Importance sampling-based estimation over AND/OR search spaces for graphical models

It is well known that the accuracy of importance sampling can be improved by reducing the variance of its sample mean and therefore variance reduction schemes have been the subject of much research. In this paper, we introduce a family of variance reduction schemes that generalize the sample mean from the conventional OR search space to the AND/OR search space for graphical models. The new AND/OR sample means allow trading time and space with variance. At one end is the AND/OR sample tree mean which has the same time and space complexity as the conventional OR sample tree mean but has smaller variance. At other end is the AND/OR sample graph mean which requires more time and space to compute but has the smallest variance. Theoretically, we show that the variance is smaller in the AND/OR space because the AND/OR sample mean is defined over a larger virtual sample size compared with the OR sample mean. Empirically, we demonstrate that the AND/OR sample mean is far closer to the true mean than the OR sample mean.

[1]  J. Geweke,et al.  Bayesian Inference in Econometric Models Using Monte Carlo Integration , 1989 .

[2]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[3]  Serafín Moral,et al.  Mixing exact and importance sampling propagation algorithms in dependence graphs , 1997, Int. J. Intell. Syst..

[4]  Nils J. Nilsson,et al.  Principles of Artificial Intelligence , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  William T. Freeman,et al.  Constructing free-energy approximations and generalized belief propagation algorithms , 2005, IEEE Transactions on Information Theory.

[6]  Vibhav Gogate,et al.  On Combining Graph-based Variance Reduction schemes , 2010, AISTATS.

[7]  Rina Dechter,et al.  Cutset Sampling for Bayesian Networks , 2011, J. Artif. Intell. Res..

[8]  L. A. Goodman.  On the Exact Variance of Products , 1960 .

[9]  Mark A. Paskin.  Sample Propagation , 2003, NIPS.

[10]  Vibhav Gogate,et al.  SampleSearch: Importance sampling in presence of determinism , 2011, Artif. Intell..

[11]  Changhe Yuan,et al.  Importance sampling algorithms for Bayesian networks: Principles and performance , 2006, Math. Comput. Model..

[12]  Enrique F. Castillo,et al.  A modified simulation scheme for inference in Bayesian networks , 1996, Int. J. Approx. Reason..

[13]  Vibhav Gogate,et al.  SampleSearch: A Scheme that Searches for Consistent Samples , 2007, AISTATS.

[14]  Augustine Kong,et al.  The properties of the cross-match estimate and split sampling , 1997 .

[15]  Philippe Jégou,et al.  Hybrid backtracking bounded by tree-decomposition of constraint networks , 2003, Artif. Intell..

[16]  Vibhav Gogate,et al.  AND/OR Importance Sampling , 2008, UAI.

[17]  Rina Dechter,et al.  AND/OR search spaces for graphical models , 2007, Artif. Intell..

[18]  Vibhav Gogate,et al.  Approximate Counting by Sampling the Backtrack-free Search Space , 2007, AAAI.

[19]  Vibhav Gogate,et al.  Sampling algorithms for probabilistic graphical models with determinism , 2009 .

[20]  Craig Boutilier,et al.  Context-Specific Independence in Bayesian Networks , 1996, UAI.

[21]  Vibhav Gogate,et al.  Studies in Lower Bounding Probabilities of Evidence using the Markov Inequality , 2007, UAI.

[22]  Michael I. Jordan,et al.  Advances in Neural Information Processing Systems 30 , 1995 .

[23]  Vibhav Gogate,et al.  Approximate Inference Algorithms for Hybrid Bayesian Networks with Discrete Constraints , 2005, UAI.

[24]  Rina Dechter,et al.  And/or search strategies for combinatorial optimization in graphical models , 2008 .

[25]  Henry A. Kautz,et al.  Heuristics for Fast Exact Model Counting , 2005, SAT.

[26]  Adnan Darwiche,et al.  Recursive conditioning , 2001, Artif. Intell..

[27]  Vibhav Gogate,et al.  Join-Graph Propagation Algorithms , 2010, J. Artif. Intell. Res..

[28]  Daniel P. Miranker,et al.  On the Space-Time Trade-off in Solving Constraint Satisfaction Problems , 1995, IJCAI.

[29]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[30]  Adnan Darwiche,et al.  On probabilistic inference by weighted model counting , 2008, Artif. Intell..

[31]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[32]  Gregory F. Cooper,et al.  The Computational Complexity of Probabilistic Inference Using Bayesian Belief Networks , 1990, Artif. Intell..

[33]  Robert A. van Engelen,et al.  Arc refractor methods for adaptive importance sampling on large Bayesian networks under evidential reasoning , 2010, Int. J. Approx. Reason..

[34]  Uffe Kjaerulff.  HUGS: Combining Exact Inference and Gibbs Sampling in Junction Trees , 2011 .

[35]  Jörg Hoffmann,et al.  From Sampling to Model Counting , 2007, IJCAI.

[36]  Michael I. Jordan,et al.  Loopy Belief Propagation for Approximate Inference: An Empirical Study , 1999, UAI.

[37]  Pedro M. Domingos,et al.  Formula-Based Probabilistic Inference , 2010, UAI.

[38]  Andrew Zisserman,et al.  Advances in Neural Information Processing Systems (NIPS) , 2007 .

[39]  Jian Cheng,et al.  AIS-BN: An Adaptive Importance Sampling Algorithm for Evidential Reasoning in Large Bayesian Networks , 2000, J. Artif. Intell. Res..

[40]  Toniann Pitassi,et al.  Value Elimination: Bayesian Interence via Backtracking Search , 2002, UAI.

[41]  Steffen L. Lauritzen,et al.  Hybrid Propagation in Junction Trees , 1994, IPMU.

[42]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[43]  Henry A. Kautz,et al.  Performing Bayesian Inference by Weighted Model Counting , 2005, AAAI.

[44]  Rina Dechter,et al.  Iterative Join-Graph Propagation , 2002, UAI.

[45]  Uffe Kjærulff.  HUGS: Combining Exact Inference and Gibbs Sampling in junction Trees , 1995, UAI.

[46]  Serafín Moral,et al.  Dynamic importance sampling in Bayesian networks based on probability trees , 2005, Int. J. Approx. Reason..

[47]  Dan Geiger,et al.  Optimizing exact genetic linkage computations , 2003, RECOMB '03.

[48]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[49]  Changhe Yuan,et al.  An Importance Sampling Algorithm Based on Evidence Pre-propagation , 2002, UAI.

[50]  Adnan Darwiche,et al.  A differential approach to inference in Bayesian networks , 2000, JACM.

[51]  Rina Dechter,et al.  A general scheme for automatic generation of search heuristics from specification dependencies , 2001, Artif. Intell..

[52]  Vibhav Gogate,et al.  Approximate Solution Sampling (and Counting) on AND/OR Spaces , 2008, CP.

[53]  M. Spence,et al.  Analysis of human genetic linkage , 1986 .

[54]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .