luste tees and message popagaton 371 Advanced A Tomas ngla Outlne mple gaphs: tees and polytees luste gaphs and clque tees unnng ntesecton sepsetsmessage popagaton VE Message passng VE n detal achng out-of-clque quees P ncemental updatng onstuctng clque tees vaable elmnaton VE and BP: Pos & cons and tadeoffs
Tees and polytees Tee: one dected path fom the oot to each node PolyTee: one undected tal fom the oot to each node Undected epesentaton: a tee gaph teewdth1 lque tees VE woks on factos Make facto a data stuctue ends and eceves messages luste gaph fo set of factos each node s assocated wth a subset cluste of. Famly-pesevng: each facto s vaables ae completely embedded n a cluste
lque tee popetes epset sepaaton set: Vaables X on one sde of sepset ae sepaated fom the vaables Y on the othe sde n the facto gaph gven vaables n Runnng ntesecton f and both contan X then all clques on the unque path between them do lque tees P P P Runnng ntesecton: lques nvolvng fom a connected subtee. P P P P ntal potentals : Assgn factos to clques and multply them. ave espect fo famles! P Tee need not be mnmal
Message Passng VE Quey fo P Elmnate : Message eceved at [] -- [] updates: τ 1 1 [ ] Message sent fom [] to [] [ ] τ1 [ ] Message Passng VE Quey fo P Elmnate : τ [ ] Message eceved at [] -- [] updates: 3[ ] τ 3[ ] Message sent fom [] to []
Message Passng VE Quey fo P Elmnate : τ 3 3[ ] Message eceved at [] -- [] updates: Message sent fom [] to [] 4[ ] τ3 4[ ]! [] s not eady! Message Passng VE Quey fo P Elmnate : τ 4 5[ ] Message sent fom [] to [] All messages eceved at [] [] updates: 4[ ] τ3 τ4 4[ ] And so on
Message Passng VE hose [] as the oot clque ould we have chosen othewse? Message Passng VE hoose [] as the oot clque Obsevaton: ome messages dd not change! [] [] Notaton: numbe the clques and denote the messages
oectness of VE on clque tees Message summazes nfomaton n the pat of tee t sepaates V p φ F p Poof s by nducton fom leaves Base case leaf clque φ φ φ F oectness of VE on clque tees nducton case: non-leaf clque sendng to wth chlden... 1 k φ... φ φ... φ Vp φ F p Y Vp V F k F F 1 p φ φ p 1 φ p k by ntesecton popety the unons ae dsont V Then + p Y UVp m m 1.. k φ F + p F U Fp m m 1.. k φ φ... V p φ F p Y φ F Vp φ F V F 1 p 1 p k φ p k φ
oectness of VE on clque tees By nducton hypothess V F p p φ φ Y k m V F m φ φ.. 1 p p X V k k k X hlden hlden X hlden V F X hlden \... φ φ p p Then the oot clque has the coect magnal: Message passng VE Message ode s only patal omputes magnals fo any node Y Results n a calbated clque tee Often many magnals desed neffcent to e-un nfeence One dstnct message pe edge & decton Recap: thee knds of facto obects ntal fnal potentals and messages
Message Passng VE hafe-henoy algothm asynchonous mplementaton of two passes: upwad and downwad Asynchonously do: node eady to send m to node when t has eceved a message fom all othe nodes end message Magnalze oot clque s ancllay vas k k N Message Passng: BP aphcal model of a dstbuton Moe edges lage expessve powe lque tee also a model of dstbuton Message passng peseves model but changes paametezaton ffeent but equvalent algothm
Facto dvson A1 B1.5 A1 B1.5/.41.5 A1 A A A3 B B1 B B1.4.8..6 A1 A A3.4.4.5 A1 A A A3 B B1 B B1.4/.41..8/.4../.4..6/.51. A3 B.5 A3 B.5/.51. nvese of facto poduct Message Passng: BP Each node: multply all the messages and dvde by the one comng fom node we send to lealy the same as VE k k N k k N \
Message Passng: BP AB B B B B 3 3 3 B B 1 B A B 3 3 3 B B B B 3 3 3 3 3 3 toe the last message on the edge and dvde each passng message by the last stoed. Message Popagaton: BP autzen-pegelhalte algothm Two knds of obects ntal potentals not kept mpoved stablty of asynchonous algothm epeated messages cancel out stbuton epesentaton clque tee nvaant X P F T T T µ
Multple quees Much cachng possble ove a clque tee Example: compute PXY fo each X Y ynamc pogammng Base case X and Y ae n neghbo clques P µ P Take advantage of condtonal ndependence: l l : P P P l l l ncemental updates Fully-nfomed: all neghbos have sent the messages albated -- messages and clques agee on magnals: fxed pont of MP µ Evdence avalable n peces Re-unnng nfeence neffcent Expess evdence n ndcato vecto multply nto some clque un one pass away fom to nfom the est woks fo soft evdence as well
Out-of-clque quees want PB no clque wth both B and! Buld a new clque tee expensve o o vaable elmnaton ove calbated tee AB B B P B P B B 3 µ 3 P B P Ths s back to VE we save f vaables of nteest ae close n the clque tee. efnng clque tees VE defnes clques Each facto s subset of a clque of Evey max clque n s a facto Each clque n s a subclque n Each clque n s a clque n Non-maxmal clques can be elmnated hodal gaphs Maxmal clques of any c.g. that s a supeset of can be aanged nto a clque tee fo tangulaton
lque tees geneated by VE aph nduced by odeng: VE constuctng a clque tee
VE constuctng a clque tee VE constuctng a clque tee
VE constuctng a clque tee VE constuctng a clque tee
VE constuctng a clque tee VE constuctng a clque tee
VE constuctng a clque tee ummay lque tees factos assgned to clques many-to-one unnng ntesecton Message passng on clque tees Vaable Elmnaton Belef popagaton dffeent vews algebacally the same VE defnes clques Tme and space tadeoff spectum
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