaching smoke. By doing this the write argument Anthony’s High, experimenter can break the thinking, relationship between smoking and any hidden factor causing both smoking and lung cancer. By comparing the cancer rates in the group who were forced to smoke to those who were forced not to smoke, it would then be possible determine whether or not there is truly a causal connection between smoking and lung cancer.

This kind of randomized, controlled experiment is highly desirable when it can be done, but experimenters often don’t have this power. In the case of how to write a written essay smoking, this kind of experiment would probably be illegal today, and, I suspect, even decades into the past. And even when it’s legal, in many cases it would be impractical, as in the case of the Civil Rights Act, and for many other important political, legal, medical, and econonomic questions. To help address problems like the thinking Appleby, two example problems just discussed, Pearl introduced a causal calculus. In the remainder of this post, I will explain the rules of the writing essays Marangoni Milano, causal calculus, and use them to analyse the smoking-cancer connection. We’ll see that even without doing a randomized controlled experiment it’s possible (with the aid of teaching creative some reasonable assumptions) to infer what the outcome of a randomized controlled experiment would have been, using only relatively easily accessible experimental data, data that doesn’t require experimental intervention to force people to smoke or not, but which can be obtained from purely observational studies. To state the rules of the causal calculus, we’ll need several background ideas. I’ll explain those ideas over the next three sections of this post. The ideas are causal models (covered in write article (Navitas) this section), causal conditional probabilities , and creative Appleby College, d-separation , respectively.

It’s a lot to argument essay Saint Anthony’s School swallow, but the ideas are powerful, and worth taking the time to understand. Teaching Creative Thinking Appleby College? With these notions under our belts, we’ll able to understand the rules of the causal calculus. To understand causal models, consider the help a college essay, following graph of possible causal relationships between smoking, lung cancer, and some unknown hidden factor (say, a hidden genetic factor): This is creative, a quite general model of causal relationships, in the sense that it includes both the suggestion of the US Surgeon General (smoking causes cancer) and also the suggestion of the essays, tobacco companies (a hidden factor causes both smoking and cancer). Indeed, it also allows a third possibility: that perhaps both smoking and some hidden factor contribute to lung cancer. This combined relationship could potentially be quite complex: it could be, for example, that smoking alone actually reduces the chance of lung cancer, but the hidden factor increases the chance of lung cancer so much that someone who smokes would, on average, see an teaching creative Appleby increased probability of lung cancer. This sounds unlikely, but later we’ll see some toy model data which has exactly this property. Of course, the model depicted in the graph above is not the most general possible model of causal relationships in this system; it’s easy to imagine much more complex causal models.

But at the very least this is an writing essays Istituto Marangoni Milano interesting causal model, since it encompasses both the US Surgeon General and the tobacco company suggestions. Teaching Thinking? I’ll return later to writing a college essay Kendall the possibility of more general causal models, but for now we’ll simply keep this model in mind as a concrete example of a causal model. Mathematically speaking, what do the arrows of causality in the diagram above mean? We’ll develop an answer to that question over the next few paragraphs. It helps to teaching start by moving away from the specific smoking-cancer model to allow a causal model to be based on a more general graph indicating possible causal relationships between a number of variables: Each vertex in writing college admission essays Forest Academy this causal model has an creative thinking Appleby associated random variable, . For example, in the causal model above could be a two-outcome random variable indicating the presence or absence of some gene that exerts an influence on write argument essay, whether someone smokes or gets lung cancer, indicates “smokes” or “does not smoke”, and indicates “gets lung cancer” or “doesn’t get lung cancer”. The other variables and would refer to creative other potential dependencies in how to write a seo article University this (somewhat more complex) model of the smoking-cancer connection.

A notational convention that we’ll use often is to interchangeably use to refer to a random variable in the causal model, and also as a way of labelling the corresponding vertex in the graph for teaching thinking Appleby, the causal model. It should be clear from college essays Lake context which is meant. We’ll also sometimes refer interchangeably to the causal model or to the associated graph. For the teaching creative thinking, notion of causality to make sense we need to constrain the class of how to argument Saint Anthony’s School graphs that can be used in a causal model. Obviously, it’d make no sense to have loops in the graph: We can’t have causing causing causing ! At least, not without a time machine. Because of Appleby College this we constrain the graph to be a directed acyclic graph, meaning a (directed) graph which has no loops in it. By the way, I must admit that I’m not a fan of the term directed acyclic graph. It sounds like a very complicated notion, at least to my ear, when what it means is very simple: a graph with no loops. I’d really prefer to call it a “loop-free graph”, or something like that.

Unfortunately, the “directed acyclic graph” nomenclature is pretty standard, so we’ll go with it. Our picture so far is writing essays Marangoni, that a causal model consists of a directed acyclic graph, whose vertices are labelled by random variables . To complete our definition of teaching thinking Appleby College causal models we need to writing a college essay capture the allowed relationships between those random variables. Intuitively, what causality means is that for any particular the creative, only random variables which directly influence the value of are the parents of , i.e., the collection of how to article Griffith University random variables which are connected directly to . For instance, in the graph shown below (which is the thinking College, same as the complex graph we saw a little earlier), we have : Now, of course, vertices further back in the graph – say, the parents of the parents – could, of course, influence the value of . But it would be indirect, an influence mediated through the parent vertices. Note, by the way, that I’ve overloaded the how to a seo Griffith University, notation, using to denote a collection of random variables. I’ll use this kind of overloading quite a bit in the rest of teaching Appleby this post. In particular, I’ll often use the a written essay, notation (or , or ) to denote a subset of random variables from the graph. Motivated by the above discussion, one way we could define causal influence would be to require that be a function of its parents: where is some function. In fact, we’ll allow a slightly more general notion of causal influence, allowing to not just be a deterministic function of the Appleby College, parents, but a random function. We do this by requiring that be expressible in the form: where is a function, and is a collection of random variables such that: (a) the are independent of writing Marangoni one another for different values of ; and (b) for each , is independent of all variables , except when is itself, or a descendant of . The intuition is that the are a collection of auxiliary random variables which inject some extra randomness into (and, through , its descendants), but which are otherwise independent of the variables in creative Appleby the causal model. Summing up, a causal model consists of a directed acyclic graph, , whose vertices are labelled by random variables, , and each is expressible in the form for writing essays Istituto, some function . Creative Appleby? The are independent of one another, and writing college admission essays Lake Forest Academy, each is independent of all variables , except when is or a descendant of teaching creative Appleby College . In practice, we will not work directly with the functions or the auxiliary random variables . Instead, we’ll work with the following equation, which specifies the causal model’s joint probability distribution as a product of conditional probabilities:

I won’t prove this equation, but the expression should be plausible, and is pretty easy to prove; I’ve asked you to essays in college Milano prove it as an optional exercise below. Prove the above equation for the joint probability distribution. (Simpson’s paradox in causal models) Consider the causal model of smoking introduced above. Suppose that the hidden factor is a gene which is teaching creative Appleby College, either switched on or off. If on, it tends to make people both smoke and get lung cancer.

Find explicit values for conditional probabilities in the causal model such that , and yet if the additional genetic factor is taken into account this relationship is reversed. That is, we have both and . An alternate, equivalent approach to defining causal models is as follows: (1) all root vertices (i.e., vertices with no parents) in the graph are labelled by independent random variables. (2) augment the graph by introducing new vertices corresponding to the . These new vertices have single outgoing edges, pointing to . How To Argument Essay Anthony’s High? (3) Require that non-root vertices in the augmented graph be deterministic functions of their parents. The disadvantage of this definition is that it introduces the overhead of dealing with the augmented graph. But the definition also has the advantage of cleanly separating the stochastic and deterministic components, and I wouldn’t be surprised if developing the thinking College, theory of causal inference from this point of view was stimulating, at help essay Kendall College the very least, and may possibly have some advantages compared to the standard approach. So the problem I set myself (and anyone else who is interested!) is to carry the consequences of Appleby this change through the rest of the theory of causal inference, looking for advantages and disadvantages. I’ve been using terms like “causal influence” somewhat indiscriminately in the discussion above, and article University, so I’d like to pause to teaching creative discuss a bit more carefully about what is how to argument Saint High, meant here, and what nomenclature we should use going forward. All the arrows in a causal model indicate are the possibility of a direct causal influence. This results in two caveats on how we think about causality in creative thinking Appleby these models. How To Write Essay Saint Anthony’s? First, it may be that a child random variable is actually completely independent of the value of one (or more) of its parent random variables.

This is, admittedly, a rather special case, but is perfectly consistent with the definition. For example, in a causal model like. it is possible that the outcome of cancer might be independent of the hidden causal factor or, for that matter, that it might be independent of whether someone smokes or not. (Indeed, logically, at least, it may be independent of both, although of course that’s not what we’ll find in teaching Appleby College the real world.) The second caveat in how we think about the arrows and causality is that the arrows only capture the direct causal influences in the model. Write Essay Chigwell School? It is possible that in a causal model like. will have a causal influence on through its influence on teaching creative thinking College, and . This would be an indirect causal influence, mediated by other random variables, but it would still be a causal influence. In the writing college admission Academy, next section I’ll give a more formal definition of causal influence that can be used to teaching creative make these ideas precise. In this section I’ll explain what I think is the most imaginative leap underlying the causal calculus. It’s the writing essays Istituto Marangoni, introduction of the concept of causal conditional probabilities . The notion of ordinary conditional probabilities is no doubt familiar to teaching Appleby you. It’s pretty straightforward to do experiments to estimate conditional probabilities such as , simply by write School looking at the population of creative thinking Appleby people who smoke, and figuring out how to School what fraction of those people develop cancer. Unfortunately, for the purpose of Appleby College understanding the causal relationship between smoking and cancer, isn’t the quantity we want. As the tobacco companies pointed out, there might well be a hidden genetic factor that makes it very likely that you’ll see cancer in anyone who smokes, but that wouldn’t therefore mean that smoking causes cancer.

As we discussed earlier, what you’d really like to do in this circumstance is a randomized controlled experiment in which it’s possible for the experimenter to Anthony’s School force someone to smoke (or not smoke), breaking the causal connection between the hidden factor and smoking. Appleby College? In such an help writing Kendall College experiment you really could see if there was a causal influence by looking at what fraction of teaching people who smoked got cancer. In particular, if that fraction was higher than in the overall population then you’d be justified in a college College concluding that smoking helped cause cancer. In practice, it’s probably not practical to do this kind of randomized controlled experiment. Teaching Thinking? But Pearl had what turns out to be a very clever idea: to imagine a hypothetical world in how to essay Saint Anthony’s School which it really is possible to force someone to (for example) smoke, or not smoke. In particular, he introduced a conditional causal probability , which is the thinking Appleby, conditional probability of cancer in this hypothetical world. Expository? This should be read as the (causal conditional) probability of cancer given that we “do” smoking, i.e., someone has been forced to smoke in creative thinking a (hypothetical) randomized experiment. Now, at first sight this appears a rather useless thing to do. How To Chigwell School? But what makes it a clever imaginative leap is that although it may be impossible or impractical to do a controlled experiment to determine , Pearl was able to establish a set of rules – a causal calculus – that such causal conditional probabilities should obey.

And, by thinking College making use of this causal calculus, it turns out to sometimes be possible to infer the value of probabilities such as , even when a controlled, randomized experiment is impossible. And that’s a very remarkable thing to be able to help a college Kendall do, and why I say it was so clever to have introduced the teaching creative thinking Appleby, notion of causal conditional probabilities. We’ll discuss the rules of the causal calculus later in this post. For now, though, let’s develop the notion of write Saint High causal conditional probabilities. Suppose we have a causal model of some phenomenon: Now suppose we introduce an external experimenter who is able to intervene to deliberately set the value of a particular variable to . In other words, the experimenter can override the other causal influences on that variable. This is creative thinking Appleby College, equivalent to having a new causal model: In this new causal model, we’ve represented the experimenter by a new vertex, which has as a child the vertex . How To Essay School? All other parents of teaching thinking Appleby are cut off, i.e., the edges from the parents to are deleted from the graph. In this case that means the edge from to has been deleted. This represents the fact that the experimenter’s intervention overrides the other causal influences. (Note that the edges to the children of how to a written essay Chigwell are left undisturbed.) In fact, it’s even simpler (and equivalent) to consider a causal model where the parents have been cut off from , and no extra vertex added: This model has no vertex explicitly representing the experimenter, but rather the Appleby, relation is replaced by the relation . We will denote this graph by , indicating the graph in which all edges pointing to how to essay Chigwell have been deleted.

We will call this a perturbed graph , and the corresponding causal model a perturbed causal model . In the teaching Appleby, perturbed causal model the only change is to delete the edges to , and to replace the relation by writing Forest Academy the relation . Our aim is to use this perturbed causal model to compute the conditional causal probability . In this expression, indicates that the term is omitted before the , since the value of is set on the right. By definition, the thinking College, causal conditional probability is just the value of the probability distribution in the perturbed causal model, . To compute the how to essay Chigwell, value of the probability in the perturbed causal model, note that the probability distribution in thinking the original causal model was given by. where the write expository essays, product on creative Appleby College, the right is over all vertices in the causal model. This expression remains true for the perturbed causal model, but a single term on the right-hand side changes: the conditional probability for the term. In particular, this term gets changed from to , since we have fixed the value of to be . As a result we have: This equation is a fundamental expression, capturing what it means for an experimenter to intervene to set the value of a college essay College some particular variable in a causal model. Creative Thinking College? It can easily be generalized to a situation where we partition the variables into two sets, and , where are the write expository College, variables we suppose have been set by teaching thinking Appleby intervention in a (possibly hypothetical) randomized controlled experiment, and how to argument essay School, are the remaining variables:

Note that on the right-hand side the values for are assumed to be given by the appropriate values from and . The expression [1] can be viewed as a definition of causal conditional probabilities. But although this expression is fundamental to understanding the causal calculus, it is not always useful in practice. The problem is that the values of some of the thinking, variables on the right-hand side may not be known, and cannot be determined by experiment. Consider, for example, the case of smoking and cancer. Recall our causal model: What we’d like is to compute . Unfortunately, we immediately run into writing essays Istituto a problem if we try to use the expression on the right of equation [1]: we’ve got no way of estimating the conditional probabilities for smoking given the hidden common factor. So we can’t obviously compute . Teaching Thinking? And, as you can perhaps imagine, this is the kind of problem that will come up a lot whenever we’re worried about the possible influence of some hidden factor. All is not lost, however. How To Write Expository Appleby College? Just because we can’t compute the teaching Appleby College, expression on essays in college Istituto Milano, the right of [1] directly doesn’t mean we can’t compute causal conditional probabilities in other ways, and we’ll see below how the causal calculus can help solve this kind of problem. It’s not a complete solution – we shall see that it doesn’t always make it possible to compute causal conditional probabilities. But it does help.

In particular, we’ll see that although it’s not possible to compute for teaching creative thinking College, this causal model, it is possible to writing essay Kendall compute in teaching creative a very similar causal model, one that still has a hidden factor. With causal conditional probabilities defined, we’re now in position to define more precisely what we mean by causal influence. Suppose we have a causal model, and and are distinct random variables (or disjoint subsets of random variables). Then we say has a causal influence over if there are values and college essays Academy, of and of such that . Teaching Creative Thinking College? In other words, an external experimenter who can intervene to change the write Anthony’s High, value of can cause a corresponding change in the distribution of values at creative Appleby . The following exercise gives an information-theoretic justification for this definition of how to write expository essays College causal influence: it shows that an teaching thinking College experimenter who can intervene to set can transmit information to if and only if the above condition for causal inference is met. (The causal capacity) This exercise is for people with some background in information theory.

Suppose we define the write School, causal capacity between and to be , where is the mutual information, the maximization is over possible distributions for creative, (we use the hat to indicate that the how to write a seo Griffith, value of is being set by creative College intervention), and is the how to essays College, corresponding random variable at teaching creative thinking , with distribution . How To Write Argument School? Shannon’s noisy channel coding theorem tells us that an teaching creative thinking College external experimenter who can intervene to set the value of how to argument essay Saint High School can transmit information to an observer at teaching creative thinking Appleby College at a maximal rate set by the causal capacity. Show that the causal capacity is greater than zero if and only if has a causal influence over . We’ve just defined a notion of causal influence between two random variables in a causal model. What about when we say something like “Event A” causes “Event B”? What does this mean? Returning to the smoking-cancer example, it seems that we would say that smoking causes cancer provided , so that if someone makes the choice to smoke, uninfluenced by other causal factors, then they would increase their chance of cancer. Intuitively, it seems to how to write argument Saint Anthony’s High School me that this notion of thinking College events causing one another should be related to the notion of causal influence just defined above. But I don’t yet see quite how to do that. The first problem below suggests a conjecture in this direction: Suppose and are random variables in a causal model such that for some pair of values and . Does this imply that exerts a causal influence on ? (Sum-over-paths for causal conditional probabilities?) I believe a kind of sum-over-paths formulation of causal conditional probabilities is possible, but haven’t worked out write Griffith University (Navitas) details.

The idea is as follows (the details may be quite wrong, but I believe something along these lines should work). Supose and are single vertices (with corresponding random variables) in a causal model. Then I would like to show first that if is not an ancestor of then , i.e., intervention does nothing. Second, if is an teaching creative College ancestor of then may be obtained by summing over all directed paths from to in , and computing for each path a contribution to the sum which is writing essay, a product of creative Appleby conditional probabilities along the path. Write Argument Essay Saint? (Note that we may need to consider the same path multiple times in the sum, since the random variables along the path may take different values). Creative Thinking Appleby College? We used causal models in our definition of causal conditional probabilities. But our informal definiton – imagine a hypothetical world in which it’s possible to force a variable to take a particular value – didn’t obviously require the use of a causal model. Argument Essay Saint Anthony’s High School? Indeed, in a real-world randomized controlled experiment it may be that there is no underlying causal model. Creative Thinking Appleby College? This leads me to writing essays in college wonder if there is some other way of formalizing the informal definition we’ve given? Another way of teaching creative thinking framing the how to a seo article Griffith University, last problem is that I’m concerned about the empirical basis for causal models. How should we go about constructing such models?

Are they fundamental, representing true facts about the teaching creative, world, or are they modelling conveniences? (This is by write Saint High no means a dichotomy.) It would be useful to teaching thinking work through many more examples, considering carefully the origin of the functions and writing admission essays Lake Forest Academy, of the auxiliary random variables . In this section we’ll develop a criterion that Pearl calls directional separation ( d-separation , for short). What d-separation does is let us inspect the Appleby, graph of a causal model and conclude that a random variable in the model can’t tell us anything about the value of another random variable in the model, or vice versa. To understand d-separation we’ll start with a simple case, and argument Anthony’s High School, then work through increasingly complex cases, building up our intuition. I’ll conclude by teaching College giving a precise definition of d-separation, and by explaining how d-separation relates to the concept of a written essay School conditional independence of teaching creative random variables. Here’s the first simple causal model: Clearly, knowing can in general tell us something about in this kind of causal model, and so in this case and writing Istituto Marangoni Milano, are not d-separated. We’ll use the term d-connected as a synonym for “not d-separated”, and teaching, so in essays College this causal model and teaching Appleby College, are d-connected. By contrast, in how to write Chigwell the following causal model and don’t give us any information about each other, and so they are d-separated: A useful piece of terminology is to say that a vertex like the middle vertex in this model is thinking College, a collider for the path from to , meaning a vertex at which both edges along the path are incoming.

What about the causal model: In this case, it is possible that knowing will tell us something about writing Istituto , because of teaching creative Appleby College their common ancestry. Essays Marangoni Milano? It’s like the way knowing the genome for Appleby College, one sibling can give us information about the genome of another sibling, since similarities between the genomes can be inferred from the common ancestry. We’ll call a vertex like the middle vertex in how to a seo Griffith University this model a fork for teaching creative, the path from to write expository essays Appleby , meaning a vertex at teaching Appleby College which both edges are outgoing. Construct an explicit causal model demonstrating the assertion of the last paragraph. Help Writing Kendall College? For example, you may construct a causal model in which and are joined by a fork, and where is actually a function of . Suppose we have a path from to in a causal model. Let be the number of colliders along the path, and let be the number of forks along the creative Appleby, path.

Show that can only take the values or , i.e., the number of forks and colliders is either the writing college Lake Forest, same or differs by at most one. We’ll say that a path (of any length) from to teaching creative Appleby College that contains a collider is a blocked path. By contrast, a path that contains no colliders is called an unblocked path. (Note that by the above exercise, an unblocked path must contain either one or no forks.) In general, we define and to Kendall College be d-connected if there is an unblocked path between them. We define them to be d-separated if there is no such unblocked path. It’s worth noting that the concepts of d-separation and d-connectedness depend only on the graph topology and on which vertices and have been chosen.

In particular, they don’t depend on the nature of the random variables and , merely on the identity of the corresponding vertices. As a result, you can determine d-separation or d-connectdness simply by teaching creative Appleby inspecting the graph. This fact – that d-separation and help writing essay College, d-connectdness are determined by the graph – also holds for the more sophisticated notions of d-separation and d-connectedness we develop below. With that said, it probably won’t surprise you to College learn that the a college essay, concept of d-separation is closely related to whether or not the random variables and are independent of one another. This is a connection you can (optionally) develop through the teaching, following exercises. I’ll state a much more general connection below. Suppose that and are d-separated. Show that and are independent random variables, i.e., that . Suppose we have two vertices which are d-connected in a graph . Explain how to construct a causal model on that graph such that the random variables and corresponding to those two vertices are not independent. The last two exercises almost but don’t quite claim that random variables and in a causal model are independent if and only if they are d-separated.

Why does this statement fail to be true? How can you modify the statement to make it true? So far, this is pretty simple stuff. It gets more complicated, however, when we extend the notion of d-separation to cases where we are conditioning on writing admission Lake Forest Academy, already knowing the value of one or more random variables in the causal model. Consider, for example, the graph: Now, if we know , then knowing doesn’t give us any additional information about , since by our original definition of a causal model is already a function of and some auxiliary random variables which are independent of Appleby . So it makes sense to say that blocks this path from to , even though in how to write a written essay School the unconditioned case this path would not have been considered blocked. Creative Appleby? We’ll also say that and are d-separated, given . It is helpful to give a name to vertices like the admission Lake Forest, middle vertex in Figure A, i.e., to vertices with one ingoing and one outgoing edge.

We’ll call such vertices a traverse along the path from to . Creative Appleby College? Using this language, the lesson of the above discussion is Griffith (Navitas), that if is in a traverse along a path from to , then the path is blocked. By contrast, consider this model: In this case, knowing will in general give us additional information about , even if we know . This is thinking Appleby, because while blocks one path from to there is another unblocked path from to . Writing Lake Forest? And so we say that and are d-connected, given . Another case similar to Figure A is the model with a fork: Again, if we know , then knowing as well doesn’t give us any extra information about thinking Appleby (or vice versa). So we’ll say that in this case is blocking the write a written essay, path from to , even though in the unconditioned case this path would not have been considered blocked. Thinking? Again, in this example and are d-separated, given . The lesson of this model is that if is admission Lake Forest, located at a fork along a path from to , then the path is blocked. A subtlety arises when we consider a collider: In the unconditioned case this would have been considered a blocked path. And, naively, it seems as though this should still be the case: at first sight (at least according to my intuition) it doesn’t seem very likely that can give us any additional information about (or vice versa), even given that is creative thinking Appleby College, known. Yet we should be cautious, because the argument we made for the graph in Figure A breaks down: we can’t say, as we did for Figure A, that is a function of and some auxiliary independent random variables. In fact, we’re wise to be cautious because and really can tell us something extra about how to expository essays College one another, given a knowledge of . This is a phenomenon which Pearl calls Berkson’s paradox . He gives the example of a graduate school in music which will admit a student (a possibility encoded in the value of ) if either they have high undergraduate grades (encoded in ) or some other evidence that they are exceptionally gifted at music (encoded in teaching thinking Appleby ). It would not be surprising if these two attributes were anticorrelated amongst students in the program, e.g., students who were admitted on the basis of exceptional gifts would be more likely than otherwise to have low grades.

And so in this case knowledge of (exceptional gifts) would give us knowledge of (likely to have low grades), conditioned on knowledge of (they were accepted into how to Chigwell School the program). Another way of seeing Berkson’s paradox is to Appleby College construct an explicit causal model for the graph in Figure B. Consider, for example, a causal model in which and are independent random bits, or , chosen with equal probabilities . We suppose that , where is how to expository Appleby, addition modulo . This causal model does, indeed, have the structure of Figure B. Teaching Creative Thinking Appleby College? But given that we know the how to a seo University, value , knowing the value of tells us everything about teaching thinking Appleby , since . As a result of this discussion, in the causal graph of Figure B we’ll say that unblocks the path from to , even though in the unconditioned case the path would have been considered blocked. And we’ll also say that in college admission essays this causal graph and are d-connected, conditional on teaching creative thinking Appleby College, . The immediate lesson from the graph of Figure B is how to a seo article Griffith, that and can tell us something about thinking College one another, given , if there is how to write a written essay Chigwell School, a path between and where the only collider is at . In fact, the teaching creative thinking Appleby College, same phenomenon can occur even in this graph: To see this, suppose we choose and as in the example just described above, i.e., independent random bits, or , chosen with equal probabilities . How To Essays Appleby? We will let the teaching creative Appleby, unlabelled vertex be . And, finally, we choose . Then we see as before that can tell us something about , given that we know , because . The general intuition about graphs like that in Figure C is that knowing allows us to infer something about the a seo article Griffith, ancestors of , and so we must act as though those ancestors are known, too. Thinking College? As a result, in this case we say that unblocks the path from to , since has an ancestor which is a collider on the path from to . Essays? And so in this case is d-connected to , given . Given the discussion of Figure C that we’ve just had, you might wonder why forks or traverses which are ancestors of can’t block a path, for similar reasons? For instance, why don’t we consider and to be d-separated, given , in creative College the following graph: The reason, of course, is that it’s easy to construct examples where tells us something about in addition to what we already know from . And so we can’t consider and to be d-separated, given , in this example.

These examples motivate the Anthony’s School, following definition: Definition: Let , and teaching creative thinking, be disjoint subsets of vertices in a causal model. Consider a path from a vertex in to a vertex in Saint High School . We say the teaching College, path is blocked by if the path contains either: (a) a collider which is not an ancestor of write essays Appleby , or (b) a fork which is in teaching creative Appleby College , or (c) a traverse which is in . We say the path is unblocked if it is not blocked. We say that and are d-connected , given , if there is an unblocked path between some vertex in college admission essays Forest and some vertex in . and are d-separated , given , if they are not d-connected. Saying “ and are d-separated given ” is a bit of a mouthful, and so it’s helpful to have an abbreviated notation. We’ll use the abbreviation . Note that this notation includes the graph ; we’ll sometimes omit the graph when the context is clear. We’ll write to denote unconditional d-separation. As an teaching College aside, Pearl uses a similar but slightly different notation for d-separation, namely . Unfortunately, while the write essay, symbol looks like a LaTeX symbol, it’s not, but is teaching thinking Appleby, most easily produced using a rather dodgy LaTeX hack.

Instead of using that hack over and over again, I’ve adopted a more standard LaTeX notation. While I’m making asides, let me make a second: when I was first learning this material, I found the “d” for “directional” in writing essays Istituto Marangoni Milano d-separation and d-connected rather confusing. It suggested to me that the key thing was having a directed path from one vertex to the other, and that the complexities of colliders, forks, and creative College, so on were a sideshow. Of course, they’re not, they’re central to how to write a written essay School the whole discussion. For this reason, when I was writing these notes I considered changing the terminology to teaching creative College i-separated and i-connected, for informationally-separated and how to write Anthony’s, informationally-connected. Ultimately I decided not to do this, but I thought mentioning the issue might be helpful, in thinking Appleby part to reassure readers (like me) who thought the write a written, “d” seemed a little mysterious. Okay, that’s enough asides, let’s get back to the main track of discussion. We saw earlier that (unconditional) d-separation is closely connected to the independence of thinking College random variables.

It probably won’t surprise you to learn that conditional d-separation is Istituto Milano, closely connected to conditional independence of random variables. Recall that two sets of random variables and teaching thinking, are conditionally independent , given a third set of random variables , if . The following theorem shows that d-separation gives a criterion for when conditional independence occurs in a causal model: Theorem (graphical criterion for conditional independence): Let be a graph, and let , and be disjoint subsets of vertices in that graph. How To Essay? Then and are d-separated, given , if and teaching thinking Appleby College, only if for all causal models on expository essays Appleby, the random variables corresponding to and are conditionally independent, given . (Update: Thanks to creative thinking Appleby College Rob Spekkens for how to write a seo article University, pointing out an error in teaching creative Appleby my original statement of write essay School this theorem.) I won’t prove the theorem here.

However, it’s not especially difficult if you’ve followed the discussion above, and is a good problem to work through: The concept of d-separation plays a central role in the causal calculus. My sense is that it should be possible to find a cleaner and more intuitive definition that substantially simplifies many proofs. Teaching Thinking College? It’d be good to spend some time trying to find such a definition. We’ve now got all the concepts we need to state the rules of the causal calculus. There are three rules. The rules look complicated at how to article Griffith (Navitas) first, although they’re easy to use once you get familiar with them.

For this reason I’ll start by explaining the College, intuition behind the first rule, and how you should think about writing essays in college Istituto Marangoni Milano that rule. Having understood how to think about the first rule it’s easy to get the hang of all three rules, and so after that I’ll just outright state all three rules. In what follows, we have a causal model on a graph , and are disjoint subsets of the variables in the causal model. Teaching? Recall also that denotes the perturbed graph in which all edges pointing to from the parents of essays Lake have been deleted. This is the graph which results when an experimenter intervenes to creative Appleby College set the value of writing college Lake Academy , overriding other causal influences on . Rule 1: When can we ignore observations: I’ll begin by stating the first rule in all its glory, but don’t worry if you don’t immediately grok the creative, whole rule. Instead, just take a look, and try to start getting your head around it. What we’ll do then is look at some simple special cases, which are easily understood, and gradually build up to an understanding of what the full rule is saying. Okay, so here’s the first rule of the causal calculus. What it tells us is that when , then we can ignore the observation of in computing the probability of help writing College , conditional on both and an intervention to set : To understand why this rule is true, and what it means, let’s start with a much simpler case. Let’s look at what happens to teaching creative thinking College the rule when there are no or variables in the mix.

In this case, our starting assumption simply becomes that is d-separated from in the original (unperturbed) graph . There’s no need to worry about because there’s no variable whose value is being set by intervention. In this circumstance we have , so is independent of . But the statement of the rule in this case is High School, merely that , which is, indeed, equivalent to the standard definition of and being independent. In other words, the first rule is simply a generalization of what it means for and to be independent. The full rule generalizes the notion of independence in Appleby two ways: (1) by adding in a written Chigwell School an extra variable whose value has been determined by passive observation; and (2) by adding in an extra variable whose value has been set by intervention. We’ll consider these two ways of generalizing separately in the next two paragraphs. We begin with generalization (1), i.e., there is no variable in teaching creative Appleby College the mix. Argument High? In this case, our starting assumption becomes that is d-separated from , given , in creative Appleby the graph . By the graphical criterion for conditional independence discussed in the last section this means that is conditionally independent of , given , and how to Anthony’s High School, so , which is exactly the statement of the rule.

And so the first rule can be viewed as a generalization of what it means for and to teaching College be independent, conditional on . Now let’s look at a seo the other generalization, (2), in which we’ve added an teaching creative thinking Appleby extra variable whose value has been set by intervention, and where there is no variable in the mix. In this case, our starting assumption becomes that is d-separated from , given , in the perturbed graph . In this case, the graphical criterion for conditional indepenence tells us that is independent from , conditional on the value of being set by experimental intervention, and so . Again, this is exactly the statement of the rule. The full rule, of course, merely combines both these generalizations in the obvious way. It is really just an explicit statement of the content of the graphical criterion for conditional independence, in a context where has been observed, and the value of help writing essay Kendall College set by experimental intervention. The rules of the causal calculus: All three rules of the causal calculus follow a similar template to the first rule: they provide ways of using facts about the causal structure (notably, d-separation) to make inferences about conditional causal probabilities. I’ll now state all three rules.

The intuition behind rules 2 and 3 won’t necessarily be entirely obvious, but after our discussion of rule 1 the remaining rules should at least appear plausible and creative Appleby, comprehensible. I’ll have bit more to say about intuition below. As above, we have a causal model on a graph , and are disjoint subsets of the variables in writing Istituto Marangoni the causal model. Teaching Creative? denotes the perturbed graph in which all edges pointing to from the parents of have been deleted. denotes the graph in how to University (Navitas) which all edges pointing out from to the children of have been deleted. We will also freely use notations like to denote combinations of these operations. Rule 1: When can we ignore observations: Suppose . Then: Rule 2: When can we ignore the act of intervention: Suppose . Then: Rule 3: When can we ignore an intervention variable entirely: Let denote the Appleby College, set of how to write article (Navitas) nodes in which are not ancestors of College . Write Essay Chigwell? Suppose . Then: In a sense, all three rules are statements of conditional independence.

The first rule tells us when we can ignore an thinking observation. The second rule tells us when we can ignore the act of intervention (although that doesn’t necessarily mean we can ignore the value of the variable being intervened with). And the a written, third rule tells us when we can ignore an intervention entirely, both the act of intervention, and the value of the variable being intervened with. I won’t prove rule 2 or rule 3 – this post is already quite long enough. (If I ever significantly revise the post I may include the teaching thinking Appleby, proofs). The important thing to take away from these rules is that they give us conditions on the structure of causal models so that we know when we can ignore observations, acts of intervention, or even entire variables that have been intervened with. This is writing Lake Forest Academy, obviously a powerful set of tools to be working with in manipulating conditional causal probabilities! Indeed, according to Pearl there’s even a sense in which this set of rules is complete , meaning that using these rules you can identify all causal effects in a causal model. I haven’t yet understood the proof of this result, or even exactly what it means, but thought I’d mention it. The proof is in papers by Shpitser and Pearl and Huang and Valtorta. If you’d like to see the proofs of the teaching creative thinking Appleby, rules of the how to write argument essay Saint Anthony’s, calculus, you can either have a go at proving them yourself, or you can read the teaching Appleby College, proof. Suppose the conditions of a written essay Chigwell School rules 1 and 2 hold.

Can we deduce that the conditions of rule 3 also hold? Using the causal calculus to analyse the smoking-lung cancer connection. We’ll now use the causal calculus to analyse the connection between smoking and lung cancer. Earlier, I introduced a simple causal model of this connection: The great benefit of this model was that it included as special cases both the hypothesis that smoking causes cancer and teaching, the hypothesis that some hidden causal factor was responsible for admission essays Academy, both smoking and cancer. It turns out, unfortunately, that the causal calculus doesn’t help us analyse this model. I’ll explain why that’s the creative thinking College, case below. However, rather than worrying about how to write a written essay Chigwell School this, at this stage it’s more instructive to work through an example showing how the causal calculus can be helpful in analysing a similar but slightly modified causal model. So although this modification looks a little mysterious at teaching creative thinking Appleby first, for now I hope you’ll be willing to argument School accept it as given. The way I’m going to modify the teaching Appleby, causal model is by introducing an extra variable, namely, whether someone has appreciable amounts of tar in Kendall their lungs or not: (By tar, I don’t mean “tar” literally, but rather all the material deposits found as a result of smoking.)

This causal model is a plausible modification of the original causal model. Creative Appleby College? It is at least plausible to suppose that smoking causes tar in the lungs and that those deposits in turn cause cancer. But if the hidden causal factor is genetic, as the in college Milano, tobacco companies argued was the case, then it seems highly unlikely that the genetic factor caused tar in creative Appleby College the lungs, except by the indirect route of writing in college Istituto Marangoni Milano causing those people to teaching thinking smoke. (I’ll come back to how to a seo Griffith what happens if you refuse to accept this line of teaching creative College reasoning. For now, just go with it.) Our goal in this modified causal model is to compute probabilities like . What we’ll show is that the causal calculus lets us compute this probability entirely in terms of how to a seo Griffith University (Navitas) probabilities like and other probabilities that don’t involve an intervention, i.e., that don’t involve . This means that we can determine without needing to know anything about the hidden factor. We won’t even need to know the creative, nature of the hidden factor. It also means that we can determine without needing to intervene to force someone to writing essays in college Istituto Marangoni Milano smoke or not smoke, i.e., to set the teaching creative, value for . In other words, the causal calculus lets us do something that seems almost miraculous: we can figure out the probability that someone would get cancer given that they are in the smoking group in a randomized controlled experiment, without needing to do the randomized controlled experiment. And this is true even though there may be a hidden causal factor underlying both smoking and cancer. Okay, so how do we compute ? The obvious first question to ask is whether we can apply rule 2 or rule 3 directly to the conditional causal probability . If rule 2 applies, for example, it would say that intervention doesn’t matter, and so . Intuitively, this seems unlikely. Write University? We’d expect that intervention really can change the probability of cancer given smoking, because intervention would override the hidden causal factor.

If rule 3 applies, it would say that , i.e., that an intervention to force someone to smoke has no impact on whether they get cancer. This seems even more unlikely than rule 2 applying. However, as practice and a warm up, let’s work through the details of seeing whether rule 2 or rule 3 can be applied directly to . For rule 2 to creative Appleby apply we need . To check whether this is true, recall that is the graph with the edges pointing out from deleted: Obviously, is not d-separated from in college essays Lake Academy this graph, since and have a common ancestor. This reflects the fact that the Appleby College, hidden causal factor indeed does influence both and . So we can’t apply rule 2.

What about rule 3? For this to apply we’d need . Recall that is the graph with the edges pointing toward deleted: Again, is not d-separated from , in this case because we have an unblocked path directly from to . How To A Written School? This reflects our intuition that the value of can influence , even when the teaching College, value of has been set by intervention. So we can’t apply rule 3. Okay, so we can’t apply the rules of the causal calculus directly to determine . Is there some indirect way we can determine this probability? An experienced probabilist would at this point instinctively wonder whether it would help to condition on essays Istituto Marangoni, the value of , writing: Of course, saying an experienced probabilist would instinctively do this isn’t quite the teaching creative thinking Appleby College, same as explaining why one should do this! However, it is at least a moderately obvious thing to do: the only extra information we potentially have in the problem is , and so it’s certainly somewhat natural to try to introduce that variable into the problem. As we shall see, this turns out to be a wise thing to do.

I used without proof the write a written Chigwell School, equation . This should be intuitively plausible, but really requires proof. Creative? Prove that the equation is correct. To simplify the right-hand side of how to write argument School equation [2], we first note that we can apply rule 2 to teaching College the second term on the right-hand side, obtaining . To check this explicitly, note that the condition for rule 2 to apply is that . We already saw the graph above, and, indeed, is d-separated from in that graph, since the only path from to how to essay Chigwell is blocked at . Teaching Creative Appleby College? As a result, we have: At this point in the presentation, I’m going to speed the discussion up, telling you what rule of the calculus to apply at each step, but not going through the process of write a seo Griffith (Navitas) explicitly checking that the conditions of the rule hold. Teaching Creative College? (If you’re doing a close read, you may wish to check the how to write a seo article Griffith University, conditions, however.) The next thing we do is to creative Appleby College apply rule 2 to the first term on the right-hand side of equation [3], obtaining . We then apply rule 3 to remove the (Navitas), , obtaining . Substituting back in gives us:

So this means that we’ve reduced the computation of to the computation of . This doesn’t seem terribly encouraging: we’ve merely substituted the thinking, computation of one causal conditional probability for another. Still, let us continue plugging away, and see if we can make progress. How To Argument Anthony’s? The obvious first thing to try is to creative thinking Appleby apply rule 2 or rule 3 to simplify . Unfortunately, though not terribly surprisingly, neither rule applies. So what do we do? Well, in a repeat of our strategy above, we again condition on the other variable we have available to us, in essay this case : Now we’re cooking!

Rule 2 lets us simplify the first term to , while rule 3 lets us simplify the creative thinking College, second term to , and so we have . To substitute this expression back into equation [4] it helps to writing essays Milano change the thinking College, summation index from to college admission Lake Academy , since otherwise we would have a duplicate summation index. Teaching Creative Thinking Appleby? This gives us: This is the promised expression for (i.e., for how to expository Appleby College, probabilities like , assuming the causal model above) in terms of quantities which may be observed directly from experimental data, and which don’t require intervention to teaching creative Appleby do a randomized, controlled experiment. Once is determined, we can compare it against . If is larger than then we can conclude that smoking does, indeed, play a causal role in cancer. Something that bugs me about the derivation of equation [5] is that I don’t really know how to “see through” the calculations. How To Write Article Griffith University (Navitas)? Yes, it all works out in the end, and it’s easy enough to teaching thinking follow along. Yet that’s not the how to essay High School, same as having a deep understanding. Too many basic questions remain unanswered: Why did we have to condition as we did in the calculation?

Was there some other way we could have proceeded? What would have happeed if we’d conditioned on teaching creative thinking College, the value of the hidden variable? (This is not obviously the write argument Saint Anthony’s High, wrong thing to teaching thinking do: maybe the hidden variable would ultimately drop out of the calculation). Why is it possible to compute causal probabilities in a written essay Chigwell this model, but not (as we shall see) in the model without tar? Ideally, a deeper understanding would make the answers to some or all of thinking these questions much more obvious. Why is it so much easier to compute than in the model above? Is there some way we could have seen that this would be the case, without needing to go through a detailed computation? Suppose we have a causal model , with a subset of vertices for write essay, which all conditional probabilities are known.

Is it possible to give a simple characterization of for thinking Appleby, which subsets and write essays College, of vertices it is possible to compute using just the conditional probabilities from ? Unfortunately, I don’t know what the experimentally observed probabilities are in the smoking-tar-cancer case. If anyone does, I’d be interested to know. In lieu of actual data, I’ll use some toy model data suggested by Pearl; the data is quite unrealistic, but nonetheless interesting as an thinking Appleby College illustration of the use of equation [5]. The toy model data is as follows: (1) 47.5 percent of the population are nonsmokers with no tar in their lungs, and 10 percent of how to write expository essays these get cancer. (2) 2.5 percent are smokers with no tar, and 90 percent get cancer. (3) 2.5 percent are nonsmokers with tar, and 5 percent get cancer. (4) 47.5 percent are smokers with tar, and 85 percent get cancer. In this case, we get:

By contrast, percent, and so if this data was correct (obviously it’s not even close) it would show that smoking actually somewhat reduces a person’s chance of getting lung cancer. This is despite the fact that percent, and so a naive approach to creative thinking causality based on correlations alone would suggest that smoking causes cancer. In fact, in this imagined world smoking might actually be useable as a preventative treatment for cancer! Obviously this isn’t truly the case, but it does illustrate the power of this method of analysis. Summing up the general lesson of the smoking-cancer example, suppose we have two competing hypotheses for the causal origin of some effect in a system, A causes C or B causes C, say.

Then we should try to construct a realistic causal model which includes both hypotheses, and writing college Lake Academy, then use the causal calculus to attempt to distinguish the relative influence of the two causal factors, on the basis of experimentally accessible data. Incidentally, the kind of analysis of smoking we did above obviously wasn’t done back in the 1960s. I don’t actually know how causality was established over the protestations that correlation doesn’t impy causation. But it’s not difficult to think of ways you might have come up with truly convincing evidence that smoking was a causal factor. One way would have been to look at the incidence of lung cancer in populations where smoking had only recently been introduced. Teaching Creative Thinking Appleby College? Suppose, for example, that cigarettes had just been introduced into write High the (fictional) country of Nicotinia, and that this had been quickly followed by a rapid increase in rates of lung cancer. If this pattern was seen across many new markets then it would be very difficult to argue that lung cancer was being caused solely by some pre-existing factor in teaching creative thinking the population. Construct toy model data where smoking increases a person’s chance of getting lung cancer. Let’s leave this model of smoking and lung cancer, and come back to our original model of smoking and lung cancer: What would have happened if we’d tried to write a written Chigwell School use the causal calculus to creative analyse this model? I won’t go through all the details, but you can easily check that whatever rule you try to apply you quickly run into a dead end.

And so the causal calculus doesn’t seem to be any help in analysing this problem. This example illustrates some of the limitations of the causal calculus. In order to compute we needed to assume a causal model with a particular structure: While this model is plausible, it is not beyond reproach. You could, for example, criticise it by saying that it is not the presence of tar deposits in the lungs that causes cancer, but maybe some other factor, perhaps something that is currently unknown. This might lead us to consider a causal model with a revised structure: So we could try instead to use the writing Kendall College, causal calculus to analyse this new model. I haven’t gone through this exercise, but I strongly suspect that doing so we wouldn’t be able to use the teaching creative thinking, rules of the causal calculus to compute the relevant probabilities. Lake Forest Academy? The intuition behind this suspicion is that we can imagine a world in which the tar may be a spurious side-effect of smoking that is in fact entirely unrelated to lung cancer. What causes lung cancer is really an entirely different mechanism, but we couldn’t distinguish the two from the statistics alone.

The point of this isn’t to say that the causal calculus is useless. It’s remarkable that we can plausibly get information about the outcome of a randomized controlled experiment without actually doing anything like that experiment. But there are limitations. To get that information we needed to make some presumptions about the causal structure in the system. Those presumptions are plausible, but not logically inevitable. If someone questions the presumptions then it may be necessary to revise the model, perhaps adopting a more sophisticated causal model. One can then use the causal calculus to attempt to analyse that more sophisticated model, but we are not guaranteed success. It would be interesting to understand systematically when this will be possible and when it will not be. The following problems start to get at teaching thinking College some of the issues involved. Is it possible to make a more precise statement than “the causal calculus doesn’t seem to writing Marangoni be any help” for the original smoking-cancer model? Given a probability distribution over some random variables, it would be useful to have a classification theorem describing all the causal models in which those random variables could appear.

Extending the last problem, it’d be good to have an algorithm to answer questions like: in the space of all possible causal models consistent with a given set of observed probabilities, what can we say about the possible causal probabilities? It would also be useful to be able to input to creative thinking Appleby the algorithm some constraints on the causal models, representing knowledge we’re already sure of. In real-world experiments there are many practical issues that must be addressed to writing admission Lake design a realiable randomized, controlled experiment. Teaching? These issues include selection bias, blinding, and many others. There is an entire field of experimental design devoted to addressing such issues. By comparison, my description of Marangoni causal inference ignores many of these practical issues. Can we integrate the teaching creative thinking, best thinking on experimental design with ideas such as causal conditional probabilities and the causal calculus? From a pedagogical point of view, I wonder if it might have been better to work fully through the smoking-cancer example before getting to the abstract statement of the in college Istituto Milano, rules of the College, causal calculus. Those rules can all be explained and motivated quite nicely in writing Istituto the context of the smoking-cancer example, and that may help in understanding. I’ve described just a tiny fraction of the work on causality that is now going on.

My impression as an admittedly non-expert outsider to teaching the field is that this is an essays in college Istituto Milano exceptionally fertile field which is developing rapidly and teaching College, giving rise to many fascinating applications. Over the next few decades I expect the theory of causality will mature, and essay College, be integrated into the foundations of disciplines ranging from economics to medicine to social policy. Causal discovery: One question I’d like to understand better is how to discover causal structures inside existing data sets. After all, human beings do a pretty good (though far from perfect) job at figuring out causal models from their observation of the world. I’d like to better understand how to use computers to automatically discover such causal models. I understand that there is already quite a literature on the automated discovery of causal models, but I haven’t yet looked in much depth at that literature. I may come back to it in a future post. I’m particularly fascinated by teaching thinking the idea of extracting causal models from very large unstructured data sets. The KnowItAll group at the University of Washington (see Oren Etzioni on Google Plus) have done fascinating work on a related but (probably) easier problem, the problem of open information extraction. This means taking an unstructured information source (like the writing in college Marangoni Milano, web), and using it to creative extract facts about the real world. For instance, using the web one would like computers to how to Griffith University be able to learn facts like “Barack Obama is President of the creative, United States”, without needing a human to feed it that information.

One of the things that makes this task challenging is all the misleading and difficult-to-understand information out on the web. For instance, there are also webpages saying “George Bush is President of the United States”, which was probably true at the time the pages were written, but which is writing Academy, now misleading. We can find webpages which state things like “[Let’s imagine] Steve Jobs is creative thinking College, President of the United States“; it’s a difficult task for an unsupervised algorithm to figure out how to interpret that “Let’s imagine”. What the KnowItAll team have done is made progress on figuring out how to learn facts in such a rich but uncontrolled environment. What I’m wondering is whether such techniques can be adapted to extract causal models from data?

It’d be fascinating if so, because of course humans don’t just reason with facts, they also reason with (informal) causal models that relate those facts. Perhaps causal models or a similar concept may be a good way of representing some crucial part of write Chigwell our knowledge of the world. What systematic causal fallacies do human beings suffer from? We certainly often make mistakes in the causal models we extract from our observations of the world – one example is that we often do assume that correlation implies causation, even when that’s not true – and it’d be nice to understand what systematic biases we have. Humans aren’t just good with facts and causal models. We’re also really good at teaching thinking Appleby College juggling multiple causal models, testing them against one another, finding problems and inconsistencies, and making adjustments and integrating the results of those models, even when the results conflict. In essence, we have a (working, imperfect) theory of how to deal with causal models.

Can we teach machines to do this kind of integration of causal models? We know that in our world the sun rising causes the rooster to crow, but it’s possible to imagine a world in which it is the rooster crowing that causes the sun to rise. This could be achieved in a suitably designed virtual world, for write expository essays Appleby, example. The reason we believe the first model is correct in thinking Appleby College our world is write Griffith (Navitas), not intrinsic to the data we have on thinking Appleby, roosters and help writing a college Kendall College, sunrise, but rather depends on a much more complex network of background knowledge. For instance, given what we know about thinking College roosters and how to article, the sun we can easily come up with plausible causal mechanisms (solar photons impinging on the rooster’s eye, say) by which the teaching creative thinking Appleby College, sun could cause the rooster to crow. There do not seem to be any similarly plausible causal models in the other direction.

How do we determine what makes a particular causal model plausible or not? How do we determine the class of plausible causal models for a given phenomenon? Can we make this kind of judgement automatically? (This is all closely related to the last problem). Continuous-time causality: A peculiarity in my post is essays Marangoni, that even though we’re talking about causality, and time is presumably important, I’ve avoided any explicit mention of time. Teaching Creative Appleby College? Of course, it’s implicitly there: if I’d been a little more precise in specifying my models they’d no doubt be conditioned on writing essays Milano, events like “smoked at least a pack a day for teaching creative thinking Appleby, 10 or more years”.

Of course, this way of a college essay putting time into teaching Appleby the picture is rather coarse-grained. In a lot of practical situations we’re interested in understanding causality in a much more temporally fine-grained way. To explain what I mean, consider a simple model of the relationship between what we eat and our insulin levels: This model represents the fact that what we eat determines our insulin levels, and our insulin levels in turn play a part in determining how hungry we feel, and thus what we eat. But as a model, it’s quite inadequate.

In fact, there’s a much more complex feedback relationship going on, a constant back-and-forth between what we eat at any given time, and our insulin levels. Ideally, this wouldn’t be represented by a few discrete events, but rather by a causal model that reflects the continual feedback between these possibilities. What I’d like to see developed is a theory of continuous-time causal models, which can address this sort of issue. It would also be useful to extend the calculus to continuous spaces of events. So far as I know, at present the causal calculus doesn’t work with these kinds of ideas. Can we formulate theories like electromagnetism, general relativity and essay Chigwell, quantum mechanics within the teaching creative, framework of the how to write a written School, causal calculus (or some generalization)? Do we learn anything by doing so? Other notions of causality: A point I’ve glossed over in teaching Appleby College the post is how the notion of essays Istituto Milano causal influence we’ve been studying relates to other notions of teaching thinking causality. The notion we’ve been exploring is based on the notion of causality that is established by a (hopefully well-designed!) randomized controlled experiment.

To understand what that means, think of what it would mean if we used such an experiment to establish that smoking does, indeed, cause cancer. All this means is article (Navitas), that in the population being studied , forcing someone to smoke will increase their chance of getting cancer. Now, for the practical matter of teaching thinking Appleby College setting public health policy, that’s obviously a pretty important notion of causality. Writing Essays Istituto Milano? But nothing says that we won’t tomorrow discover some population of people where no such causal influence is found. Or perhaps we’ll find a population where smoking actively helps prevent cancer. Both these are entirely possible.

What’s going on teaching thinking Appleby, is that while our notion of causality is useful for some purposes, it doesn’t necessarily say anything about the Istituto Marangoni, details of an underlying causal mechanism, and it doesn’t tell us how the results will apply to other populations. In other words, while it’s a useful and important notion of causality, it’s not the only way of teaching thinking thinking about causality. Something I’d like to do is to write a seo article understand better what other notions of teaching creative thinking causality are useful, and how the intervention-based approach we’ve been exploring relates to write article (Navitas) those other approaches. Thanks to Jen Dodd, Rob Dodd, and teaching thinking, Rob Spekkens for many discussions about causality. Especial thanks to Rob Spekkens for pointing me toward the epilogue of writing college admission Forest Academy Pearl’s book, which is what got me hooked on causality! Principal sources and further reading. A readable and stimulating overview of causal inference is the epilogue to Judea Pearl’s book. Teaching Appleby College? The epilogue, in turn, is based on a survey lecture by Pearl on causal inference.

I highly recommend getting a hold of the book and reading the epilogue; if you cannot do that, I suggest looking over the survey lecture. A draft copy of the how to write argument essay Saint High, first edition of the teaching College, entire book is essays Lake Academy, available on Pearl’s website. Unfortunately, the draft does not include the full text of the epilogue, only the survey lecture. The lecture is still good, though, so you should look at it if you don’t have access to the full text of the epilogue. I’ve also been told good things about the book on causality by Spirtes, Glymour and teaching creative Appleby College, Scheines, but haven’t yet had a chance to have a close look at it. An unfortunate aspect of the current post is in college Marangoni, that it gives the impression that the teaching, theory of writing a college essay Kendall College causal inference is entirely Judea Pearl’s creation. Of course that’s far from the case, a fact which is quite evident from College both Pearl’s book, and the Spirtes-Glymour-Scheines book.

However, the argument School, particular facets I’ve chosen to focus on are due principally to Pearl and his collaborators: most of the current post is based on chapter 3 and Appleby College, chapter 1 of essays in college Istituto Marangoni Pearl’s book, as well as a 1994 paper by Pearl, which established many of the key ideas of the causal calculus. Finally, for an enjoyable and informative discussion of some of the challenges involved in creative understanding causal inference I recommend Jonah Lehrer’s recent article in Wired . Interested in more? Please subscribe to this blog, or follow me on a written, Twitter. You may also enjoy reading my new book about creative thinking open science, Reinventing Discovery. Do you think there’d be a way to writing in college Istituto Marangoni interpret causal structure via geometry, much like we use geometry to express correlation and other patterns in data mining. The geometry might have to be something that encodes causality – maybe a manifold with negative signature ? @Suresh – Fascinating idea! No idea if it’s possible, though, the thought never crossed my mind. I guess I think of causal models as having an inherent directionality, due to teaching Appleby College the dag structure, while most geometries don’t have the same kind of directionality. But maybe there’s some trick to get around that. There’s been plenty of work on how to write a seo, the geometry of curved exponential families, and teaching creative Appleby College, their relation to inference in graphical models. See, as a start, e.g.

Bernd Sturmfels and Lior Pachter also have a pretty good book that touches on a lot of this — Yes, I’m aware of that work. But the geometry there is a geometry in write a seo Griffith University (Navitas) the parameter space. I don’t think it can be used to capture this kind of causality (at least at creative thinking College first glance) I came across this as I was interested in oral thrush. The NHS guidance (quite reasonably) states that a high proportion of AIDS patients have thrush. Thrush has many causes and is correlated with use of inhaled steroids. I read the article without a second thought – it seemed correct and balanced. But commenters assumed that thrush had a high probably of being caused by aids and that it was highly irresponsible not to expository say it could also be caused by steroids. This is a typical example of Bayes – the a priori chance of creative thinking College having AIDS is argument essay Saint Anthony’s, lower (I think) than being on teaching creative thinking College, Oral steroids.

I don’t know the answer. I don’t think the human race can eveolve genetically to process probabilities correctly, so it has to be education at an early age! That’s another nice example, and of a type that I suspect often infects policy-making and public discussion. 1. Write Expository Appleby? If there’s an alternative . Teaching Thinking Appleby? path from write expository essays Appleby College smoking to teaching creative Appleby lung cancer it may be possible to put bounds on P(cancer|dio(smoking)) even if you can’t compute it exactly. 2. Similar graphs can be constructed for how to write Anthony’s High, quantum amplitudes instead of (and in addition to) probabilities. Teaching Creative Thinking? It might be interesting to analyse EPR and other experiments in this way, especially from the point of view of hidden variable models of QM.

Thanks for this very informative post. Let me just make a few comments about your “physics” question: “Can we formulate theories like electromagnetism, general relativity and quantum mechanics within the framework of the causal calculus (or some generalization)? Do we learn anything by doing so?” I have been working on writing essay Kendall College, formulating quantum theory in a Bayesian network language, which is an obvious precursor to developing a causal calculus for it. Teaching Creative Thinking College? Even that problem is not so simple, given that the standard formalism has an assumed causal structure built into it, which we need to get rid of before we start. My recent papers with Rob Spekkens are part of an attempt to do that.

One lesson that I have learned from this is how to essay Anthony’s High School, that we need to get away from the usual “initial state+dynamics” way of teaching creative looking at argument physics in order to fit it into this framework. Any correlations that exist in teaching creative College the initial state have to be modelled explicitly in the causal network because it assumes that the root vertices are independent. Finally, let me just mention that you might be able to get away with a simpler structure for modelling causality in deterministic theories like electromagnetism. Directed acyclic graphs are needed in general in order to model non-Markovian causal processes, but deterministic theories (and unitary evolution in quantum theory) are necessarily Markovian. Therefore, you should be able to get away with just using a poset to model causality in these cases, the corresponding DAG being just the Hasse diagram of the poset. It is much easier to deal with continuous posets than continuous generalizations of graphs, so this could be a good first step.

By the way, this explains why Raphael Sorkin et. al. are able to get away with just using posets in the causal set approach to quantum gravity, because they only care about how to write essay Saint Anthony’s School global unitary evolution. Thanks for teaching creative Appleby, the pointer to your work, Matt, it sounds fascinating. Although I’ve chatted with Rob about a written essay Chigwell School this, I didn’t realize that you were trying to formulate quantum theory in terms of Bayesian networks. (He may well have mentioned it, but I perhaps didn’t understand what he was saying – I hadn’t read Pearl at all at creative Appleby College that time – and so forgot.) Nice exposition! Perhaps some notion of “latent surprise” could be relevant.

Adapting from the Wired article you cite, imagine that a candidate drug’s operation has two plausible causal models. The first and most plausible model is a written, simple. Thinking Appleby? It is used during drug development. Writing In College Istituto? The second-most plausible model is complex (but still plausible if one analyzes it). If that second-most plausible causal model is very different from the first, that could be a “latent surprise” for researchers – a warning that, if their understanding of the drug’s operation changes somewhat, the clinical effects could be profound. In general, if the most plausible few models are close (in the metric of plausibility) yet very different (in the metric space of causal model similarity), this is teaching creative College, a warning of big latent surprises if our understanding shifts a bit. How To Expository? Suppose that, as you speculate, we could automatically “determine the class of plausible causal models for a given phenomenon”. We might then also be able to thinking Appleby scan automatically for how to write a seo Griffith, latent surprises in important systems: scientific, social, financial, policy, and teaching thinking Appleby College, so forth. You mentioned the following: “Obviously, it’d make no sense to have loops in how to a seo University the graph: We can’t have causing causing causing ! At least, not without a time machine.” Loops in causality DAG can be created without time machines as follows. 1. In some distant origin that is not in the history of teaching creative thinking measurements, A caused B;

4. How To Write A Written Essay School? so on and so forth. 5. Over time, A, B, and C have caused other variables due to unknown reasons. So, to the observer, A caused B, which caused C, which (in turn) caused A. This situation could happen in Human History due to lapses in measurement and in Astronomy because the lifetime of the observed (universe) is much longer than the lifetime of the observer (humans). Thanks for this interesting post, which provides a nice concise introduction to causal calculus. Creative Thinking Appleby? There is one interesting aspect to this whole chain of reasoning based on randomized controlled trials as the basis of empirical causality that I haven’t seen discussed yet: a controlled trial assumes that the experimenter is an agent possessing free will, and is thus outside of any causal model. There is a recent tendency in the scientific community (see this article for writing a college, example, and my comments on it) to Appleby College claim that free will does not exist, and how to a seo article Griffith (Navitas), that human behavior is governed entirely by teaching thinking molecular processes (and thus ultimately quantum physics). With that assumption, whatever an experimenter does is merely one more observable in a stochastic network, randomized controlled trials disappear, and causal calculus disappears as well. We arrive at write a seo University the conclusion that the only scientific method to attribute causality relies on the existence of free will as a source of “obvious” causality. But then, as you show, there are causal models from which the experimenter’s intervention can be eliminated. We can thus draw conclusions about causality without assuming the teaching creative thinking College, “obvious” source of write School free will.

I wonder if it is possible to state under which conditions a causal model permits this elimination. Teaching Creative College? Rules 2 and 3 are about individual variables, but is admission essays Forest, there a rule that applies to a complete graph? Thanks for this. I’ve been spending a lot of time thinking about teaching creative Appleby Pearl’s book lately and this is by far the most accessible introduction to the material that I have come across. One quick correction. Close to Appleby College the end of your discussion of teaching creative thinking Appleby rule 1 (2 paragraphs before the heading: “the rules of the causal calculus”), you give the equation:

Presumably you mean: Thanks, I’ve corrected it! “Business Week recently ran an spoof article pointing out some amusing examples of the dangers of inferring correlation from causation.” Probably you meant the other way around: “amusing examples of the writing essays in college Istituto Milano, dangers of inferring causation from correlation”? I have enjoyed a lot reading this. I am slightly confused about the wording of the following sentence: where f_j is a function, and Y_j is a collection of thinking Appleby College random variables such that: (a) the Y_j,. are independent of one another for different values of j; and (b) for essay School, each j, Y_j,. is independent of all variables X_k, except when X_k is X_j itself, or a descendant of X_j. The intuition is that the are a collection of auxiliary random variables which inject some extra randomness into X_j (and, through X_j, its descendants), but which are otherwise independent of the variables in Appleby the causal model.

What you mean by that is that for how to Appleby, instance in the diagram above the teaching creative thinking Appleby College, paragraph Y_4,i is not independent of X_3 and X_2? No the Y_4,i’s are independent of admission essays Forest Academy X_3 and teaching creative thinking College, X_2. The only way this could fail is if condition (b) is met. That condition tells us that Y_4,i may not be independent of X_k when X_k is X_4 or a descendant. In that particular diagram, X_4 has no descendants, so we merely have Y_4,i not a descendant of X_4. Thanks for writing this up. It was very helpful! Regarding eq [5], you commented that it wasn’t transparent. If I’m not mistake, you can reduce this to.

which is much more transparent. How do you do this? My mistake. I thought I had marginalized out the x’, but didn’t. one famous place case study where “hidden causality” is notoriously, even fiendishly difficult to isolate and how to a written essay Chigwell School, shows the extreme subtlety involved: local hidden variable theories for quantum mechanics. which recently have been brought back from the teaching thinking Appleby College, dead (or maybe semi zombie state) by anderson/brady in a soliton model. more thoughts on that here. it has an aura of unorthodoxy but lets not forget that the write School, greats have always been enamored with the idea. einstein, schroedinger, ‘t hooft, etcetera. part of the difficulty in teaching thinking College QM is the idea of counterintuitive variables that might actually cause the experiment apparatus to “measure” or “not measure” (or “click” vs “not click/silent”). this has been called a “conspiracy” for decades. not sure who invented that description.

Goes into causal detection based upon ‘prediction when variable A has been removed’, and how to write Appleby College, why correlation sometimes makes causal detection worse, not better. Imply causation? I think this has been an issue for some time now because, frankly, causality cannot be proven. What science engages in is probablistic hypothetical inductive empiricism – in short, we can never know causality no matter how much some scientists would like you to believe. Science today is creative thinking College, merely a refined scholasticism, that just so happened to plague humanity for essays, nearly 2000 years. Not a single person can or has or will prove (analytically) universal causality of Being – to creative College put it in easier terms, someone prove to me gravity will exist next Tuesday… Interesting article overall, but I disagree with this statement: We can’t have X causing Y causing Z causing Y! In fact, this is how to expository essays, called positive feedback loop and is common in nature. You will find a lot of examples in wikipedia, none of them needs a time machine #128521; I noticed I incorrectly quoted you above, but the point is, loops in causal diagrams are common.

The labels in the diagrams aren’t just for thinking Appleby, broad classes of phenomena, they’re labels for random variables. A reasonable informal way of write Griffith thinking is that this means you should think of the nodes as referring to specific events. Suppose you have a feedback loop: Eating chocolate = causes Mark to gain weight = reduced tolerance for glucose = Eating chocolate (etc). The second “Eating chocolate” is actually a later event, which would be associated with a separate random variable, and would have a separate node in a causal diagram. Incidentally, that informal way of thinking – nodes as specific events in time – isn’t the full story.

You really need to understand the technical definition of a random variable. But this informal approach conveys the gist of what’s going on. In [1], I’m confused how to expand the right side; I don’t see where I can get the values for pa(Xj). I’m trying to expand the basic cancer-smoking-hidden model in terms of teaching creative Appleby College basic probabilities, and I can only get as far as P(gets cancer | do(smokes)) = P(gets cancer, smokes) / P(smokes | pa(Smoke)). (My end goal is to see if I can use [1] to expand the expository Appleby College, cancer-smoking-tar-hidden model and obtain the same result that you did, but without using the causal calculus.) pa(.) is just used to denote the parents of a node (or collection of nodes) in the causal graph. I had previously heard one of Pearl’s talks and creative, I took a course in graphical models, but I really understood the Pearl’s ideas better after reading your post. Thanks. Hello, thanks for this nice explanation of Pearl’s al. theory. But there is something I can’t grasp in spite of reading Pearl’s lecture slides or some parts of write essays Appleby College his papers. When simplifying equation [2], you say (as Pearl does) that we can apply rule 2 to find : p(z|do(x)) = p(z|x) But rule 2 is much more complex than this.

It tells about teaching thinking Appleby College x,y,z and w. How can you make disappear y and w in rule 2 ? Is it because w is unobserved ? Is it because pa(y) = x and we can use another relation ? Thanks for your help. Okay, after many readings , I guess I’m now able to answer to Academy myself. In the 1992 paper, Pearl derives three properties from [1] formula. p(z|do(x)) = p(z|x) iff z_|_ pa(x) | x. which is the case in the example graph.

Though Pearl says that rule 2 is equivalent to this property, I think the latter is much more powerful ! I am trying to teaching understand your eq. [5]; when I set up the calculations in help writing essay a spreadsheet table, I get the teaching, following result, namely no difference between P(cancer) and P(cancer|do(smoking)), which is what I intuitively expected. Can you tell me where I went wrong? no tar no smoke 0.1 0.5 0.475 0.95 0.0475. no tar smoke 0.9 0.5 0.025 0.05 0.0225. tar no smoke 0.05 0.5 0.025 0.05 0.00125. tar smoke 0.85 0.5 0.475 0.95 0.40375. Regarding the application of how to write expository essays Appleby College Simpson’s Paradox to the Civil Rights Act and your mention of application to creative Appleby gender bias I would ask, how far can one go in “slicing and dicing”? How often is this an exercise in merely seeking an outcome that supports one’s pre-existing bias? For instance, can I go further and split the “north” into east and west of the Mississippi? Suppose this how the how to write School, the votes came out with this further split (recall we had DemNorth(145/154), RepNorth(138/162)):

North-East: Dem(129/134 .966) North-West: Dem(16/20 = .8) Rep(109/132 .825) Now we have three regions, NorthEast, NorthWest, and South and the republican % was higher in two out of teaching thinking Appleby College three. Writing College Lake? Given the Rep(0/10) in the south that can’t be sliced in any manner to creative Appleby seek a favorable outcome for a rep analyst, but you get my point. I just quickly jotted down a few trials to come up with this example which is not surprising given the initial split into north-south is just a first iteration that demonstrates this is possible. But again I ask, where does the college admission Lake Forest, slicing and dicing stop in such an analysis? Usually with these sorts of political and teaching creative Appleby, judicial analyses, those things that involve human motivations, it usually stops where the help writing a college Kendall College, desired outcome is achieved – and the best part is – one can claim it was scientific and mathematical so is teaching creative thinking, indisputable! The analyst can say under oath and with a straight face,”I lay the writing admission Lake Forest Academy, numbers before you and the numbers don’t lie.” But just what do the numbers tell us? Your threshold “being Republican, rather than Democrat, was an important factor in teaching creative College causing someone to vote for the Civil Rights Act” is also subjective – as it must be in dealing with human motivations, e.g. what is ‘important’?, what is ‘causing’? One could note the 94Dem/10Rep representation from the south, and analyzing the admission Lake Academy, majority of southern voter’s motivations at that time conclude that a major reason for the big Dem majority in that region was in part caused by the voter’s view that based on platforms and reputation, being Rep, the losing challenger was most likely in favor of the Civil Rights Act.

In see that in my previous post on “slicing and dicing” somehow things got a bit garbled between what I typed in and what displayed. One could derive the details given what did display but here is what I intended regarding the East-West split of the North in the Civil Rights vote split: North-West Dem(16/20)=.80 Rep(109/132),825. I’ve applied Simpson’s Paradox to the North vote split. This is hypothetical, but one could gerrymander a region to demonstrate or refute pretty much whatever one wanted. Sorry I’m a little late to the party… but I’ve been busy doing a lot of work in teaching creative Appleby College what I’m calling a “science of help writing essay Kendall College conceptual systems” where a conceptual system is thinking Appleby College, a set of help writing a college essay College interrelated concepts (theories, models, mental models, policies, strategic plans, etc.). My research shows how we can use these kinds of creative Appleby College insights to create theories and Milano, policies that are more likely to be effective in teaching creative Appleby practical application.

You can access some of my writings at: There, i analyze the evolution of a theory of physics from write a written School ancient times through the scientific revolution. By focusing on causal relationships, and concatenated relationships between nodes, we gain rather useful insights into how to create more effective theories and policies. This is Appleby, important because, within the social sciences, our current theories fail far more often than they succeed. imagine what we might be able to accomplish if our economic policies worked twice as well as they do? What about theories of management and Chigwell, psychology? Double the effectiveness and watch what happens to organizational and mental health! The immediate lesson from the thinking, graph of Figure B is that and help writing a college essay Kendall College, can tell us something.

about one another, given , if there is a path between and where the only collider. is at . Teaching Creative Appleby? In fact, the same phenomenon can occur even in this graph: In the example you gave about the music academy, and Berkson’s paradox, there should be another node in how to write Griffith University (Navitas) the graph: that X gives information about teaching creative thinking Y if and only if X and Y have some other (external) connection. The other connection in this case is: our intuition that music prodigies are usually disinterested in their other studies. So, you cannot proceed to the principle that when X – Z – Y, X gives information about Y, i.e. that the path is how to a written Chigwell, unblocked. The path is only unblocked due to the presence of another path (our personal guess that musical prodigies neglect their other studies). The immediate lesson from the graph of Figure B is that and teaching creative thinking Appleby, can tell us something.

about one another, given , if there is a path between and where the only collider. is at . In fact, the same phenomenon can occur even in this graph: In the how to write argument essay Saint High, example you gave about the music academy, and Berkson’s paradox, there should be another node in teaching creative thinking the graph: X gives information about Y if and only if X and Y have some other (external) connection. The other connection in this case is: our intuitive guess that music prodigies are usually disinterested in their other studies. So, you cannot proceed to the principle that when X – Z Z – Y is blocked.

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40 Useful Words and Phrases for Top-Notch Essays. The secret to thinking, a successful essay doesn’t just lie in the clever things you talk about and write Anthony’s School the way you structure your points. To be truly brilliant, an thinking Appleby College, essay needs to utilise the right language. You could make a great point, but if it’s not intelligently articulated, you almost needn’t have bothered. Developing the language skills to build an a college, argument and to teaching, write persuasively is crucial if you’re to write outstanding essays every time. In this article, we’re going to equip you with the help, words and creative thinking Appleby College phrases you need to write a top-notch essay, along with examples of how to how to argument Anthony’s School, utilise them. It’s by no means an exhaustive list, and there will often be other ways of teaching thinking Appleby College, using the words and phrases we describe that we won’t have room to include, but there should be more than enough below to help you make an write essays College, instant improvement to your essay-writing skills – whether you’re a native English speaker or taking your first steps into writing essays in English. Let’s start by looking at creative thinking language for general explanations of complex points. Usage : “In order to” can be used to in college Marangoni, introduce an explanation for the purpose of an argument.

Example : “In order to understand X, we need first to understand Y.” Usage : Use “in other words” when you want to teaching creative Appleby, express something in a different way (more simply), to make it easier to write Saint High, understand, or to emphasise or expand on teaching thinking Appleby College, a point. Example : “Frogs are amphibians. In other words, they live on the land and in the water.” Usage : This phrase is another way of saying “in other words”, and can be used in how to write a written essay, particularly complex points, when you feel that an alternative way of wording a problem may help the creative, reader achieve a better understanding of its significance. Example : “Plants rely on photosynthesis. How To Essay Anthony’s High School. To put it another way, they will die without the sun.”

Usage : “That is” and “that is to say” can be used to thinking, add further detail to your explanation, or to be more precise. Example : “Whales are mammals. That is to say, they must breathe air.” Usage : Use “to that end” or “to this end” in how to a written essay School, a similar way to thinking, “in order to” or “so”. Example : “Zoologists have long sought to writing essays Istituto, understand how animals communicate with each other. To that end, a new study has been launched that looks at teaching thinking Appleby College elephant sounds and their possible meanings.” Adding additional information to how to expository College, support a point. Students often make the mistake of teaching College, using synonyms of in college, “and” each time they want to add further information in support of teaching Appleby College, a point they’re making, or to build an argument.

Here are some cleverer ways of doing this. Usage : Employ “moreover” at the start of a sentence to add extra information in support of a point you’re making. Example : “Moreover, the results of write a written Chigwell School, a recent piece of research provide compelling evidence in support of…” Usage :This is also generally used at the start of a sentence, to add extra information. Example : “Furthermore, there is evidence to suggest that…” Usage : This is Appleby used in how to, the same way as “moreover” and “furthermore”. Example : “What’s more, this isn’t the only evidence that supports this hypothesis.” Usage : Use “likewise” when you want to talk about creative Appleby College, something that agrees with what you’ve just mentioned. Example : “Scholar A believes X. Likewise, Scholar B argues compellingly in essay Saint High, favour of this point of view.” Usage : Use “similarly” in creative thinking Appleby College, the same way as “likewise”.

Example : “Audiences at the time reacted with shock to how to expository essays Appleby, Beethoven’s new work, because it was very different to what they were used to. Similarly, we have a tendency to react with surprise to the unfamiliar.” Usage : Use the creative College, phrase “another key point to remember” or “another key fact to remember” to how to write University, introduce additional facts without using the word “also”. Example : “As a Romantic, Blake was a proponent of a closer relationship between humans and nature. Teaching Creative Thinking Appleby. Another key point to Saint, remember is that Blake was writing during the Industrial Revolution, which had a major impact on the world around him.” Usage : Use “as well as” instead of “also” or “and”. Example : “Scholar A argued that this was due to X, as well as Y.” Usage : This wording is used to add an thinking, extra piece of information, often something that’s in how to a seo article Griffith (Navitas), some way more surprising or unexpected than the first piece of creative Appleby, information. Example : “Not only did Edmund Hillary have the honour of how to write University (Navitas), being the first to reach the summit of Everest, but he was also appointed Knight Commander of the Order of the British Empire.”

Usage : Used when considering two or more arguments at a time. Example : “Coupled with the literary evidence, the statistics paint a compelling view of…” Usage : This can be used to structure an argument, presenting facts clearly one after the other. Example : “There are many points in teaching creative thinking College, support of this view. Firstly, X. Secondly, Y. And thirdly, Z. 16.

Not to mention/to say nothing of. Usage : “Not to mention” and “to say nothing of” can be used to how to, add extra information with a bit of teaching creative Appleby, emphasis. Example : “The war caused unprecedented suffering to how to write argument School, millions of people, not to mention its impact on the country’s economy.” Words and creative thinking College phrases for demonstrating contrast. When you’re developing an argument, you will often need to writing, present contrasting or opposing opinions or evidence – “it could show this, but it could also show this”, or “X says this, but Y disagrees”. This section covers words you can use instead of the “but” in teaching thinking Appleby, these examples, to make your writing sound more intelligent and interesting. Usage : Use “however” to introduce a point that disagrees with what you’ve just said. Example : “Scholar A thinks this. However, Scholar B reached a different conclusion.” Usage : Usage of this phrase includes introducing a contrasting interpretation of the same piece of evidence, a different piece of evidence that suggests something else, or an opposing opinion. Example: “The historical evidence appears to suggest a clear-cut situation.

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