#Variational calculus examples full

Even though q is simple, it may be able to adequately model the variation in those samples for a given, typical x.īut in contrast to the above, variational methods don't require any sampling, so they are fast and deterministic. The complex model p is still used to define what we're trying to approximate by q(y), namely p(y | x), which may differ for each input x.Ī similar approach would be to do MCMC sampling of y values from p(y | x) and then train a simpler model q(y) on those samples. We never define anything like q(x,y), only q(y) for a given input x. Notice that the method is not the same as just throwing away the complex model p(x,y) and using a simpler one q(x,y) in its place. A particular q in Q is specified by setting some variational parameters - the knobs on Q that you can turn to get a good approximation. The term variational is used because you pick the best q in Q - the term derives from the 'calculus of variations,' which deals with optimization problems that pick the best function (in this case, a distribution q).

Q is called the variational approximation to the posterior. Some examples of variational methods include the mean-field approximation, loopy belief propagation, tree-reweighted belief propagation, and expectation propagation. Case (2) leads to variational Bayes, which combines training and decoding into a single optimization problem. Then you pick the q in Q that best approximates the posterior (under some measure of 'best').Ĭase (1) above leads to variational decoding, which can be used within variational EM training. Which simpler distribution q? Well, you define a whole family Q of distributions that would be computationally easy to work with.

#Variational calculus examples full

This is more or less 'okay' because q only has to model y for the particular x that we actually saw, not the full relation between x and y. Typically, q makes more independence assumptions than p. Solution: Approximate that complicated posterior p(y | x) with a simpler distribution q(y). Or (2) Given a training corpus x, the posterior probability distribution over parameters y is too complicated to work with. Overview Problem: (1) Given an input x, the posterior probability distribution over outputs y is too complicated to work with. In general, this page assumes familiarity with models like Markov Random Fields. One or more specific variational methods, such as the mean-fieldĪpproximation, which is used in variational Bayes and elsewhere. Some people may find this page more valuable after they have learned The setting is familiar in NLP, and it illustrates all the important points. The part-of-speech tagging example at the end is a good one to think about whenever you are trying to remember how variational inference works. The variational stuff is near the start and end of the message.

Those parts will be old hat to some of you. I threw in some background on what we are trying to do with inference in general. So please do spend 30 minutes reading it, if only to make it worth my while to have written it.

This should be easy reading since I've left out almost all the math. Inference, to read before our unit in the NLP reading group. High-Level Explanation of Variational Inference by Jason Eisner (2011) īy popular demand, here is a high-level explanation of variational