MAP estimation, linear programming and belief propagation with convex free energies

Yair Weiss*, Yanover Chen, Talya Meltzer

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

86 Scopus citations

Abstract

Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the MAP solution if the beliefs have no ties. In this paper we extend the setting under which BP can be used to provably extract the MAP. We define Convex BP as BP algorithms based on a convex free energy approximation and show that this class includes ordinary BP with single-cycle, tree reweighted BP and many other BP variants. We show that when there are no ties, fixed-points of convex max-product BP will provably give the MAP solution. We also show that convex sum-product BP at sufficiently small temperatures can be used to solve linear programs that arise from relaxing the MAP problem. Finally, we derive a novel condition that allows us to derive the MAP solution even if some of the convex BP beliefs have ties. In experiments, we show that our theorems allow us to find the MAP in many real-world instances of graphical models where exact inference using junction-tree is impossible.

Original languageAmerican English
Title of host publicationProceedings of the 23rd Conference on Uncertainty in Artificial Intelligence, UAI 2007
Pages416-425
Number of pages10
StatePublished - 2007
Event23rd Conference on Uncertainty in Artificial Intelligence, UAI 2007 - Vancouver, BC, Canada
Duration: 19 Jul 200722 Jul 2007

Publication series

NameProceedings of the 23rd Conference on Uncertainty in Artificial Intelligence, UAI 2007

Conference

Conference23rd Conference on Uncertainty in Artificial Intelligence, UAI 2007
Country/TerritoryCanada
CityVancouver, BC
Period19/07/0722/07/07

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