By Peter Norvig, Stuart Russell

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Extra resources for Artificial Intelligence: A Modern Approach (3rd Edition)

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8) defines each δi→j in terms of δk→j other than δi→j . The other parameters are all defined in a noncyclic way in terms of the δi→j ’s. The form of the equations resulting from the theorem suggest an iterative procedure for finding a fixed point, in which we view the equations as assignments, and iteratively apply equations to the current values of the left-hand side to define a new value for the right-hand side. 8), computing the left-hand side δi→j of each equality in terms of the right-hand side (essentially converting each equality sign to an assignment).

2 Exact Inference as Optimization Before considering approximate inference methods, we illustrate the use of a variational approach to derive an exact inference procedure. The concepts we introduce here will serve in discussion of the following approximate inference methods. The goal of exact inference here will be to compute marginals of the distribution. To achieve this goal, we will need to make sure that the set of distributions Q is expressive enough to represent the target distribution PF .

There are various conditions that suffice to guarantee this property. The condition most commonly used is a fairly technical one, that the chain be ergodic. 32 A Markov chain is said to be regular if there exists some number k such that, for every x, x ∈ Val(X), the probability of getting from x to x in exactly k steps is greater than 0. 33 A finite-state Markov chain T has a unique stationary distribution if and only if it is regular. Ensuring regularity is usually straightforward. Two simple conditions that guarantee regularity in finite-state Markov chains are: It is possible to get from any state to any state using a positive probability path in the state graph.

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