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# Machine Learning CS 5350

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This 2 page Class Notes was uploaded by Marian Kertzmann DVM on Monday October 26, 2015. The Class Notes belongs to CS 5350 at University of Utah taught by Harold Daume in Fall. Since its upload, it has received 11 views. For similar materials see /class/229971/cs-5350-university-of-utah in ComputerScienence at University of Utah.

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Date Created: 10/26/15

Machine Learning CS 5350CS 6350 05 Apr 2007 Bayesian inference The general problem we face in Bayesian inference is to compute the expectation of some function with respect to a probabilistic model with unknowns In the simplest case we want the expectation of a single variable In more complex cases we want an expectation of a complex function of all variables Suppose p9 is our distribution of interest some 9 will be known some not We want 2 EM m depr For some cases this integral will be available in closed form eg HMMs For many most cases however it will not Lets say that 9 is discrete univariate Then we can compute the expectation by just summing over all possible values Obviously though this won t scale well to highdimensional or nondiscrete variables But let s see what happens if we try Integration by Summation Suppose 9 is univariate bounded continuous Wlog 9 E 01 If we remember how we rst learned integration we can break 0 1 into R equallysized rectangles Then we have Pd z 7 piRfiR 0 As R A 0 Z becomes increasingly more accurate One way of thinking about this is that we have a set S containing Rmany equally spaced points and the integral is approximated y 1 Z R W mw Unfortunately if 9 is Ddimensional then we need to sum RD values of 9 Uniform Sampling Instead of spacing 9 E S evenly let s space them randomly This is the idea of Monte Carlo77 integration which essentially means randomized integration Uniform sampling is the simplest case Let S be a random sampling of 9s Then we still have m prm ass Machine Learning CS 535005 6350 2 This scales better computationally but still the number of samples required to guarantee that we get a close approximation is huge It s worth thinking about how hard this problem is Think of a boat on a lake We want to estimate the volume of the lake but cannot see the bottom We can drive the boat to any position in the lake and drop an anchor thereby measuring the depth there How can we approximate the volume Uniform sampling says to drive randomly around the lake dropping at the ip of a coin But there are many chases in which we can do better Importance Sampling Here we use prior knowledge in the form of a helper distribution 11 that we expect to be similar to p and from which we can sample It must have the same support as p ie not zero too often Then we compute Z EeNplfWH d6p9f9 209 d6 6 6 qlt q9f gt 209 E N 6 9 1 law So instead of computing an expectation wrt p we compute wrt 11 And then we weight each example Rejection Sampling The idea in rejection sampling is similar to importance sampling Let 1 be a proposal distribution that satis es px S Mqx for M lt 0 Now draw points from 11 and accept them with probability Compute expectations only over the accepted points

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