Exact Inference: Variable Elimination

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1 Readings: K&F Exact nerence: Variable Elimination ecture 6-7 Apr 1/ E 515 tatistical Methods pring 2011 nstructor: u-n ee University o Washington eattle et s revisit the tudent Network Assumptions Binary RVs oherence iiculty ntelligence rade AT etter ob appy ocal probabilistic models: table s arameters and structure are given. Notations & abbreviations : a random variable : a set o random variables Val : a set o values on j : a value on : size o Val j : j 2 1

2 nerence Tasks in tudent Network onditional probability queries l 1 or l 1 h 0 or h 0 j 1 or j 1 j 1 i 1 d 1 or j 1 i 0 d 1 j 1 h 0 i 1 or j 0 h 0 i 1 j 1 i 0 h 0 : Query RVs Evidence RVs ow to compute the probabilities? Use joint distribution Naïve Approach Use ull joint distribution omputing j 1 j 1 c 0 d 0 i 0 g 0 s 0 l 0 j 1 h 0 + c 0 d 0 i 0 g 0 s 0 l 0 j 1 h 1 + c 0 d 0 i 0 g 0 s 0 l 1 j 1 h 0 + c 0 d 0 i 0 g 0 s 0 l 1 j 1 h 1 + c 0 d 0 i 0 g 0 s 1 l 0 j 1 h 0 + c 0 d 0 i 0 g 0 s 1 l 0 j 1 h 1 + c 0 d 0 i 0 g 0 s 1 l 1 j 1 h 0 + c 0 d 0 i 0 g 0 s 1 l 1 j 1 h 1 + c 0 d 0 i 0 g 1 s 0 l 0 j 1 h 0 + c 0 d 0 i 0 g 1 s 0 l 0 j 1 h 1 + c 0 d 0 i 0 g 1 s 0 l 1 j 1 h 0 + c 0 d 0 i 0 g 1 s 0 l 1 j 1 h 1 + c 0 d 0 i 0 g 1 s 1 l 0 j 1 h 0 + c 0 d 0 i 0 g 1 s 1 l 0 j 1 h 1 + c 0 d 0 i 0 g 1 s 1 l 1 j 1 h 0 + c 0 d 0 i 0 g 1 s 1 l 1 j 1 h 1 + : omputing i 0 j 1 i 0 j 1 c 0 d 0 i 0 g 0 s 0 l 0 j 1 h 0 + c 0 d 0 i 0 g 0 s 0 l 0 j 1 h 1 + c 0 d 0 i 0 g 0 s 0 l 1 j 1 h 0 + c 0 d 0 i 0 g 0 s 0 l 1 j 1 h 1 + c 0 d 0 i 0 g 0 s 1 l 0 j 1 h 0 + c 0 d 0 i 0 g 0 s 1 l 0 j 1 h 1 + c 0 d 0 i 0 g 0 s 1 l 1 j 1 h 0 + c 0 d 0 i 0 g 0 s 1 l 1 j 1 h 1 + c 0 d 0 i 0 g 1 s 0 l 0 j 1 h 0 + c 0 d 0 i 0 g 1 s 0 l 0 j 1 h 1 + c 0 d 0 i 0 g 1 s 0 l 1 j 1 h 0 + c 0 d 0 i 0 g 1 s 0 l 1 j 1 h 1 + c 0 d 0 i 0 g 1 s 1 l 0 j 1 h 0 + c 0 d 0 i 0 g 1 s 1 l 0 j 1 h 1 + c 0 d 0 i 0 g 1 s 1 l 1 j 1 h 0 + c 0 d 0 i 0 g 1 s 1 l 1 j 1 h 1 + : omputational complexity: exponential blowup Exploiting the independence properties? 4 2

3 Naïve Approach omputing j 1 c 0 d 0 c 0 i 00 g [ g 0 i 0 d 0 i 0 s d 0 0 s i 0 l 0 i 0 l g 0 j 0 g 01 j l 0 s 1 l 0 h 0 s 0 h 0 g 0 j 0 g 1 0 j 1 + c 0 d 0 c 0 i 0 g + g 0 i d s 0 i l 0 g j 1 l 0 s h 1 g 0 j 1 ertain terms are 0 i 0 d 0 s 0 i 0 l 0 g 0 j 1 l 0 s 0 h 1 g 0 j 1 + c 0 d 0 c 0 i 0 g + g 0 i 0 d i 0 s d 0 s 0 i 0 l i 0 l 1 g 10 g j 0 j 1 l 11 s l 01 h s 0 h 0 g 0 g j 10 j 1 + c 0 d 0 c 0 i 0 g + g 0 i d s 0 i l 1 g 0 j 1 l 1 s 0 h 1 g 0 j 1 repeated several 0 i 0 d 0 s 0 i 0 l 1 g 0 j 1 l 1 s 0 h 1 g 0 j 1 + c 0 d 0 c 0 i 0 g + g 0 i 0 d i 0 s d 0 s 1 i 01 l i 0 l 0 g 0 g j 0 j 1 l 01 s l 10 h s 1 h 0 g 0 g j 10 j 1 times + c 0 d 0 c 0 i 0 g + g 0 i 0 d i 0 s d 0 s 1 i 01 l i 0 l 0 g 0 g j 0 j 1 l 01 s l 10 h s 1 h 1 g 10 g j 10 j 1 + c 0 d 0 c 0 i 0 g + g 0 i 0 d i 0 s d 0 s 1 i 01 l i 0 l 1 g 10 g j 0 j 1 l 11 s l 1 h s 1 h 0 g 0 g j 10 j 1 + c 0 d 0 c 0 i 0 g + g 0 i 0 d i 0 s d 0 s 1 i 01 l i 0 l 1 g 10 g j 0 j 1 l 11 s l 1 h s 1 h 1 g 10 g j 10 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 0 i 0 l i 0 l 0 g 01 g j 1 j 1 l 01 s l 0 h s 0 h 0 g 01 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 0 i 0 l i 0 l 0 g 01 g j 1 j 1 l 01 s l 0 h s 0 h 1 g 1 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 0 i 0 l i 0 l 1 g 1 g j 1 j 1 l 11 s l 01 h s 0 h 0 g 01 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 0 i 0 l i 0 l 1 g 1 g j 1 j 1 l 11 s l 01 h s 0 h 1 g 1 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 1 i 01 l i 0 l 0 g 01 g j 1 j 1 l 01 s l 10 h s 1 h 0 g 01 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 1 i 01 l i 0 l 0 g 01 g j 1 j 1 l 01 s l 10 h s 1 h 1 g 1 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 1 i 01 l i 0 l 1 g 1 g j 1 j 1 l 11 s l 1 h s 1 h 0 g 01 g j 11 j 1 + c 0 d 0 c 0 i 0 g + g 1 i 01 d i 0 s d 0 s 1 i 01 l i 0 l 1 g 1 g j 1 j 1 l 11 s l 1 h s 1 h 1 g 1 g j 11 j 1 + : Exploiting the structure can reduce computation. et s systematically analyze computational complexity. 5 et s start with the simplest network 6

4 Exact nerence Variable Elimination nerence in a simple chain omputing x1 2 x1 2 2 x1 x1 x1 All the numbers or this computation are in the s o the original Bayesian network O 1 2 operations 7 Exact nerence Variable Elimination nerence in a simple chain omputing omputing x 2 x1 2 x1 x1 x1 x2 x2 x2 x2 x is a given 2 was computed above O operations 8 4

5 Exact nerence: Variable Elimination n nerence in a general chain omputing n ompute each i rom i-1 k 2 operations or each computation or i assuming i k Onk 2 operations or the inerence ompare to k n operations required in summing over all possible entries in the joint distribution over 1... n nerence in a general chain can be done in linear time! 9 Exact nerence: Variable Elimination ushing summations ynamic programming 10 5

6 nerence With a oop omputing ierences 2 4 ummations are not pushed in as ar as beore. The scope o includes two variables not one. epends on network structure 11 Eicient nerence in Bayesnets roperties that allow us to avoid exponential blowup in the joint distribution Bayesian network structure some subexpressions depend on a small number o variables omputing these subexpressions and caching the results avoids generating them exponentially many times 12 6

7 Variable Elimination: Factors nerence algorithm deined in terms o actors Factors generalize the notion o s A actor is a unction rom value assignments o a set o random variables to real positive numbers R + The set o variables is the scope o the actor Thus the algorithm we describe applies both to Bayesian networks and Markov networks 1 Operations on Factors : roduct et Y Z be three sets o disjoint sets o RVs and let 1 Y and 2 YZ be two actors We deine the actor product 1 x 2 operation to be a actor ψ:valyz R as ψyz 1 Y 2 YZ Y 1 [Y] x 0 y 0 2 x 0 y 1 x 1 y 0 10 x 1 y 1 5 Y Z 2 [Y] y 0 z 0 6 y 0 z 1 1 y 1 z 0 7 y 1 z 1 12 Y Z Ψ [YZ] x 0 y 0 z 0 2 x 6 12 x 0 y 0 z 1 2 x 1 2 x 0 y 1 z 0 x 7 21 x 0 y 1 z 1 x 12 6 x 1 y 0 z 0 10 x 6 60 x 1 y 0 z 1 10 x 1 10 x 1 y 1 z 0 5 x 7 5 x 1 y 1 z 1 5 x

8 Operations on Factors : Marginalization et be a set o RVs Y a RV and Y a actor We deine the actor marginalization o Y in to be a actor ψ:val R as ψ Σ Y Y Also called summing out n a Bayesian network summing out all variables 1 n a Markov network summing out all variables is the partition unction 15 More on Factors For actors 1 and 2 : Factors are commutative 1 x 2 1 x 2 Σ Σ Y Y Σ Y Σ Y roducts are associative 1 x 2 x 1 x 2 x cope[ 1 ] we used this in elimination above Σ 1 x 2 1 x Σ

9 nerence in hain by Factors cope o and 4 does not contain 1 cope o 4 does not contain 2 17 um-roduct nerence et Y be the query RVs and Z be all other RVs We can generalize this task as that o computing the value o an expression o the orm: Y all it sum-product inerence task. Eective computation Z ' F The scope o the actors is limited. ush in some o the summations perorming them over the product o only a subset o actors ' 18 9

10 um-roduct Variable Elimination Algorithm iven an ordering o variables Z 1 Z n um out the variables one at a time When summing out each variable Z Multiply all the actors s that mention the variable generating a product actor Ψ um out the variable rom the combined actor Ψ generating a new actor without the variable Z um out age um-roduct Variable Elimination Theorem et be a set o RVs et Y be a set o query RVs et Z-Y For any ordering α over Z the above algorithm returns a actor Y such that Y ' Bayesian network query Y F consists o all s in F Each i i a i { } n i i Z ' F Apply variable elimination or ZU-Y summing out Z

11 11 Example et s consider a little more complex network 21 A More omplex Network oal: Eliminate: 22

12 12 A More omplex Network oal: Eliminate: ompute: A More omplex Network oal: Eliminate: ompute:

13 1 A More omplex Network oal: Eliminate: ompute: A More omplex Network oal: Eliminate: ompute:

14 14 A More omplex Network oal: Eliminate: ompute: A More omplex Network oal: Eliminate: ompute:

15 15 A More omplex Network oal: Eliminate: ompute: A More omplex Network oal: Eliminate: dierent ordering Note: intermediate actors tend to be large 1 ordering matters

16 16 nerence With Evidence omputing Y Ee et Y be the query RVs et E be the evidence RVs and e their assignment et Z be all other RVs U-Y-E The general inerence task is Z Y e E Z e E e e Y U U 1 nerence With Evidence oal: hi Eliminate: Below compute hi i h i i i h 1 i h i i 2 i i h i i ierences ess number o variables to be eliminated and are excluded cope o actors tend to be smaller.

17 What s the complexity o variable elimination? omplexity o Variable Elimination Variable elimination consists o enerating the actors s that will be summed out umming out age 1 enerating the actor i 1 x...x ki through actor product operation et i be the scope o i Each entry requires k i multiplications to generate enerating actor i is Ok i Val i umming out Addition operations at most Val i er actor: OkN where Nmax i Val i kmax i k i 4 17

18 omplexity o Variable Elimination tart with n actors nnumber o variables enerate exactly one actor at each iteration there are at most 2n actors enerating actors ay Nmax i Val i At most Σ i Val i k i NΣ i k i N 2n since each actor is multiplied in exactly once and there are 2n actors umming out Σ i Val i N n since we have n summing outs to do Total work is linear in N and n where Nmax i Val i Exponential blowup can be in N i which or actor i can be v m i actor i has m variables with v values each nterpretation: maximum scope size is important. 5 Factors and Undirected raphs The algorithm does not care whether the graph that generated the actors is directed or undirected. The algorithm s input is a set o actors and the only relevant aspect to the computational is the scope o the actors. et s view the algorithm as operating on an undirected graph. For Bayesian networks we consider the moralized Markov network o the original BNs. ow does the network structure change in each variable elimination step? 6 18

19 19 VE as raph Transormation At each step we are computing lot a graph where there is an undirected edge Y i variables and Y appear in the same actor Note: this is the Markov network o the probability on the variables that were not eliminated yet j j j i i Z 7 VE as raph Transormation oal: Eliminate: 8

20 20 VE as raph Transormation oal: Eliminate: ompute: VE as raph Transormation oal: Eliminate: ompute:

21 21 VE as raph Transormation oal: Eliminate: ompute: ill edge VE as raph Transormation oal: Eliminate: ompute:

22 22 VE as raph Transormation oal: Eliminate: ompute: VE as raph Transormation oal: Eliminate: ompute:

23 2 VE as raph Transormation oal: Eliminate: ompute: The nduced raph The induced graph Fα over actors F and ordering α: Union o all o the graphs resulting rom the dierent steps o the variable elimination algorithm. i and j are connected i they appeared in the same actor throughout the VE algorithm using α as the ordering Original graph nduced graph 46

24 The nduced raph The induced graph Fα over actors F and ordering α: Undirected i and j are connected i they appeared in the same actor throughout the VE algorithm using α as the ordering The width o an induced graph width Kα is the number o nodes in the largest clique in the graph minus 1 Minimal induced width o a graph K is min α width Kα Minimal induced width provides a lower bound on best perormance by applying VE to a model that actorized on K ow can we compute the minimal induced width o the graph and the elimination ordering achieving that width? No easy way to answer this question. 47 The nduced raph Finding the optimal ordering is N-hard opeless? No heuristic techniques can ind good elimination orderings reedy search using heuristic cost unction We eliminate variables one at a time in a greedy way so that each step tends to lead to a small blowup in size. At each point ind the node with smallest cost ossible costs: number o neighbors in current graph neighbors o neighbors number o illing edges 48 24

25 nerence should be eicient or certain kinds o graphs 49 Elimination On Trees Tree Bayesian network Each variable has at most one parent All actors involve at most two variables Elimination Eliminate lea variables Maintains tree structure nduced width

26 Elimination on olytrees olytree Bayesian network At most one path between any two variables Theorem: inerence is linear in the network representation size 51 For a ixed graph structure is there any way to reduce the induced width? 52 26

27 nerence By onditioning eneral idea Enumerate the possible values o a variable Apply Variable Elimination in a simpliied network Aggregate the results Transorm s o and to eliminate as parent Observe 5 E 515 tatistical Methods pring