Max-Sum Inference Algorithm
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1 Ma-Sum Inference Algorithm Sargur Srihari 1
2 The ma-sum algorithm Sum-product algorithm Takes joint distribution epressed as a factor graph Efficiently finds marginals over component variables Ma-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 Ma-sum is an application of dynamic programming to graphical models 2
3 Finding latent variable values having high probability Consider simple approach Use sum-product to obtain marginals for every variable i For each variable find value that maimizes marginal This would give set of values that are individually most probable * i However we wish to find vector maimizes joint distribution, i.e. ma = arg ma p( p( i ma that With join probability p( ma = ma p( 3
4 Eample Maimum of joint distribution Occurs at =1, y=0 With p(=1,y=0=0.4 Marginal p( p(=0 = p(=0,y=0+p(=0,y=0=0.6 p(=1 = p(=1,y=0+p(=1,y=1=0.4 Marginal p(y P(y=0=0.7 P(y=1=0.3 p(,y Marginals are maimized by =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 =0 =1 y= y=
5 Ma-sum principle Seek efficient algorithm for Finding value of that maimizes p( Find value of joint distribution at that Second task is written ma p( = ma... ma M p( where M is total number of variables 1 Make use of distributive law for ma operator ma (ab,ac = a ma (bc Which holds for a 0 Allows echange of products with maimizations 5
6 Chain eample Markov chain joint distribution has form p( = 1 Z ψ (, ψ (...ψ Evaluation of probability maimum has form ma p( = The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been Echanging ma and product operators ma p( Results in More efficient computation Interpreted as messages passed from node N to node 1 ( 1, , 3 2, 3 N 1,N N 1 1 ma... ma ψ Z 1 N 1, 2 (, 1 2 ψ 2, 3 ( 1 ma ψ 1, 2(1,2... ma ψ Z 1 N 2, 3...ψ, N N 1,N corrupted. Restart your computer, and then open the file again. If the red still appears, you may have to delete the image and then insert it again. = N 1,N(N 1,N ( N 1, N 6
7 Generalization to tree factor graph Substitution factored graph epansion Into ma p( = s f s (s p( = ma... ma 1 M p( And echanging maimizations with products Final maimization is performed over product of all messages arriving at the root node Could be called the ma-product algorithm 7
8 Use of log probabilities Products of probabilities can lead to numerical underflow problems Convenient to work with logarithm of joint distribution Has the effect of replacing products in maproduct algorithm with sums Thus we obtain the ma-sum algorithm 8
9 Message Passing formulation In sum-product we had ( =.. f(,1,...,m From factor node to variable node From variable Node to factor node Initial messages sent by leaf nodes µ f µ f (m m 1 M m ne(f\ µ µ f( = 1 f ( = µ f ( l ne(\f l µ ( f( f = By replacing sum with ma and products with sums of logarithms ( = ma ln f(, 1,..M + µ 1,..M m ne(f\ µ m f f(m µ f( = µ fl l ne(\f ( Initial messages sent by leaf nodes µ µ f f ( = 0 ( = ln f( 9
10 Maimum compution At root node in sum-product algorithm p( = s ne( µ f ( By analogy in ma-sum algorithm p ma = s ne( s µ f ( s 10
11 Finding variable configuration with maimum value In evaluating p ma we will also get ma for the most probable value for the root node as ma = arg ma It is tempting to apply the above to from the root back to leaves s ne( However there may be multiple configurations of all of which give rise to maimum value of p( Recursively repeated at every node So over all configuration need not be the one that maimizes µ f s ( 11
12 Modified message passing Different type of message passing from the root node to the leaves Keeping track of which values of the variables give rise to the maimum state of each variable Storing quantities given by φ( = arg ma[ln fn-,n(n 1,n + µ n 1 fn 1 n 1 1 (n n,n Understood better by looking at lattice or trellis diagram ] 12
13 Lattice or Trellis Diagram k=2 and k=3 each represent possible values of N ma Two paths give global maimum Can be found by tracing back along opposite direction of arrow Not a graphical model Columns represent variables Row represent states of variable 13
14 Backtracking in Trellis For each state of given variable there is a unique state of the previous variable that maimizes probability ties are broken systematically or randomly Equivalent to propagating a message back down the chain using ma n 1 = φ ( ma n Know as backtracking 14
15 Etension to general tree graphs Method is generalizable to tree-structured factor graphs If a message is sent from a factor node f to a variable node Maimization is performed over all other variable nodes 1,.., N that are neighbors of the factor node Keeping track of which values of the variables gave the maimum 15
16 Viterbi Algorithm Ma-sum algorithm gives eact maimizing configuration for variables provided factor graph is a tree Important application is in finding most probable sequence of hidden states in a HMM known as the Viterbi algorithm 16
17 Ma sum versus ICM ICM is simpler Ma sum finds global maimum for tree graphs ICM is not guaranteed to find global maimum 17
18 Eact inference in general graphs Sum-product and ma-sum algorithms are efficient and eact solutions to inference problems in tree-structured graphs In some cases we need to deal with graphs with loops Message passing framework can be generalized to arbitrary graph topologies Know as junction tree algorithm 18
19 Junction Tree Algorithm Triangulation: Find chord-less Cycles such as ACBDA and add links such as AB or CD Join tree Nodes correspond to maimal cliques of triangulated graph Links connect pairs of cliques that have variables in common Done so as to give a maimal spanning tree defined as Weight of the tree is maimum Weight is sum of weights for links Junction tree Tree is condensed so that any clique that is a subset of another clique is absorbed Tow-stage message passing algorithm equivalent to sum-product, can be applied to junction tree to find marginals and conditionals 19
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