DOCSLIB.ORG

Belief Propagation in [.1Cm] Bayesian Networks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Belief Propagation in [.1Cm] Bayesian Networks Belief Propagation in Bayesian Networks Céline Comte Reading Group “Network Theory” November 5, 2018 Introduction (First-order) logic Represent causal relations between variables by a directed acyclic graph Probabilities Weight these causal relations by probabilities that implicitly account for non-represented variables 2/40 © 2018 Nokia Public Introduction “Belief networks are directed acyclic graphs in which the nodes represent propositions (or variables), the arcs signify direct dependencies between the linked propositions, and the strengths of these de- pendencies are quantified by conditional probabilities” (Pearl, 1986) 2/40 © 2018 Nokia Public Bayesian networks are also … ² A memory-efficient way of storing a PMF ² Based on simple probability rules (more details in a few slides) ² Inspired by human causal reasoning (Pearl, 1986, 1988) ² Used for decision taking if a utility function is provided ² Applied in many fields: medecine diagnoses, turbo-codes, (programming) language detection, … ² Related to other models: Markov random fields, Markov chains, hidden Markov models, … 3/40 © 2018 Nokia Public References: Pearl’s articles and book ² J. Pearl (1982). “Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach”. In: AAAI’82 ! Belief propagation in causal trees ² J. Pearl (1986). “Fusion, propagation, and structuring in belief networks”. In: Artificial Intelligence ! Belief propagation in causal trees and polytrees ² J. Pearl (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann ! A complete reference (thanks Achille for providing me with this book) 4/40 © 2018 Nokia Public References: Textbooks ² T. D. Nielsen and F. V. Jensen (2007). Bayesian Networks and Decision Graphs. Springer-Verlag ! A lot of examples in Chapters 2 and 3 ² D. Koller, N. Friedman, and F. Bach (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press ² M. Jordan (Last modified in 2015). An Introduction to Probabilistic Graphical Models. ! Definition and belief probagation (thanks Nathan for pointing this reference) 5/40 © 2018 Nokia Public Outline Reminders on probability theory Bayesian networks Belief propagation in trees Belief propagation in polytrees 6/40 © 2018 Nokia Public Outline Reminders on probability theory Bayesian networks Belief propagation in trees Belief propagation in polytrees 7/40 © 2018 Nokia Public ² A and B are marginally independent (written A ?? B) if one of these three equivalent conditions is satisfied: ¡ P(A,B) Æ P(A)P(B) ¡ P(A j B) Æ P(A) ¡ P(B j A) Æ P(B) ² A and B are conditonally independent given C (written A ?? B j C) if one of these three equivalent conditions is satisfied: ¡ P(A,B j C) Æ P(A j C)P(B j C) ¡ P(A j B,C) Æ P(A j C) ¡ P(B j A,C) Æ P(B j C) Independence and conditional independence Remark: We work exclusively with discrete random variables 8/40 © 2018 Nokia Public ² A and B are conditonally independent given C (written A ?? B j C) if one of these three equivalent conditions is satisfied: ¡ P(A,B j C) Æ P(A j C)P(B j C) ¡ P(A j B,C) Æ P(A j C) ¡ P(B j A,C) Æ P(B j C) Independence and conditional independence Remark: We work exclusively with discrete random variables ² A and B are marginally independent (written A ?? B) if one of these three equivalent conditions is satisfied: ¡ P(A,B) Æ P(A)P(B) ¡ P(A j B) Æ P(A) ¡ P(B j A) Æ P(B) 8/40 © 2018 Nokia Public Independence and conditional independence Remark: We work exclusively with discrete random variables ² A and B are marginally independent (written A ?? B) if one of these three equivalent conditions is satisfied: ¡ P(A,B) Æ P(A)P(B) ¡ P(A j B) Æ P(A) ¡ P(B j A) Æ P(B) ² A and B are conditonally independent given C (written A ?? B j C) if one of these three equivalent conditions is satisfied: ¡ P(A,B j C) Æ P(A j C)P(B j C) ¡ P(A j B,C) Æ P(A j C) ¡ P(B j A,C) Æ P(B j C) 8/40 © 2018 Nokia Public ² Law of total probability If A and B are two random variables, X P(B) Æ P(B j A)P(A) A Useful rules ² The chain rule of probabilities If A1,...,An are random variables, we have P(A1,...,An) Æ P(A1) £ P(A2 j A1) £ P(A3 j A1,A2) £ ¢¢¢ £ P(An j A1,...,An¡1) 9/40 © 2018 Nokia Public Useful rules ² The chain rule of probabilities If A1,...,An are random variables, we have P(A1,...,An) Æ P(A1) £ P(A2 j A1) £ P(A3 j A1,A2) £ ¢¢¢ £ P(An j A1,...,An¡1) ² Law of total probability If A and B are two random variables, X P(B) Æ P(B j A)P(A) A 9/40 © 2018 Nokia Public Useful rules ² Bayes’ rule If A and B are two random variables, P(A j B)P(B) P(B j A) Æ P(A) We can see P(A) as a normalizing constant: we can first compute P(B j A) / P(A j B)P(B) for each value of B and then normalize to obtain P(B j A) without computing P(A) 10/40 © 2018 Nokia Public ² Observe a random variable ² Evidence (piece of evidence) The set of random variables that have been observed Glossary ² Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables) 11/40 © 2018 Nokia Public ² Evidence (piece of evidence) The set of random variables that have been observed Glossary ² Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables) ² Observe a random variable 11/40 © 2018 Nokia Public Glossary ² Belief in a random variable (conviction in french) Marginal distribution of this random variable (given the value of some observed variables) ² Observe a random variable ² Evidence (piece of evidence) The set of random variables that have been observed 11/40 © 2018 Nokia Public Outline Reminders on probability theory Bayesian networks Belief propagation in trees Belief propagation in polytrees 12/40 © 2018 Nokia Public D I G B L The Student example Course Student difficulty intelligence Grade Baccalauréat Reference letter Borrowed from (Koller, Friedman, and Bach, 2009) 13/40 © 2018 Nokia Public The Student example D I G B Course Student difficulty intelligence L Grade Baccalauréat Reference letter Borrowed from (Koller, Friedman, and Bach, 2009) 13/40 © 2018 Nokia Public ² Chain rule of Bayesian networks By the chain rule of probabilities: P(D,I,G,B,L) Æ P(D)P(I j D)P(G j D,I)P(B j D,I,G)P(L j D,I,G,B), Æ P(D)P(I)P(G j D,I)P(B j I)P(L j G) These two definitions are equivalent The Student example D I ² Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L D ?? {I,B}, I ?? D, G ?? B j {D,I}, B ?? {D,G,L} j I, L ?? {D,I,B} j G 14/40 © 2018 Nokia Public These two definitions are equivalent The Student example D I ² Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L D ?? {I,B}, I ?? D, G ?? B j {D,I}, B ?? {D,G,L} j I, L ?? {D,I,B} j G ² Chain rule of Bayesian networks By the chain rule of probabilities: P(D,I,G,B,L) Æ P(D)P(I j D)P(G j D,I)P(B j D,I,G)P(L j D,I,G,B), Æ P(D)P(I)P(G j D,I)P(B j I)P(L j G) 14/40 © 2018 Nokia Public The Student example D I ² Local Markov property G B Each node is conditionally independent of its non-descendants given its parents: L D ?? {I,B}, I ?? D, G ?? B j {D,I}, B ?? {D,G,L} j I, L ?? {D,I,B} j G ² Chain rule of Bayesian networks By the chain rule of probabilities: P(D,I,G,B,L) Æ P(D)P(I j D)P(G j D,I)P(B j D,I,G)P(L j D,I,G,B), Æ P(D)P(I)P(G j D,I)P(B j I)P(L j G) These two definitions are equivalent 14/40 © 2018 Nokia Public The Student example D I G B Course Student L P(D) P(I) difficulty intelligence P(G j D,I) Grade Baccalauréat P(B j I) P(L j G) Reference letter 15/40 © 2018 Nokia Public Bayesian networks in general D I G B Described by L ² A directed acyclic graph ¡ Nodes » (discrete) random variables X1,...,Xn ¡ Arrows » conditional (in)dependencies ² Local conditional probability tables (CPT) ¡ P(Xi j parents(Xi)) for each node Xi 16/40 © 2018 Nokia Public Bayesian networks in general D I G B Two equivalent definitions L ² Local Markov property Each node is conditionally independent of its non-descendants given its parents ² Chain rule of Bayesian networks Yn P(X1,...,Xn) Æ P(Xi j parents(Xi)) iÆ1 Proof of the equivalence: Corollary 4 p.20 of (Pearl, 1988) 17/40 © 2018 Nokia Public ² Interpretation: chain of causality X “causes” Y that “causes” Z y // ² Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Flow of information unless Y is observed ² Example: Markov chains Base case: serial connection X ?? Z j YP(X,Y,Z) Æ P(X)P(Y j X)P(Z j Y) X Y Z 18/40 © 2018 Nokia Public y // ² Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Flow of information unless Y is observed ² Example: Markov chains Base case: serial connection X ?? Z j YP(X,Y,Z) Æ P(X)P(Y j X)P(Z j Y) X ² Interpretation: chain of causality X “causes” Y that “causes” Z Y Z 18/40 © 2018 Nokia Public y // Flow of information ² Example: Markov chains Base case: serial connection X ?? Z j YP(X,Y,Z) Æ P(X)P(Y j X)P(Z j Y) X ² Interpretation: chain of causality X “causes” Y that “causes” Z Y ² Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Z unless Y is observed 18/40 © 2018 Nokia Public y // ² Example: Markov chains Base case: serial connection X ?? Z j YP(X,Y,Z) Æ P(X)P(Y j X)P(Z j Y) X ² Interpretation: chain of causality X “causes” Y that “causes” Z Y ² Information can flow between X and Z through Y (that is, observing X changes our belief in Z and vice versa), Z Flow of
Load more

Recommended publications
  • Inference Algorithm for Finite-Dimensional Spin
    PHYSICAL REVIEW E 84, 046706 (2011) Inference algorithm for finite-dimensional spin glasses: Belief propagation on the dual lattice Alejandro Lage-Castellanos and Roberto Mulet Department of Theoretical Physics and Henri-Poincare´ Group of Complex Systems, Physics Faculty, University of Havana, La Habana, Codigo Postal 10400, Cuba Federico Ricci-Tersenghi Dipartimento di Fisica, INFN–Sezione di Roma 1, and CNR–IPCF, UOS di Roma, Universita` La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy Tommaso Rizzo Dipartimento di Fisica and CNR–IPCF, UOS di Roma, Universita` La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy (Received 18 February 2011; revised manuscript received 15 September 2011; published 24 October 2011) Starting from a cluster variational method, and inspired by the correctness of the paramagnetic ansatz [at high temperatures in general, and at any temperature in the two-dimensional (2D) Edwards-Anderson (EA) model] we propose a message-passing algorithm—the dual algorithm—to estimate the marginal probabilities of spin glasses on finite-dimensional lattices. We use the EA models in 2D and 3D as benchmarks. The dual algorithm improves the Bethe approximation, and we show that in a wide range of temperatures (compared to the Bethe critical temperature) our algorithm compares very well with Monte Carlo simulations, with the double-loop algorithm, and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreover, it is usually 100 times faster than other provably convergent methods, as the double-loop algorithm. In 2D and 3D the quality of the inference deteriorates only where the correlation length becomes very large, i.e., at low temperatures in 2D and close to the critical temperature in 3D.
    [Show full text]
  • Many-Pairs Mutual Information for Adding Structure to Belief Propagation Approximations
    Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) Many-Pairs Mutual Information for Adding Structure to Belief Propagation Approximations Arthur Choi and Adnan Darwiche Computer Science Department University of California, Los Angeles Los Angeles, CA 90095 {aychoi,darwiche}@cs.ucla.edu Abstract instances of the satisfiability problem (Braunstein, Mezard,´ and Zecchina 2005). IBP has also been applied towards a We consider the problem of computing mutual informa- variety of computer vision tasks (Szeliski et al. 2006), par- tion between many pairs of variables in a Bayesian net- work. This task is relevant to a new class of Generalized ticularly stereo vision, where methods based on belief prop- Belief Propagation (GBP) algorithms that characterizes agation have been particularly successful. Iterative Belief Propagation (IBP) as a polytree approx- The successes of IBP as an approximate inference algo- imation found by deleting edges in a Bayesian network. rithm spawned a number of generalizations, in the hopes of By computing, in the simplified network, the mutual in- understanding the nature of IBP, and further to improve on formation between variables across a deleted edge, we the quality of its approximations. Among the most notable can estimate the impact that recovering the edge might are the family of Generalized Belief Propagation (GBP) al- have on the approximation. Unfortunately, it is compu- gorithms (Yedidia, Freeman, and Weiss 2005), where the tationally impractical to compute such scores for net- works over many variables having large state spaces. “structure” of an approximation is given by an auxiliary So that edge recovery can scale to such networks, we graph called a region graph, which specifies (roughly) which propose in this paper an approximation of mutual in- regions of the original network should be handled exactly, formation which is based on a soft extension of d- and to what extent different regions should be mutually con- separation (a graphical test of independence in Bayesian sistent.
    [Show full text]
  • Variational Message Passing
    Journal of Machine Learning Research 6 (2005) 661–694 Submitted 2/04; Revised 11/04; Published 4/05 Variational Message Passing John Winn [email protected] Christopher M. Bishop [email protected] Microsoft Research Cambridge Roger Needham Building 7 J. J. Thomson Avenue Cambridge CB3 0FB, U.K. Editor: Tommi Jaakkola Abstract Bayesian inference is now widely established as one of the principal foundations for machine learn- ing. In practice, exact inference is rarely possible, and so a variety of approximation techniques have been developed, one of the most widely used being a deterministic framework called varia- tional inference. In this paper we introduce Variational Message Passing (VMP), a general purpose algorithm for applying variational inference to Bayesian Networks. Like belief propagation, VMP proceeds by sending messages between nodes in the network and updating posterior beliefs us- ing local operations at each node. Each such update increases a lower bound on the log evidence (unless already at a local maximum). In contrast to belief propagation, VMP can be applied to a very general class of conjugate-exponential models because it uses a factorised variational approx- imation. Furthermore, by introducing additional variational parameters, VMP can be applied to models containing non-conjugate distributions. The VMP framework also allows the lower bound to be evaluated, and this can be used both for model comparison and for detection of convergence. Variational message passing has been implemented in the form of a general purpose inference en- gine called VIBES (‘Variational Inference for BayEsian networkS’) which allows models to be specified graphically and then solved variationally without recourse to coding.
    [Show full text]
  • Linearized and Single-Pass Belief Propagation
    Linearized and Single-Pass Belief Propagation Wolfgang Gatterbauer Stephan Gunnemann¨ Danai Koutra Christos Faloutsos Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University [email protected] [email protected] [email protected] [email protected] ABSTRACT DR TS HAF D 0.8 0.2 T 0.3 0.7 H 0.6 0.3 0.1 How can we tell when accounts are fake or real in a social R 0.2 0.8 S 0.7 0.3 A 0.3 0.0 0.7 network? And how can we tell which accounts belong to F 0.1 0.7 0.2 liberal, conservative or centrist users? Often, we can answer (a) homophily (b) heterophily (c) general case such questions and label nodes in a network based on the labels of their neighbors and appropriate assumptions of ho- Figure 1: Three types of network effects with example coupling mophily (\birds of a feather flock together") or heterophily matrices. Shading intensity corresponds to the affinities or cou- (\opposites attract"). One of the most widely used methods pling strengths between classes of neighboring nodes. (a): D: for this kind of inference is Belief Propagation (BP) which Democrats, R: Republicans. (b): T: Talkative, S: Silent. (c): H: iteratively propagates the information from a few nodes with Honest, A: Accomplice, F: Fraudster. explicit labels throughout a network until convergence. A well-known problem with BP, however, is that there are no or heterophily applies in a given scenario, we can usually give known exact guarantees of convergence in graphs with loops.
    [Show full text]
  • Α Belief Propagation for Approximate Inference
    α Belief Propagation for Approximate Inference Dong Liu, Minh Thanh` Vu, Zuxing Li, and Lars K. Rasmussen KTH Royal Institute of Technology, Stockholm, Sweden. e-mail: fdoli, mtvu, zuxing, [email protected] Abstract Belief propagation (BP) algorithm is a widely used message-passing method for inference in graphical models. BP on loop-free graphs converges in linear time. But for graphs with loops, BP’s performance is uncertain, and the understanding of its solution is limited. To gain a better understanding of BP in general graphs, we derive an interpretable belief propagation algorithm that is motivated by minimization of a localized α-divergence. We term this algorithm as α belief propagation (α-BP). It turns out that α-BP generalizes standard BP. In addition, this work studies the convergence properties of α-BP. We prove and offer the convergence conditions for α-BP. Experimental simulations on random graphs validate our theoretical results. The application of α-BP to practical problems is also demonstrated. I. INTRODUCTION Bayesian inference provides a general mathematical framework for many learning tasks such as classification, denoising, object detection, and signal detection. The wide applications include but not limited to imaging processing [34], multi-input-multi-output (MIMO) signal detection in digital communication [2], [10], inference on structured lattice [5], machine learning [19], [14], [33]. Generally speaking, a core problem to these applications is the inference on statistical properties of a (hidden) variable x = (x1; : : : ; xN ) that usually can not be observed. Specifically, practical interests usually include computing the most probable state x given joint probability p(x), or marginal probability p(xc), where xc is subset of x.
    [Show full text]
  • Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: a Survey Chaohui Wang, Nikos Komodakis, Nikos Paragios
    Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey Chaohui Wang, Nikos Komodakis, Nikos Paragios To cite this version: Chaohui Wang, Nikos Komodakis, Nikos Paragios. Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey. Computer Vision and Image Under- standing, Elsevier, 2013, 117 (11), pp.Page 1610-1627. 10.1016/j.cviu.2013.07.004. hal-00858390v1 HAL Id: hal-00858390 https://hal.archives-ouvertes.fr/hal-00858390v1 Submitted on 5 Sep 2013 (v1), last revised 6 Sep 2013 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey Chaohui Wanga,b, Nikos Komodakisc, Nikos Paragiosa,d aCenter for Visual Computing, Ecole Centrale Paris, Grande Voie des Vignes, Chˆatenay-Malabry, France bPerceiving Systems Department, Max Planck Institute for Intelligent Systems, T¨ubingen, Germany cLIGM laboratory, University Paris-East & Ecole des Ponts Paris-Tech, Marne-la-Vall´ee, France dGALEN Group, INRIA Saclay - Ileˆ de France, Orsay, France Abstract In this paper, we present a comprehensive survey of Markov Random Fields (MRFs) in computer vision and image understanding, with respect to the modeling, the inference and the learning.
    [Show full text]
  • Belief Propagation for the (Physicist) Layman
    Belief Propagation for the (physicist) layman Florent Krzakala ESPCI Paristech Laboratoire PCT, UMR Gulliver CNRS-ESPCI 7083, 10 rue Vauquelin, 75231 Paris, France [email protected] http: // www. pct. espci. fr/ ∼florent/ (Dated: January 30, 2011) These lecture notes have been prepared for a series of course in a master in Lyon (France), and other teaching in Beijing (China) and Tokyo (Japan). It consists in a short introduction to message passing algorithms and in particular Belief propagation, using a statistical physics formulation. It is intended as a first introduction to such ideas which start to be now widely used in both physics, constraint optimization, Bayesian inference and quantitative biology. The reader interested to pursur her journey beyond these notes is refer to [1]. I. WHY STATISTICAL PHYSICS ? Why should we speak about statistical physics in this lecture about complex systems? A. Physics First of all, because of physics itself. Statistical physics is a fondamental tool and one of the most formidable success of modern physics. Here, we shall however use it only as a probabilist toolbox for other problems. It is perhaps better to do a recap before we start. In statistical physics, we consider N discrete (not always, actually, but for the matter let us assume discrete) variables σi and a cost function H({σ}) which we call the Hamiltonian. We introduce the temperature T (and the inverse temperature β = 1/T and the idea is that each configurations, or assignments, of the variable ({σ}) has a weight e−βH({σ}), which is called the
    [Show full text]
  • Efficient Belief Propagation for Higher Order Cliques Using Linear
    * 2. Manuscript Efficient Belief Propagation for Higher Order Cliques Using Linear Constraint Nodes Brian Potetz ∗, Tai Sing Lee Department of Computer Science & Center for the Neural Basis of Cognition Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213 Abstract Belief propagation over pairwise connected Markov Random Fields has become a widely used approach, and has been successfully applied to several important computer vision problems. However, pairwise interactions are often insufficient to capture the full statistics of the problem. Higher-order interactions are sometimes required. Unfortunately, the complexity of belief propagation is exponential in the size of the largest clique. In this paper, we introduce a new technique to compute belief propagation messages in time linear with respect to clique size for a large class of potential functions over real-valued variables. We discuss how this technique can be generalized to still wider classes of potential functions at varying levels of efficiency. Also, we develop a form of nonparametric belief representation specifically designed to address issues common to networks with higher-order cliques and also to the use of guaranteed-convergent forms of belief propagation. To illustrate these techniques, we perform efficient inference in graphical models where the spatial prior of natural images is captured by 2 2 cliques. This approach × shows significant improvement over the commonly used pairwise-connected models, and may benefit a variety of applications using belief propagation to infer images or range images, including stereo, shape-from-shading, image-based rendering, seg- mentation, and matting. Key words: Belief propagation, Higher order cliques, Non-pairwise cliques, Factor graphs, Continuous Markov Random Fields PACS: ∗ Corresponding author.
    [Show full text]
  • 13 : Variational Inference: Loopy Belief Propagation and Mean Field
    10-708: Probabilistic Graphical Models 10-708, Spring 2012 13 : Variational Inference: Loopy Belief Propagation and Mean Field Lecturer: Eric P. Xing Scribes: Peter Schulam and William Wang 1 Introduction Inference problems involve answering a query that concerns the likelihood of observed data. For example, to P answer the query on a marginal p(xA), we can perform the marginalization operation to derive C=A p(x). Or, for queries concern the conditionals, such as p(xAjxB), we can first compute the joint, and divide by the marginals p(xB). Sometimes to answer a query, we might also need to compute the mode of density x^ = arg maxx2X m p(x). So far, in the class, we have covered the exact inference problem. To perform exact inference, we know that brute force search might be too inefficient for large graphs with complex structures, so a family of message passing algorithm such as forward-backward, sum-product, max-product, and junction tree, was introduced. However, although we know that these message-passing based exact inference algorithms work well for tree- structured graphical models, it was also shown in the class that they might not yield consistent results for loopy graphs, or, the convergence might not be guaranteed. Also, for complex graphical models such as the Ising model, we cannot run exact inference algorithm such as the junction tree algorithm, because it is computationally intractable. In this lecture, we look at two variational inference algorithms: loopy belief propagation (yww) and mean field approximation (pschulam). 2 Loopy Belief Propagation The general idea for loopy belief propagation is that even though we know graph contains loops and the messages might circulate indefinitely, we still let it run anyway and hope for the best.
    [Show full text]
  • Characterization of Belief Propagation and Its Generalizations
    MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Characterization of belief propagation and its generalizations Jonathan S. Yedidia, William T. Freeman, Yair Weiss TR2001-15 March 2001 Abstract Graphical models are used in many scientific disciplines, including statistical physics, machine learning, and error-correcting coding. One typically seeks the marginal probability of selected variables (nodes of the graph) given observations at other variables. Belief propagation (BP) is a fast marginalization method, based on passing local messages. Designed for singly-connected graphs, BP nonetheless works well in many applications involving graphs with loops, for rea- sons that were not well understood. We characterize the BP solutions, showing that BP can only converge to a stationary point of an approximate free energy, known as the Bethe free energy in statistical physics. This understanding lets us for construct new message-passing algorithms based on improvements to Bethe’s approximation introduced by Kikuchi and others. The new generalized belief propagation (GBP) algorithms are much more accurate than ordinary BP for some problems, and permit solutions to Kikuchi approximations for otherwise intractable inho- mogeneous systems. We illustrate GBP with a spin-glass example and an error-correcting code, showing dramatically improved estimates of local magnetizations and decoding performance using GBP. This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice.
    [Show full text]
  • Particle Belief Propagation
    Particle Belief Propagation Alexander Ihler David McAllester [email protected] [email protected] Dept. of Computer Science Toyota Technological Institite, Chicago University of California, Irvine Abstract passing” algorithms such as belief propagation (Pearl, 1988). Traditionally, most such work has focused on systems of many variables, each of which has a rela- The popularity of particle filtering for infer- tively small state space (number of possible values), ence in Markov chain models defined over or particularly nice parametric forms (such as jointly random variables with very large or contin- Gaussian distributions). For systems with continuous- uous domains makes it natural to consider valued variables, or discrete-valued variables with very sample–based versions of belief propagation large domains, one possibility is to reduce the effec- (BP) for more general (tree–structured or tive state space through gating, or discarding low- loopy) graphs. Already, several such al- probability states (Freeman et al., 2000; Coughlan and gorithms have been proposed in the litera- Ferreira, 2002), or through random sampling (Arulam- ture. However, many questions remain open palam et al., 2002; Koller et al., 1999; Sudderth et al., about the behavior of particle–based BP al- 2003; Isard, 2003; Neal et al., 2003). The best-known gorithms, and little theory has been devel- example of the latter technique is particle filtering, de- oped to analyze their performance. In this fined on Markov chains, in which each distribution is paper, we describe a generic particle belief represented using a finite collection of samples, or par- propagation (PBP) algorithm which is closely ticles. It is therefore only natural to consider general- related to previously proposed methods.
    [Show full text]
  • Belief Propagation Neural Networks
    Belief Propagation Neural Networks Jonathan Kuck1, Shuvam Chakraborty1, Hao Tang2, Rachel Luo1, Jiaming Song1, Ashish Sabharwal3, and Stefano Ermon1 1Stanford University 2Shanghai Jiao Tong University 3Allen Institute for Artificial Intelligence {kuck,shuvamc,rsluo,tsong,ermon}@stanford.edu, [email protected], [email protected] Abstract Learned neural solvers have successfully been used to solve combinatorial opti- mization and decision problems. More general counting variants of these problems, however, are still largely solved with hand-crafted solvers. To bridge this gap, we introduce belief propagation neural networks (BPNNs), a class of parameterized operators that operate on factor graphs and generalize Belief Propagation (BP). In its strictest form, a BPNN layer (BPNN-D) is a learned iterative operator that provably maintains many of the desirable properties of BP for any choice of the parameters. Empirically, we show that by training BPNN-D learns to perform the task better than the original BP: it converges 1.7x faster on Ising models while pro- viding tighter bounds. On challenging model counting problems, BPNNs compute estimates 100’s of times faster than state-of-the-art handcrafted methods, while returning an estimate of comparable quality. 1 Introduction Probabilistic inference problems arise in many domains, from statistical physics to machine learning. There is little hope that efficient, exact solutions to these problems exist as they are at least as hard as NP-complete decision problems. Significant research has been devoted across the fields of machine learning, statistics, and statistical physics to develop variational and sampling based methods to approximate these challenging problems [13, 34, 48, 6, 38].
    [Show full text]