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Ch8.3.3-Max-SumAlg (1)

# Ch8.3.3-Max-SumAlg (1) - Machine Learning Srihari Max-Sum...

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Machine Learning Srihari 1 Max-Sum Inference Algorithm Sargur Srihari

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Machine Learning Srihari 2 The max-sum algorithm Sum-product algorithm Takes joint distribution expressed as a factor graph Efficiently finds marginals over component variables Max-sum addresses two other tasks 1. Setting of the variables that has the highest probability 2. Find value of that probability Algorithms are closely related Max-sum is an application of dynamic programming to graphical models
Machine Learning Srihari 3 Finding latent variable values having high probability Consider simple approach Use sum-product to obtain marginals for every variable For each variable find value that maximizes marginal This would give set of values that are individually most probable However we wish to find vector that maximizes joint distribution, i.e. With join probability ) p(x i i x * i x max x p(x) x x max arg max = p(x) ) p(x x max max =

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Machine Learning Srihari 4 Example Maximum of joint distribution Occurs at x=1, y=0 – With p(x=1,y=0)=0.4 Marginal p(x) p(x=0) = p(x=0,y=0)+p(x=0,y=0)=0.6 p(x=1) = p(x=1,y=0)+p(x=1,y=1)=0.4 Marginal p(y) P(y=0)=0.7 P(y=1)=0.3 Marginals are maximized by x=0 and y=0 which corresponds to 0.3 of joint distribution In fact, set of individually most probable values can have probability zero in joint p(x,y ) x=0 x=1 y=0 0.3 0.4 y=1 0.3 0.0
Machine Learning Srihari 5 Max-sum principle Seek efficient algorithm for – Finding value of x that maximizes p(x) – Find value of joint distribution at that x Second task is written where M is total number of variables Make use of distributive law for max operator – Which holds for – Allows exchange of products with maximizations p(x) ... p(x) M x x x max max max 1 = (bc) a (ab,ac) max max = 0 a

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Machine Learning Srihari 6 Chain example
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Ch8.3.3-Max-SumAlg (1) - Machine Learning Srihari Max-Sum...

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