Variable and Value Ordering for MPE Search

Variable and Value Ordering for MPE Search Sajjad Siddiqi and Jinbo Huang

Most Probable Explanation (MPE) N : Bayesian network X : {X,Y,Z} e : {X=x} Y X Z

Most Probable Explanation (MPE) N : Bayesian network X : {X,Y,Z} e : {X=x} max Y X Z

Exact 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 Searching X x Y y y Z Z z z z z 0.1 0.2 0.1 0.1

Exact 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 Search X x current solution = 0.2 Y if upper bound <= 0.2 y y Z Z z z z z 0.1 0.2 0.1 0.1

Computing Bounds: Mini-Buckets • 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 Splitting Y Q splitting Y Q Y1 Y2 Q1 Q2 ^ ^ ^ ^ X Z R X Z R (N) (N`) ^ ^ are clones of Y: fully split and Y1 Y2 split variables = {Q,Y}

Node 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} then MPEp (N, e) <= MPEp (N´, e, e)  = total number of instantiations of clone variables   ^ ^ 

Computing Bounds: Node Splitting (Choi et. Al 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 MPE MPE(N`,{X=x}) Y bound ^ ^ y y ^ ^ MPE(N`,{X=x, Y=y, Y1=y, Y2=y}) MPE(N`,{X=x, Y=y, Y1=y, Y2=y}) Q Q q bound bound ^ ^ ^ ^ MPE(N`,{X=x, Y=y, Y1=y, Y2=y,Q=q, Q1=q, Q2=q}) exact solution  = 4, for two split variables with binary domain

B-n-B Search for MPE • Leaves of the search treegivecandidate MPE solutions. • Elsewhere we get upper bounds to prune the search. • A branch gets pruned if bound <= current solution.

Choice 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 Ordering • 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 Ordering Pr(Y=y), Pr(Y=y) Pr(Q=q), Pr(Q=q) entropy(Y), entropy(Q) Computation Do the same for clones and get average probabilities: Pr (Y=y) = [Pr(Y=y)+Pr(Y1=y)+Pr(Y2=y)]/3 Y y y Q Q ^ ^

Entropy-based Ordering Computation 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. Y y y Q Q Favor those instantiations that are more likely to be MPEs.

Entropy-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 Learning g = {X=x, Y=y, Z=z} is a nogood if MPEp (N’, g, g) <= current solution  X current solution=1.0 bound=1.5 x Y bound=1.3 y let g’ = g \ {Y=y} & MPEp (N’, g’, g’) <= current solution then g = g’  Z bound=1.2 z bound=0.5

Nogood-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 Ordering A more effective approach based on nogoods. Scores of variables/values tell how can a nogood be obtained quickly (backtrack early). X S(X=x) += 1.5-1.3=0.2 bound=1.5 x Y bound=1.3 S(Y=y) += 1.3-1.2=0.1 y Z S(Z=z) += 1.2-0.5=0.7 bound=1.2 z bound=0.5

Improved Heuristic • Periodically reinitialize scores (focus on recent past). • Use static entropy-based order as the initial order of variables/values.

Experimental 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 networks

Comparing search time on grid networks

Comparing nogood learning and score-based DVO on grid networks

Results on grid networks, 25 queries per network

Random networks 20 queries per network

Networks for genetic linkage analysis, which are some of the hardest networksOnly SC-DVO succeeded

Comparison with SamIam on grid networks

Comparison with (Marinescu & Dechter, 2007) on grid networks (SMBBF – Static mini-bucket best first)Parameter ‘i=20’, where ‘i’ controls the size of the mini-bucketWe tried a few cases from random and genetic linkage analysis networks which SMBBF could not solve (4 random networks of sizes 100, 110, 120, and 130 and pedigree13 from the genetic linkage analysis network).

Conclusion • Novel 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.

Variable and Value Ordering for MPE Search