Graphical)Models - University of Massachusetts Amherst PDF
Preview Graphical)Models - University of Massachusetts Amherst
Graphical Models Lecture 12: Belief Update Message Passing Andrew McCallum [email protected] Thanks to Noah Smith and Carlos Guestrin for slide materials. 1 Today’s Plan • Quick Review: Sum Product Message Passing (also known as “Shafer-‐Shenoy”) • Today: Sum Product Divide Message Passing (also known as “belief update” message passing “Lauritzen-‐ Spiegelhalter” and “belief propagaFon”) • MathemaFcally equivalent, but different intuiFons. • Moving toward approximate inference. ϕ ϕ Quick Review F A ϕ Flu All. • {H, R, S, A, F} FAS S.I. ϕ ϕ SR SH ϕ SR ψ R.N. H. SR SR τ S ϕ F ϕ A ν ϕ ϕ FAS SH FAS ψ SH HS SAF ν (C ) = φ j j τ S ψ φ Φ FAS ∈ j Message Passing (One Root) • Input: clique tree T, factors Φ, root C r • For each clique C , calculate ν i i • While C is sWll waiWng on incoming messages: r – Choose a C that has received all of its incoming i δ = ν δ i j i k i → → messages. C S k Neighbors j i\ i,j ∈ i\{ } – Calculate and send the message from C to C i upstream-‐neighbor(i)β = ν δ r r k r → k Neighbors ∈ r = φ X C φ Φ \ i ∈ = Z P(C ) r · Different Roots; Same Messages δ SAF→SR β SR δ SR SR→SAF HS SAF root β β SAF HS δ HS→SAF δ SAF→HS Sum-‐Product Message Passing • Each clique tree vertex C passes messages to i each of its neighbors once it’s ready to do so. • At the end, for all C : i β = ν δ i i k i → k Neighbors i ∈ – This is the unnormalized marginal for C . i Calibrated Clique Tree • Two adjacent cliques C and C are calibrated i j when: β = β i j C S C S ii,j ji,j \ \ = µ (S ) i,j i,j β β i j μ C i,j C i j S = i,j C ∩ C i j Calibrated Clique Tree as a • Original (unnormalized) factor model and calibrated clique tree represent the same (unnormalized) measure: clique beliefs β C C Vertices( ) φ = ∈ T µ φ Φ S ∈ S Edges( ) ∈ T sepset beliefs unnormalized Gibbs distribuWon from original factors Φ Inventory of Factors • original factors ϕ • iniWal potenWals ν • messages δ • intermediate factors ψ (no longer explicit) • clique beliefs β • sepset beliefs μ Inventory of Factors • original factors ϕ • iniWal potenWals ν • messages δ • intermediate factors ψ (no longer explicit) • clique beliefs β • sepset beliefs μ New algorithm collapses everything into beliefs!
Description:The list of books you might like
