Variable and Value Variable and Value Ordering for MPE SearchOrdering for MPE Search

Sajjad Siddiqi and Jinbo Huang

Most Probable Explanation Most Probable Explanation (MPE)(MPE)

N : Bayesian networkX : {X,Y,Z}e : {X=x}

YY

XX ZZ

X Y Z Pr(X,Y,Z)

x y z 0.05

x y z 0.3

x y z 0.05

x y z 0.1

x y z 0.1

x y z 0.2

x y z 0.1

x y z 0.1

Most Probable Explanation Most Probable Explanation (MPE)(MPE)

N : Bayesian networkX : {X,Y,Z}e : {X=x}

YY

XX ZZ

X Y Z Pr(X,Y,Z)

x y z 0.05

x y z 0.3

x y z 0.05

x y z 0.1

x y z 0.1

x y z 0.2

x y z 0.1

x y z 0.1

maxmax

Exact MPE by InferenceExact MPE by Inference

Variable Elimination– Bucket Elimination

Exponential in the treewidth of the elimination order.

Compilation– Decomposable Negation Normal Form (DNNF)

Exploits local structure so treewidth is not necessarily the limiting factor.

Both methods can either run out of time or memory

Exact MPE by SearchingExact MPE by Searching

XX

YY

ZZ ZZ

0.10.1 0.20.2 0.10.1 0.10.1

xx

yy yy

zz zz zz zz

Exact MPE by SearchingExact MPE by Searching

Depth-First Search– Exponential in the number of variables X.

Depth-First Branch-and-Bound Search– Computes an upper bound on any

extension to the current assignment.– Backtracks when upper bound >=

current solution.– Reduces complexity of search.

Exact MPE by B-n-B SearchExact MPE by B-n-B Search

XX

YY

ZZ ZZ

0.10.1 0.20.2 0.10.1 0.10.1

xx

yy yy

zz zz zz zz

if if upper boundupper bound <= 0.2 <= 0.2 current solutioncurrent solution = 0.2 = 0.2

Computing Bounds: Mini-Computing Bounds: Mini-BucketsBuckets

Ignores certain dependencies amongst variables: – New network is easier to solve.– Solution grows only in one direction.

Splits a bucket into two or more mini-buckets.– Focuses on generating tighter bounds.

Mini-buckets is a special case of node splitting.

Node SplittingNode Splitting

Y1Y1 Y2Y2^̂ ^̂

andand are clones of are clones of YY: fully split: fully split

(N)(N)

YY

XX ZZ

QQ

RR

^̂

YY

XX ZZ

QQ11

RR

YY11 YY22^̂

(N`)(N`)

split variables = {split variables = {QQ,,YY}}

QQ

QQ22^̂ ^̂

splittingsplitting

Node SplittingNode Splitting

e: an instantiation of variables X in N.e: a compatible assignment to their

clones in N´e.g. if e = {Y=y}, then e = {Y1=y,

Y2=y}

thenMPEp (N, e) <= MPEp (N´, e, e)

= total number of instantiations of clone variables

^̂ ^̂

Computing Bounds: Node Computing Bounds: Node Splitting (Choi et. Al Splitting (Choi et. Al

2007).2007).Split network is easier to solve, its MPE

computes the bound.Search performed only over the ‘split

variables’ instead of all.Focuses on good network relaxations

trying to reduce the number of splits.

B-n-B Search for MPEB-n-B Search for MPE

YY

QQ QQ

yy yy

qq

MPE(N`,{X=x})MPE(N`,{X=x})

MPE(N`,{X=x, Y=MPE(N`,{X=x, Y=yy, Y, Y11==yy, Y, Y22==yy})}) MPE(N`,{X=x, Y=MPE(N`,{X=x, Y=yy, Y, Y11==yy, Y, Y22==yy})})

MPE(N`,{X=x, Y=y, YMPE(N`,{X=x, Y=y, Y11==yy, Y, Y22==yy,Q=q, Q,Q=q, Q11=q, Q=q, Q22=q})=q})

= 4, for two split variables with binary domain= 4, for two split variables with binary domain

boundbound

exact solutionexact solution

^̂ ^̂ ^̂ ^̂

^̂ ^̂ ^̂ ^̂

boundbound

boundbound

B-n-B Search for MPEB-n-B Search for MPE

Leaves of the search tree give candidate MPE solutions.

Elsewhere we get upper bounds to prune the search.

A branch gets pruned if bound <= current solution.

Choice of Variables to SplitChoice of Variables to Split

Reduce the number of split variables.– Heuristic based on the reduction in the size

of jointree cliques and separators.Split enough variables to reduce the

treewidth to a certain threshold (when the network is easy to solve).

Variable and Value Variable and Value OrderingOrdering

Reduce search space using an efficient variable and value ordering.

(Choi et al. 2007) do not address this and use a neutral heuristic.

Several heuristics are analyzed and their powers combined to produce an effective heuristic.

Scales up the technique.

Entropy-based OrderingEntropy-based Ordering

YY

QQ QQ

yy yy

Pr(Pr(YY==yy), Pr(), Pr(YY==yy))Pr(Pr(QQ==qq), Pr(), Pr(QQ==qq))

entropy(entropy(YY), entropy(), entropy(QQ))ComputationComputation

Do the same for clonesDo the same for clones

and get average probabilities:

Pr (Y=y) = [Pr(Y=y)+Pr(Y1=y)+Pr(Y2=y)]/3

^̂ ^̂

Entropy-based OrderingEntropy-based Ordering

YY

QQ QQ

yy yy

Favor those instantiations that are more likely to be MPEs.

ComputationComputation Prefer Y over Q if

entropy(Y) < entropy(Q).

Prefer Y=y over Y=y if

Pr(Y=y) > Pr(Y=y)

Static and Dynamic versions.

Entropy-based OrderingEntropy-based Ordering

Probabilities computed using DNNF:– Evaluation and Differentiation of AC

Experiments: – Static heuristic, significantly faster

than the neutral. – Dynamic heuristic, generally too

expensive to compute and slower.

Nogood LearningNogood Learning

g = {X=x, Y=y, Z=z} is a nogood ifMPEp (N’, g, g) <= current solution

XX

YY

ZZ

current solution=1.0current solution=1.0bound=1.5bound=1.5

bound=1.3bound=1.3

bound=1.2bound=1.2

bound=0.5bound=0.5

yy

xx

zz

let g’ = g \ {Y=y} &MPEp (N’, g’, g’) <= current solution

then g = g’

Nogood-based OrderingNogood-based Ordering

Scores:S(X=x) = number of occurrences in nogoods

S(X) = [S(X=x) + S(X=x)]/2 (binary variables)

Dynamic Ordering:

Prefer higher scores.

Impractical: overhead of repeated bound computation during learning.

Score-based OrderingScore-based Ordering

A more effective approach based on nogoods.Scores of variables/values tell how can a nogood

be obtained quickly (backtrack early).

XX

YY

ZZ

bound=1.5bound=1.5

bound=1.3bound=1.3

bound=1.2bound=1.2

bound=0.5bound=0.5

yy

xx

zz

S(X=x) += 1.5-1.3=0.2S(X=x) += 1.5-1.3=0.2

S(Y=y) += 1.3-1.2=0.1S(Y=y) += 1.3-1.2=0.1

S(Z=z) += 1.2-0.5=0.7S(Z=z) += 1.2-0.5=0.7

Improved HeuristicImproved Heuristic

Periodically reinitialize scores (focus on recent past).

Use static entropy-based order as the initial order of variables/values.

Experimental SetupExperimental Setup

Intel Core Duo 2.4 GHz + AMD Athlon 64 X2 Dual Core Processor 4600+, both with 4 GB of RAM running Linux.

A memory limit of 1 GB on each MPE query.C2D DNNF compiler [Darwiche, 2004;

2005]. Trivial seed of 0 as the initial MPE solution

to start the search. Keep splitting the network variables until

treewidth <= 10.

Comparing search spaces on grid networksComparing search spaces on grid networks

Comparing search time on grid networksComparing search time on grid networks

Comparing nogood learning and score-based DVO on grid networks

Results on grid networks, 25 queries per networkResults on grid networks, 25 queries per network

Random networks Random networks 20 queries per network20 queries per network

Networks for genetic linkage analysis, which are some Networks for genetic linkage analysis, which are some of the hardest networksof the hardest networks

Only SC-DVO succeededOnly SC-DVO succeeded

Comparison with SamIam on grid networksComparison with SamIam on grid networks

Comparison with (Marinescu & Dechter, 2007) on grid networks Comparison with (Marinescu & Dechter, 2007) on grid networks (SMBBF (SMBBF –– Static mini-bucket best first) Static mini-bucket best first)

Parameter Parameter ‘‘i=20i=20’’, where , where ‘‘ii’’ controls the size of the mini-bucket controls the size of the mini-bucket

We tried a few cases from random and genetic linkage analysis We tried a few cases from random and genetic linkage analysis networks which SMBBF could not solve (4 random networks of networks which SMBBF could not solve (4 random networks of sizes 100, 110, 120, and 130 and sizes 100, 110, 120, and 130 and pedigree13pedigree13 from the genetic from the genetic

linkage analysis network).linkage analysis network).

ConclusionConclusionNovel and efficient heuristic for

dynamic variable ordering for computing the MPE in Bayesian networks.

A significant improvement in time and space over less sophisticated heuristics and other MPE tools.

Many hard network instances solved for the first time.