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32
Compiling Bayesian Networks Using Variable Elimination
, 2007
"... Compiling Bayesian networks has proven an effective approach for inference that can utilize both global and local network structure. In this paper, we define a new method of compiling based on variable elimination (VE) and Algebraic Decision Diagrams (ADDs). The approach is important for the followi ..."
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Cited by 48 (8 self)
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Compiling Bayesian networks has proven an effective approach for inference that can utilize both global and local network structure. In this paper, we define a new method of compiling based on variable elimination (VE) and Algebraic Decision Diagrams (ADDs). The approach is important for the following reasons. First, it exploits local structure much more effectively than previous techniques based on VE. Second, the approach allows any of the many VE variants to compute answers to multiple queries simultaneously. Third, the approach makes a large body of research into more structured representations of factors relevant in many more circumstances than it has been previously. Finally, experimental results demonstrate that VE can exploit local structure as effectively as state–of–the–art algorithms based on conditioning on the networks considered, and can sometimes lead to much faster compilation times.
AND/OR multivalued decision diagrams (AOMDDs) for graphical models
, 2008
"... Inspired by the recently introduced framework of AND/OR search spaces for graphical models, we propose to augment MultiValued Decision Diagrams (MDD) with AND nodes, in order to capture function decomposition structure and to extend these compiled data structures to general weighted graphical model ..."
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Cited by 18 (3 self)
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Inspired by the recently introduced framework of AND/OR search spaces for graphical models, we propose to augment MultiValued Decision Diagrams (MDD) with AND nodes, in order to capture function decomposition structure and to extend these compiled data structures to general weighted graphical models (e.g., probabilistic models). We present the AND/OR MultiValued Decision Diagram (AOMDD) which compiles a graphical model into a canonical form that supports polynomial (e.g., solution counting, belief updating) or constant time (e.g. equivalence of graphical models) queries. We provide two algorithms for compiling the AOMDD of a graphical model. The first is searchbased, and works by applying reduction rules to the trace of the memory intensive AND/OR search algorithm. The second is inferencebased and uses a Bucket Elimination schedule to combine the AOMDDs of the input functions via the the APPLY operator. For both algorithms, the compilation time and the size of the AOMDD are, in the worst case, exponential in the treewidth of the graphical model, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs). We introduce the concept of semantic treewidth, which helps explain why the size of a decision diagram is often much smaller than the worst case bound. We provide an experimental evaluation that demonstrates the potential of AOMDDs.
Efficient Solutions to Factored MDPs with Imprecise Transition Probabilities
"... When modeling realworld decisiontheoretic planning problems in the Markov decision process (MDP) framework, it is often impossible to obtain a completely accurate estimate of transition probabilities. For example, natural uncertainty arises in the transition specification due to elicitation of MDP ..."
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Cited by 12 (1 self)
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When modeling realworld decisiontheoretic planning problems in the Markov decision process (MDP) framework, it is often impossible to obtain a completely accurate estimate of transition probabilities. For example, natural uncertainty arises in the transition specification due to elicitation of MDP transition models from an expert or data, or nonstationary transition distributions arising from insufficient state knowledge. In the interest of obtaining the most robust policy under transition uncertainty, the Markov Decision Process with Imprecise Transition Probabilities (MDPIPs) has been introduced to model such scenarios. Unfortunately, while solutions to the MDPIP are wellknown, they require nonlinear optimization and are extremely timeconsuming in practice. To address this deficiency, we propose efficient dynamic programming methods to exploit the structure of factored MDPIPs. Noting that the key computational bottleneck in the solution of MDPIPs is the need to repeatedly solve nonlinear constrained optimization problems, we show how to target approximation techniques to drastically reduce the computational overhead of the nonlinear solver while producing bounded, approximately optimal solutions. Our results show up to two orders of magnitude speedup in comparison to traditional “flat ” dynamic programming approaches and up to an order of magnitude speedup over the extension of factored MDP approximate value iteration techniques to MDPIPs.
Firstorder decisiontheoretic planning in structured relational environments
, 2008
"... We consider the general framework of firstorder decisiontheoretic planning in structured relational environments. Most traditional solution approaches to these planning problems ground the relational specification w.r.t. a specific domain instantiation and apply a solution approach directly to the ..."
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Cited by 10 (2 self)
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We consider the general framework of firstorder decisiontheoretic planning in structured relational environments. Most traditional solution approaches to these planning problems ground the relational specification w.r.t. a specific domain instantiation and apply a solution approach directly to the resulting ground Markov decision process (MDP). Unfortunately, the space and time complexity of these solution algorithms scale linearly with the domain size in the best case and exponentially in the worst case. An alternate approach to grounding a relational planning problem is to lift it to a firstorder MDP (FOMDP) specification. This FOMDP can then be solved directly, resulting in a domainindependent solution whose space and time complexity either do not scale with domain size or can scale sublinearly in the domain size. However, such generality does not come without its own set of challenges and the first purpose of this thesis is to explore exact and approximate solution techniques for practically solving FOMDPs. The second purpose of this thesis is to extend the FOMDP specification to succinctly capture factored actions and additive rewards while extending the exact and approximate solution techniques to directly exploit this structure. In addition, we provide a proof of correctness of the firstorder symbolic dynamic programming approach w.r.t. its wellstudied ground MDP
Compiling Bayesian networks by symbolic probability calculation based on zerosuppressed BDDs
 In Proceedings of the Twentieth International Joint Conference on Artificial Intelligence
, 2007
"... Compiling Bayesian networks (BNs) is one of the hot topics in the area of probabilistic modeling and processing. In this paper, we propose a new method of compiling BNs into multilinear functions (MLFs) based on Zerosuppressed BDDs (ZBDDs), which are the graphbased representation of combinatorial ..."
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Cited by 8 (3 self)
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Compiling Bayesian networks (BNs) is one of the hot topics in the area of probabilistic modeling and processing. In this paper, we propose a new method of compiling BNs into multilinear functions (MLFs) based on Zerosuppressed BDDs (ZBDDs), which are the graphbased representation of combinatorial item sets. Our method is different from the original approach of Darwiche et. al which encodes BNs into Conjunctive Normal Forms (CNFs) and then translates CNFs into factored MLFs. Our approach directly translates a BN into a set of factored MLFs using ZBDDbased symbolic probability calculation. The MLF may have an exponential size, but our ZBDDbased data structure provides a compact factored form of the MLF, and arithmetic operations can be executed in a time almost linear to the ZBDD size. Our method is not necessary to generate the MLF for the whole network, but we can extract MLFs for only a part of network related to the query, to avoid unnecessary calculation of redundant terms of MLFs. We show experimental results for some typical benchmark examples. Although our algorithm is simply based on the mathematical definition of probability calculation, the performance is competitive to the existing stateoftheart method. 1
Solving #SAT and Bayesian inference with backtracking search
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2009
"... Inference in Bayes Nets (BAYES) is an important problem with numerous applications in probabilistic reasoning. Counting the number of satisfying assignments of a propositional formula (#SAT) is a closely related problem of fundamental theoretical importance. Both these problems, and others, are memb ..."
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Cited by 8 (1 self)
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Inference in Bayes Nets (BAYES) is an important problem with numerous applications in probabilistic reasoning. Counting the number of satisfying assignments of a propositional formula (#SAT) is a closely related problem of fundamental theoretical importance. Both these problems, and others, are members of the class of sumofproducts (SUMPROD) problems. In this paper we show that standard backtracking search when augmented with a simple memoization scheme (caching) can solve any sumofproducts problem with time complexity that is at least as good any other stateoftheart exact algorithm, and that it can also achieve the best known timespace tradeoff. Furthermore, backtrackingâs ability to utilize more flexible variable orderings allows us to prove that it can achieve an exponential speedup over other standard algorithms for SUMPROD on some instances. The ideas presented here have been utilized in a number of solvers that have been applied to various types of sumofproduct problems. These systemâs have exploited the fact that backtracking can naturally exploit more of the problemâs structure to achieve improved performance on a range of problem instances. Empirical evidence of this performance gain has appeared in published works describing these solvers, and we provide references to these works.
On Valued Negation Normal Form Formulas ∗
"... Subsets of the Negation Normal Form formulas (NNFs) of propositional logic have received much attention in AI and proved as valuable representation languages for Boolean functions. In this paper, we present a new framework, called VNNF, for the representation of a much more general class of function ..."
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Cited by 7 (1 self)
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Subsets of the Negation Normal Form formulas (NNFs) of propositional logic have received much attention in AI and proved as valuable representation languages for Boolean functions. In this paper, we present a new framework, called VNNF, for the representation of a much more general class of functions than just Boolean ones. This framework supports a larger family of queries and transformations than in the NNF case, including optimization ones. As such, it encompasses a number of existing settings, e.g. NNFs, semiring CSPs, mixed CSPs, SLDDs, ADD, AADDs. We show how the properties imposed on NNFs to define more “tractable ” fragments (decomposability, determinism, decision, readonce) can be extended to VNNFs, giving rise to subsets for which a number of queries and transformations can be achieved in polynomial time. 1
Interactive Cost Configuration Over Decision Diagrams
 Journal of Artificial Intelligence Research (JAIR
"... Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrackf ..."
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Cited by 6 (1 self)
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Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrackfree user interaction online. In particular, binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDDbased configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multivalued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is nonadditive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiplecost configuration is NPhard in the input MDD. However, for solving twocost configuration we develop a pseudopolynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over realworld configuration models and productcatalogue datasets. Response times are generally within a fraction of a second even for very large instances.
Probabilistic sentential decision diagrams.
 In Proceedings of the 14th International Conference on Principles of Knowledge Representation and Reasoning (KR),
, 2014
"... Abstract We propose the Probabilistic Sentential Decision Diagram (PSDD): A complete and canonical representation of probability distributions defined over the models of a given propositional theory. 1 Each parameter of a PSDD can be viewed as the (conditional) probability of making a decision in a ..."
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Cited by 5 (3 self)
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Abstract We propose the Probabilistic Sentential Decision Diagram (PSDD): A complete and canonical representation of probability distributions defined over the models of a given propositional theory. 1 Each parameter of a PSDD can be viewed as the (conditional) probability of making a decision in a corresponding Sentential Decision Diagram (SDD). The SDD itself is a recently proposed complete and canonical representation of propositional theories. PSDDs are tractable representations, and further, the parameters of a PSDD can be efficiently estimated, in closed form, from complete data. We empirically evaluate the quality of PSDDs learned from data, when we have knowledge, a priori, of the domain logical constraints.