Conditional Independence d-Separation Belief Propogation Graphical Models Steven J Zeil Old Dominion Univ. Fall 2010 1
Conditional Independence d-Separation Belief Propogation Graphical Models Conditional Independence 1 d-Separation 2 Belief Propogation 3 2
Conditional Independence d-Separation Belief Propogation Graphical Models a.k.a. Bayesian networks, probabilistic networks Nodes are hypotheses (random vars) Values are the probabilities of the observed value of that variable Arcs are direct influences between hypotheses Forms a directed acyclic graph (DAG) The parameters are the conditional probabilities in the arcs 3
Conditional Independence d-Separation Belief Propogation Example Knowing that the grass is wet, what is the probability that rain is the cause? P ( W | R ) P ( R ) P ( R | W ) = P ( W ) P ( W | R ) P ( R ) = P ( W | R ) P ( R ) + P ( W |¬ R )( P ( ¬ R ) 0 . 9 × 0 . 4 = 0 . 9 × 0 . 4 + 0 . 2 × 0 . 6) = 0 . 75 4
Conditional Independence d-Separation Belief Propogation Causes & Diagnoses Graph shows a causal relationship. Bayes rules “reverses” the arc to give a diagnosis. 5
Conditional Independence d-Separation Belief Propogation Conditional Independence X and Y are independent if P ( X , Y ) = P ( X ) P ( Y ) X and Y are conditionally independent given Z if P ( X , Y | Z ) = P ( X | Z ) P ( Y | Z ) or P ( X | Y , Z ) = P ( X | Z ) Three canonical cases: Head-to-tail, Tail-to-tail, head-to-head 6
Conditional Independence d-Separation Belief Propogation Head-to-Tail P ( X , Y , Z ) = P ( X ) P ( Y | X ) P ( Z | Y ) P ( W | C ) = P ( W | R ) P ( R | C ) + P ( W |¬ R ) P ( ¬ R | C ) 7
Conditional Independence d-Separation Belief Propogation Blocking If we know the state of Y, we know everything we can about Z without knowingthe state of X. We say that Y blocks the path from X to Z or, Y separates X and Z. 8
Conditional Independence d-Separation Belief Propogation Tail-to-Tail P ( X , Y , Z ) = P ( X ) P ( Y | X ) P ( Z | X ) An observed X blocks the path between Y and Z: P ( X , Y , Z ) P ( X , Y | X ) = P ( X ) P ( X ) P ( Y | X ) P ( Z | X ) = P ( X ) = P ( Y | X ) P ( Z | X ) 9
Conditional Independence d-Separation Belief Propogation Head-to-Head P ( X , Y , Z ) = P ( X ) P ( Y ) P ( Z | X , Y ) Z blocks the path between X and Y when it is unobserved . 10
Conditional Independence d-Separation Belief Propogation Causal vs Diagnostic Causal inference : If the sprinkler is on, what is the probability that the grass is wet? ( P ( W | S )) Diagnostic inference : If the grass is wet, what is the probability that the sprinkler is on? P ( S , W ) = P ( W | S ) P ( S ) = 0 . 35 P ( W ) 11
Conditional Independence d-Separation Belief Propogation Explaining Away Suppose that we know that it rained: P ( W | R , S ) P ( S | R ) P ( S | R , W ) = P ( W | R P ( W | R , S ) P ( S ) = P ( W | R = 0 . 21 Note that P ( S | R , W ) < P ( S | W ). Explaining Away : Knowing that it rained, the prob that the sprinkler caused the wet grass is decreased. 12
Conditional Independence d-Separation Belief Propogation Larger Systems Larger systems formed by combining the three basic subgraphs Provides a structure & explanation of complicated relationships This graph describes P ( C , S , R , W , F ) How would you compute P ( F | C )? 13
Conditional Independence d-Separation Belief Propogation Example: Classification Causal relation P ( x | C ) Bayes’ rule inverts P ( C | x ) = p ( x | C ) P ( C ) p ( x ) 14
Conditional Independence d-Separation Belief Propogation Example: Naive Bayes Classification Given C, the x j are independent P ( � x | C ) = p ( x 1 | C ) p ( x 2 | C ) . . . p ( x d | 15
Conditional Independence d-Separation Belief Propogation Example: Hidden Markov State at time t depends only on state at time t − 1 Output depends only on the current state 16
Conditional Independence d-Separation Belief Propogation Path Blocking A path from node A to node B is blocked given { C } if The directions of edges on the path meet head-to-tail (case 1) or tail-to-tail (case 2) at a node in C, or The directions of edges meet head-to-head (case 3) and neither that node nor any of its descendants is in C. Examples BCDF is blocked given C BEFG is blocked given E or F BEFD is blocked unless F (or G) is given 17
Conditional Independence d-Separation Belief Propogation d-Separation If all paths from A to B are blocked given C, A and B are d-separated (conditionally independent) given C. 18
Conditional Independence d-Separation Belief Propogation Belief Propogation Use graph-based algorithms to answer queries of the form P ( X | E ) where The query node X is any node in the graph E is a set of evidence nodes whose values are known 19
Conditional Independence d-Separation Belief Propogation Chains Evidence E + in ancestors of X will flow along as diagnostic inference Evidence E − in decendents of X will flow back as causal inference E + and E − separate X from any more nodes in the chain, so we have at most two evidence nodes to consider 20
Conditional Independence d-Separation Belief Propogation Chains: Propogated Info For each node N, λ ( N ) = P ( E − | N ) π ( N ) = P ( N | E + ) 21
Conditional Independence d-Separation Belief Propogation Chains: Meeting at the Middle P ( E | X ) P ( X ) P ( X | E ) = P ( E ) P ( E + , E − | X ) P ( X ) = P ( E ) P ( E + | X ) P ( E − | X ) P ( X ) = P ( E ) P ( X | E + ) P ( E + ) P ( E − | X ) P ( X ) = P ( X ) P ( E ) α P ( X | E + ) P ( E − | X ) = = απ ( X ) λ ( X ) 22
Conditional Independence d-Separation Belief Propogation Chains: Updating P ( X | E ) = απ ( X ) λ ( X ) � π ( X ) = P ( X | U ) π ( U ) U � λ ( X ) = P ( Y | X ) λ ( Y ) Y 23
Conditional Independence d-Separation Belief Propogation Trees P ( E − λ ( X ) = X | X ) = λ Y ( X ) λ Z ( X ) � λ X ( U ) = λ ( X ) P ( X | U ) X P ( X | E + π ( X ) = X ) � = P ( X | U ) π X ( U U π Y ( X ) = αλ Z ( X ) π ( X ) 24
Conditional Independence d-Separation Belief Propogation Polytrees P ( X | E + π ( X ) = X ) k � � � � = P ( X | U 1 , U 2 , . . . , U k ) π X ( U i ) . . . U 1 U 2 U k j =1 � π Y j ( X ) = α λ Y s ( X ) π ( X ) x � = j 25
Conditional Independence d-Separation Belief Propogation Polytrees � � � λ X ( U i ) = β λ ( X ) P ( X | U 1 , U 2 , . . . , U k ) π X ( U r ) X r � = i r � = i m � λ ( X ) = λ Y j ( X ) j =1 26
Conditional Independence d-Separation Belief Propogation Junction Trees If X does not separate E + and E − (e.g., loops in dependencies) Moralize the graph by joining all nodes that have common children Identify cliques Embed cliques into single nodes to form a junction tree Each compressed node is a separately solvable subproblem 27
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