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Graphical Models

04 Jan 2010 09:14

A.k.a. causal models, causal graphs, Bayes graphs, Bayes networks, Bayesian networks. (Here "Bayes" is a metonym for "conditional probability". There are perfectly good frequentist interpretations of these models.) I'm sticking latent-variable and path-analysis models in here, too, because they all pretty much work the same way.

Everyone who takes basic statistics has it drilled into them that "correlation is not causation." (When I took psych. 1, the professor said he hoped that, if he were to come to us on our death-beds and prompt us with "Correlation is," we would all respond "not causation.") This is a problem, because one can infer correlation from data, and would like to be able to make inferences about causation. There are typically two ways out of this. One is to perform an experiment, preferably a randomized double-blind experiment, to eliminate accidental sources of correlation, common causes, etc. That's nice when you can do it, but impossible with supernovae, and not even easy with people. The other out is to look for correlations, say that of course they don't equal causations, and then act as if they did anyway. The technical names for this latter course of action are "linear regression" and "analysis of variance," and they form the core of applied quantitative social science, e.g., The Bell Curve.

Graphical models are, in part, a way of escaping from this impasse.

The basic idea is as follows. You have a bunch of variables, and you want to represent the causal relationships, or at least the probabilistic dependencies, between them. You do so by means of a graph. Each node in the graph stands for a variable. If variable A is a cause of B, then an arrow runs from A to B. If A is a cause of B, we also say that A is one of B's parents, and B one of A's children. If there is a causal path from A to B, then A is an ancestor of B, and B is a descendant of A. If a variable has no parents in the graph, it is exogenous, otherwise it is endogenous.

Part of what we mean by "cause" is that, when we know the immediate causes, the remoter causes are irrelevant --- given the parents, remoter ancestors don't matter. The standard example is that applying a flame to a piece of cotton will cause it to burn, whether the flame came from a match, spark, lighter or what-not. Probabilistically, this is a conditional indepedence property, or a Markov property: a variable is independent of its ancestors conditional on its parents. In fact, given its parents, its children, and its childrens' other parents, a variable is conditionally independent of all other variables. This is called the graphical or causal Markov property. When this holds, we can factor the joint probability distribution for all the variables into the product of the distribution of the exogenous variables, and the conditional distribution for each endogenous variable given its parents.

(You may be wondering what happens if A is a parent of B and B is a parent of A, as can happen when there is feedback between the variables. This leads to difficulties, traditionally dealt with by explicitly limiting the discussion to acyclic graphs. I shall follow this wise precedent here.)

Now, there are certain rules which let us infer conditional independence relations from each other. For instance, if X is independent of the combination of Y and W, given Z, then X is indepdent of Y alone given Z. So, if we have a graph which obeys the causal Markov condition, there are generally other conditional independence relations which follow from the basic ones. If these are the only conditional indepences which hold in the distribution, it is said to be faithful to the graph (or vice versa); otherwise it is unfaithful. For a graph to be Markov and unfaithful, there must (as it were) be an elaborate conspiracy among the conditional distributions, so elaborate that it will generally be destroyed by any change in any of those distributions. So faithfulness is a robust property.

This may sound pretty arcane, but that's just because it is arcane. The point, however, is that if you can make the three assumptions above (no causal cycles, Markov property, faithfulness), you're in business in a really remarkable way. There are very powerful statistical techniques that will let you infer the causal structure connecting your variables. This comes in two flavors. One is the Bayesian way: cook up a prior distribution over all possible causal graphs; compute the likelihood of the data under each graph; update your distribution over graphs; iterate. This is generally computationally intractable, assuming you can come up with a meaningful prior in the first place. The other approach is to use tests for conditional independence to eliminate possible connections between variables, and so to narrow down the range of candidate structures; it is basically frequentist, and can be shown, under a broad range of circumstances, to be asymptotically reliable.

Once you have your causal graph --- whether through estimation or through simply being handed one --- you can do lots of great things with it, like predict the effects of manipulating some of the variables, or make backward inferences from effects to causes. Of course, if the graph is big, doing the necessary calculations can be very troublesome in itself, and so people work on approximation methods and even ways of doing statistical inference on models of statistical distributions...

It's probably obvious I think this is incredibly neat, and even one of the most important ideas to come out of machine learning. Of course it doesn't really solve the problem of establishing causal relations, in the way Hume objected to; it says, assuming there are causal relations, of a certain stochastic form, and that these are stable, then they can be learned. But that, and the more general questions of what we ought to mean by "cause", deserve a notebook of their own.

Things I want to understand better: frequentist inference procedures. Computational learning theory for graphical models (the paper by Janzing and Herrmann is good). How to treat systems with feedback? How to treat dynamical systems and time series? How does all of this fit together with computational mechanics?

Not even a conjecture. Back in the 1960s, Chow and Liu (reference below) gave a polynomial algorithm for finding the best approximation to a global joint probability distribution using only pairwise interactions among the variables, i.e., the one which minimized the Kullback-Leibler divergence between the true and the approximating distribution. I have read that extending this to even three-way interactions is NP, though I don't know if it's NP-complete. (1) How is the intractability result established? (2) Is this the same as the computational phase transition one finds in going from 2-SAT to 3-SAT, where the critical point is at two-point-something SAT? (Presumably the answer to (1) would shed some light on this.) (3) Even if not, is there an analogous phase transition, perhaps in a different universality class? (Update in 2009, several years later: Bento and Montanari, below, sounds relevant, but I haven't read it yet.)

(Thanks to Gustavo Lacerda for pointing out a goof.)

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