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Adnan Darwiche

Adnan Darwiche appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Artificial IntelligenceMachine LearningLogic in Computer ScienceComputational ComplexityMethodology

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Published work

40 published item(s)

preprint2026arXiv

Dsat: A Native SAT Solver for Discrete Logic

Discrete variables are common in many applications, such as probabilistic reasoning, planning and explainable AI. When symbolic reasoning techniques are brought in to bear on these applications, a standard technique for handling discrete variables is to binarize them into Boolean variables to allow the use of Boolean computational machinery such as SAT solvers. This technique can face both computational and semantical challenges though. In this work, we develop a native SAT solver for discrete logic, which is a direct extension of Boolean logic in which variables can take arbitrary values. Our proposed solver has a similar design to Boolean SAT solvers, with ingredients such as unit resolution and clause learning but ones that operate natively on discrete variables. We illustrate the merits of the developed SAT solver by comparing it empirically to CSP solvers applied to discrete CNFs, to Boolean SAT solver applied to binarized CNFs, and to some hybrid solvers.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Causal Inference Using Tractable Circuits

The aim of this paper is to discuss a recent result which shows that probabilistic inference in the presence of (unknown) causal mechanisms can be tractable for models that have traditionally been viewed as intractable. This result was reported recently to facilitate model-based supervised learning but it can be interpreted in a causality context as follows. One can compile a non-parametric causal graph into an arithmetic circuit that supports inference in time linear in the circuit size. The circuit is also non-parametric so it can be used to estimate parameters from data and to further reason (in linear time) about the causal graph parametrized by these estimates. Moreover, the circuit size can sometimes be bounded even when the treewidth of the causal graph is not, leading to tractable inference on models that have been deemed intractable previously. This has been enabled by a new technique that can exploit causal mechanisms computationally but without needing to know their identities (the classical setup in causal inference). Our goal is to provide a causality-oriented exposure to these new results and to speculate on how they may potentially contribute to more scalable and versatile causal inference.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On the Computation of Necessary and Sufficient Explanations

The complete reason behind a decision is a Boolean formula that characterizes why the decision was made. This recently introduced notion has a number of applications, which include generating explanations, detecting decision bias and evaluating counterfactual queries. Prime implicants of the complete reason are known as sufficient reasons for the decision and they correspond to what is known as PI explanations and abductive explanations. In this paper, we refer to the prime implicates of a complete reason as necessary reasons for the decision. We justify this terminology semantically and show that necessary reasons correspond to what is known as contrastive explanations. We also study the computation of complete reasons for multi-class decision trees and graphs with nominal and numeric features for which we derive efficient, closed-form complete reasons. We further investigate the computation of shortest necessary and sufficient reasons for a broad class of complete reasons, which include the derived closed forms and the complete reasons for Sentential Decision Diagrams (SDDs). We provide an algorithm which can enumerate their shortest necessary reasons in output polynomial time. Enumerating shortest sufficient reasons for this class of complete reasons is hard even for a single reason. For this problem, we provide an algorithm that appears to be quite efficient as we show empirically.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Tractable Boolean and Arithmetic Circuits

Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit various types of inference to be performed in linear-time and a feed-forward fashion like neural networks. In more recent years, the role of tractable circuits has significantly expanded as they became a computational and semantical backbone for some approaches that aim to integrate knowledge, reasoning and learning. In this article, we review the foundations of tractable circuits and some associated milestones, while focusing on their core properties and techniques that make them particularly useful for the broad aims of neuro-symbolic AI.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A New Perspective on Learning Context-Specific Independence

Local structure such as context-specific independence (CSI) has received much attention in the probabilistic graphical model (PGM) literature, as it facilitates the modeling of large complex systems, as well as for reasoning with them. In this paper, we provide a new perspective on how to learn CSIs from data. We propose to first learn a functional and parameterized representation of a conditional probability table (CPT), such as a neural network. Next, we quantize this continuous function, into an arithmetic circuit representation that facilitates efficient inference. In the first step, we can leverage the many powerful tools that have been developed in the machine learning literature. In the second step, we exploit more recently-developed analytic tools from explainable AI, for the purposes of learning CSIs. Finally, we contrast our approach, empirically and conceptually, with more traditional variable-splitting approaches, that search for CSIs more explicitly.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

An Advance on Variable Elimination with Applications to Tensor-Based Computation

We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference and learning efficiently on models that have very large treewidth. The highlight of the advance is that it works with standard (dense) factors, without the need for sparse factors or techniques based on knowledge compilation that are commonly utilized. This is significant as it permits a direct implementation of the improved variable elimination algorithm using tensors and their operations, leading to extremely efficient implementations especially when learning model parameters. Moreover, the proposed technique does not require knowledge of the specific functional dependencies, only that they exist, so can be used when learning these dependencies. We illustrate the efficacy of our proposed algorithm by compiling Bayesian network queries into tensor graphs and then learning their parameters from labeled data using a standard tool for tensor computation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On Symbolically Encoding the Behavior of Random Forests

Recent work has shown that the input-output behavior of some machine learning systems can be captured symbolically using Boolean expressions or tractable Boolean circuits, which facilitates reasoning about the behavior of these systems. While most of the focus has been on systems with Boolean inputs and outputs, we address systems with discrete inputs and outputs, including ones with discretized continuous variables as in systems based on decision trees. We also focus on the suitability of encodings for computing prime implicants, which have recently played a central role in explaining the decisions of machine learning systems. We show some key distinctions with encodings for satisfiability, and propose an encoding that is sound and complete for the given task.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On The Reasons Behind Decisions

Recent work has shown that some common machine learning classifiers can be compiled into Boolean circuits that have the same input-output behavior. We present a theory for unveiling the reasons behind the decisions made by Boolean classifiers and study some of its theoretical and practical implications. We define notions such as sufficient, necessary and complete reasons behind decisions, in addition to classifier and decision bias. We show how these notions can be used to evaluate counterfactual statements such as "a decision will stick even if ... because ... ." We present efficient algorithms for computing these notions, which are based on new advances on tractable Boolean circuits, and illustrate them using a case study.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On Tractable Representations of Binary Neural Networks

We consider the compilation of a binary neural network's decision function into tractable representations such as Ordered Binary Decision Diagrams (OBDDs) and Sentential Decision Diagrams (SDDs). Obtaining this function as an OBDD/SDD facilitates the explanation and formal verification of a neural network's behavior. First, we consider the task of verifying the robustness of a neural network, and show how we can compute the expected robustness of a neural network, given an OBDD/SDD representation of it. Next, we consider a more efficient approach for compiling neural networks, based on a pseudo-polynomial time algorithm for compiling a neuron. We then provide a case study in a handwritten digits dataset, highlighting how two neural networks trained from the same dataset can have very high accuracies, yet have very different levels of robustness. Finally, in experiments, we show that it is feasible to obtain compact representations of neural networks as SDDs.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Dual Decomposition from the Perspective of Relax, Compensate and then Recover

Relax, Compensate and then Recover (RCR) is a paradigm for approximate inference in probabilistic graphical models that has previously provided theoretical and practical insights on iterative belief propagation and some of its generalizations. In this paper, we characterize the technique of dual decomposition in the terms of RCR, viewing it as a specific way to compensate for relaxed equivalence constraints. Among other insights gathered from this perspective, we propose novel heuristics for recovering relaxed equivalence constraints with the goal of incrementally tightening dual decomposition approximations, all the way to reaching exact solutions. We also show empirically that recovering equivalence constraints can sometimes tighten the corresponding approximation (and obtaining exact results), without increasing much the complexity of inference.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

On the Role of Canonicity in Bottom-up Knowledge Compilation

We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called an Apply function). While such a polytime Apply function is known to exist for certain languages (e.g., OBDDs) and not exist for others (e.g., DNNF), its existence for certain languages remains unknown. Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs), for which a polytime Apply function exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical SDDs). We resolve this open question in this paper and consider some of its theoretical and practical implications. Some of the findings we report question the common wisdom on the relationship between bottom-up compilation, language canonicity and the complexity of the Apply function.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Query DAGs: A Practical Paradigm for Implementing Belief Network Inference

We describe a new paradigm for implementing inference in belief networks, which relies on compiling a belief network into an arithmetic expression called a Query DAG (Q-DAG). Each non-leaf node of a Q-DAG represents a numeric operation, a number, or a symbol for evidence. Each leaf node of a Q-DAG represents the answer to a network query, that is, the probability of some event of interest. It appears that Q-DAGs can be generated using any of the algorithms for exact inference in belief networks --- we show how they can be generated using clustering and conditioning algorithms. The time and space complexity of a Q-DAG generation algorithm is no worse than the time complexity of the inference algorithm on which it is based; that of a Q-DAG on-line evaluation algorithm is linear in the size of the Q-DAG, and such inference amounts to a standard evaluation of the arithmetic expression it represents. The main value of Q-DAGs is in reducing the software and hardware resources required to utilize belief networks in on-line, real-world applications. The proposed framework also facilitates the development of on-line inference on different software and hardware platforms, given the simplicity of the Q-DAG evaluation algorithm. This paper describes this new paradigm for probabilistic inference, explaining how it works, its uses, and outlines some of the research directions that it leads to.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Skolemization for Weighted First-Order Model Counting

First-order model counting emerged recently as a novel reasoning task, at the core of efficient algorithms for probabilistic logics. We present a Skolemization algorithm for model counting problems that eliminates existential quantifiers from a first-order logic theory without changing its weighted model count. For certain subsets of first-order logic, lifted model counters were shown to run in time polynomial in the number of objects in the domain of discourse, where propositional model counters require exponential time. However, these guarantees apply only to Skolem normal form theories (i.e., no existential quantifiers) as the presence of existential quantifiers reduces lifted model counters to propositional ones. Since textbook Skolemization is not sound for model counting, these restrictions precluded efficient model counting for directed models, such as probabilistic logic programs, which rely on existential quantification. Our Skolemization procedure extends the applicability of first-order model counters to these representations. Moreover, it simplifies the design of lifted model counting algorithms.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

When do Numbers Really Matter?

Common wisdom has it that small distinctions in the probabilities quantifying a Bayesian network do not matter much for the resultsof probabilistic queries. However, one can easily develop realistic scenarios under which small variations in network probabilities can lead to significant changes in computed queries. A pending theoretical question is then to analytically characterize parameter changes that do or do not matter. In this paper, we study the sensitivity of probabilistic queries to changes in network parameters and prove some tight bounds on the impact that such parameters can have on queries. Our analytical results pinpoint some interesting situations under which parameter changes do or do not matter. These results are important for knowledge engineers as they help them identify influential network parameters. They are also important for approximate inference algorithms that preprocessnetwork CPTs to eliminate small distinctions in probabilities.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A Differential Approach to Inference in Bayesian Networks

We present a new approach for inference in Bayesian networks, which is mainly based on partial differentiation. According to this approach, one compiles a Bayesian network into a multivariate polynomial and then computes the partial derivatives of this polynomial with respect to each variable. We show that once such derivatives are made available, one can compute in constant-time answers to a large class of probabilistic queries, which are central to classical inference, parameter estimation, model validation and sensitivity analysis. We present a number of complexity results relating to the compilation of such polynomials and to the computation of their partial derivatives. We argue that the combined simplicity, comprehensiveness and computational complexity of the presented framework is unique among existing frameworks for inference in Bayesian networks.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A Standard Approach for Optimizing Belief Network Inference using Query DAGs

This paper proposes a novel, algorithm-independent approach to optimizing belief network inference. rather than designing optimizations on an algorithm by algorithm basis, we argue that one should use an unoptimized algorithm to generate a Q-DAG, a compiled graphical representation of the belief network, and then optimize the Q-DAG and its evaluator instead. We present a set of Q-DAG optimizations that supplant optimizations designed for traditional inference algorithms, including zero compression, network pruning and caching. We show that our Q-DAG optimizations require time linear in the Q-DAG size, and significantly simplify the process of designing algorithms for optimizing belief network inference.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Action Networks: A Framework for Reasoning about Actions and Change under Uncertainty

This work proposes action networks as a semantically well-founded framework for reasoning about actions and change under uncertainty. Action networks add two primitives to probabilistic causal networks: controllable variables and persistent variables. Controllable variables allow the representation of actions as directly setting the value of specific events in the domain, subject to preconditions. Persistent variables provide a canonical model of persistence according to which both the state of a variable and the causal mechanism dictating its value persist over time unless intervened upon by an action (or its consequences). Action networks also allow different methods for quantifying the uncertainty in causal relationships, which go beyond traditional probabilistic quantification. This paper describes both recent results and work in progress.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Any-Space Probabilistic Inference

We have recently introduced an any-space algorithm for exact inference in Bayesian networks, called Recursive Conditioning, RC, which allows one to trade space with time at increments of X-bytes, where X is the number of bytes needed to cache a floating point number. In this paper, we present three key extensions of RC. First, we modify the algorithm so it applies to more general factorization of probability distributions, including (but not limited to) Bayesian network factorizations. Second, we present a forgetting mechanism which reduces the space requirements of RC considerably and then compare such requirmenets with those of variable elimination on a number of realistic networks, showing orders of magnitude improvements in certain cases. Third, we present a version of RC for computing maximum a posteriori hypotheses (MAP), which turns out to be the first MAP algorithm allowing a smooth time-space tradeoff. A key advantage of presented MAP algorithm is that it does not have to start from scratch each time a new query is presented, but can reuse some of its computations across multiple queries, leading to significant savings in ceratain cases.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Approximating MAP using Local Search

MAP is the problem of finding a most probable instantiation of a set of variables in a Bayesian network, given evidence. Unlike computing marginals, posteriors, and MPE (a special case of MAP), the time and space complexity of MAP is not only exponential in the network treewidth, but also in a larger parameter known as the "constrained" treewidth. In practice, this means that computing MAP can be orders of magnitude more expensive than computingposteriors or MPE. Thus, practitioners generally avoid MAP computations, resorting instead to approximating them by the most likely value for each MAP variableseparately, or by MPE.We present a method for approximating MAP using local search. This method has space complexity which is exponential onlyin the treewidth, as is the complexity of each search step. We investigate the effectiveness of different local searchmethods and several initialization strategies and compare them to otherapproximation schemes.Experimental results show that local search provides a much more accurate approximation of MAP, while requiring few search steps.Practically, this means that the complexity of local search is often exponential only in treewidth as opposed to the constrained treewidth, making approximating MAP as efficient as other computations.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Argument Calculus and Networks

A major reason behind the success of probability calculus is that it possesses a number of valuable tools, which are based on the notion of probabilistic independence. In this paper, I identify a notion of logical independence that makes some of these tools available to a class of propositional databases, called argument databases. Specifically, I suggest a graphical representation of argument databases, called argument networks, which resemble Bayesian networks. I also suggest an algorithm for reasoning with argument networks, which resembles a basic algorithm for reasoning with Bayesian networks. Finally, I show that argument networks have several applications: Nonmonotonic reasoning, truth maintenance, and diagnosis.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Conditioning Methods for Exact and Approximate Inference in Causal Networks

We present two algorithms for exact and approximate inference in causal networks. The first algorithm, dynamic conditioning, is a refinement of cutset conditioning that has linear complexity on some networks for which cutset conditioning is exponential. The second algorithm, B-conditioning, is an algorithm for approximate inference that allows one to trade-off the quality of approximations with the computation time. We also present some experimental results illustrating the properties of the proposed algorithms.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Dynamic Jointrees

It is well known that one can ignore parts of a belief network when computing answers to certain probabilistic queries. It is also well known that the ignorable parts (if any) depend on the specific query of interest and, therefore, may change as the query changes. Algorithms based on jointrees, however, do not seem to take computational advantage of these facts given that they typically construct jointrees for worst-case queries; that is, queries for which every part of the belief network is considered relevant. To address this limitation, we propose in this paper a method for reconfiguring jointrees dynamically as the query changes. The reconfiguration process aims at maintaining a jointree which corresponds to the underlying belief network after it has been pruned given the current query. Our reconfiguration method is marked by three characteristics: (a) it is based on a non-classical definition of jointrees; (b) it is relatively efficient; and (c) it can reuse some of the computations performed before a jointree is reconfigured. We present preliminary experimental results which demonstrate significant savings over using static jointrees when query changes are considerable.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Objection-Based Causal Networks

This paper introduces the notion of objection-based causal networks which resemble probabilistic causal networks except that they are quantified using objections. An objection is a logical sentence and denotes a condition under which a, causal dependency does not exist. Objection-based causal networks enjoy almost all the properties that make probabilistic causal networks popular, with the added advantage that objections are, arguably more intuitive than probabilities.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

On the Complexity and Approximation of Binary Evidence in Lifted Inference

Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities. The reason is that conditioning on evidence breaks many of the model's symmetries, which can preempt standard lifting techniques. Recent theoretical results show, for example, that conditioning on evidence which corresponds to binary relations is #P-hard, suggesting that no lifting is to be expected in the worst case. In this paper, we balance this negative result by identifying the Boolean rank of the evidence as a key parameter for characterizing the complexity of conditioning in lifted inference. In particular, we show that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization, which we investigate both theoretically and empirically.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

On the Relation between Kappa Calculus and Probabilistic Reasoning

We study the connection between kappa calculus and probabilistic reasoning in diagnosis applications. Specifically, we abstract a probabilistic belief network for diagnosing faults into a kappa network and compare the ordering of faults computed using both methods. We show that, at least for the example examined, the ordering of faults coincide as long as all the causal relations in the original probabilistic network are taken into account. We also provide a formal analysis of some network structures where the two methods will differ. Both kappa rankings and infinitesimal probabilities have been used extensively to study default reasoning and belief revision. But little has been done on utilizing their connection as outlined above. This is partly because the relation between kappa and probability calculi assumes that probabilities are arbitrarily close to one (or zero). The experiments in this paper investigate this relation when this assumption is not satisfied. The reported results have important implications on the use of kappa rankings to enhance the knowledge engineering of uncertainty models.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

A Variational Approach for Approximating Bayesian Networks by Edge Deletion

We consider in this paper the formulation of approximate inference in Bayesian networks as a problem of exact inference on an approximate network that results from deleting edges (to reduce treewidth). We have shown in earlier work that deleting edges calls for introducing auxiliary network parameters to compensate for lost dependencies, and proposed intuitive conditions for determining these parameters. We have also shown that our method corresponds to IBP when enough edges are deleted to yield a polytree, and corresponds to some generalizations of IBP when fewer edges are deleted. In this paper, we propose a different criteria for determining auxiliary parameters based on optimizing the KL-divergence between the original and approximate networks. We discuss the relationship between the two methods for selecting parameters, shedding new light on IBP and its generalizations. We also discuss the application of our new method to approximating inference problems which are exponential in constrained treewidth, including MAP and nonmyopic value of information.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Approximating the Partition Function by Deleting and then Correcting for Model Edges

We propose an approach for approximating the partition function which is based on two steps: (1) computing the partition function of a simplified model which is obtained by deleting model edges, and (2) rectifying the result by applying an edge-by-edge correction. The approach leads to an intuitive framework in which one can trade-off the quality of an approximation with the complexity of computing it. It also includes the Bethe free energy approximation as a degenerate case. We develop the approach theoretically in this paper and provide a number of empirical results that reveal its practical utility.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

EDML: A Method for Learning Parameters in Bayesian Networks

We propose a method called EDML for learning MAP parameters in binary Bayesian networks under incomplete data. The method assumes Beta priors and can be used to learn maximum likelihood parameters when the priors are uninformative. EDML exhibits interesting behaviors, especially when compared to EM. We introduce EDML, explain its origin, and study some of its properties both analytically and empirically.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Exploiting Evidence in Probabilistic Inference

We define the notion of compiling a Bayesian network with evidence and provide a specific approach for evidence-based compilation, which makes use of logical processing. The approach is practical and advantageous in a number of application areas-including maximum likelihood estimation, sensitivity analysis, and MAP computations-and we provide specific empirical results in the domain of genetic linkage analysis. We also show that the approach is applicable for networks that do not contain determinism, and show that it empirically subsumes the performance of the quickscore algorithm when applied to noisy-or networks.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Lifted Relax, Compensate and then Recover: From Approximate to Exact Lifted Probabilistic Inference

We propose an approach to lifted approximate inference for first-order probabilistic models, such as Markov logic networks. It is based on performing exact lifted inference in a simplified first-order model, which is found by relaxing first-order constraints, and then compensating for the relaxation. These simplified models can be incrementally improved by carefully recovering constraints that have been relaxed, also at the first-order level. This leads to a spectrum of approximations, with lifted belief propagation on one end, and exact lifted inference on the other. We discuss how relaxation, compensation, and recovery can be performed, all at the firstorder level, and show empirically that our approach substantially improves on the approximations of both propositional solvers and lifted belief propagation.

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preprint2012arXiv

New Advances and Theoretical Insights into EDML

EDML is a recently proposed algorithm for learning MAP parameters in Bayesian networks. In this paper, we present a number of new advances and insights on the EDML algorithm. First, we provide the multivalued extension of EDML, originally proposed for Bayesian networks over binary variables. Next, we identify a simplified characterization of EDML that further implies a simple fixed-point algorithm for the convex optimization problem that underlies it. This characterization further reveals a connection between EDML and EM: a fixed point of EDML is a fixed point of EM, and vice versa. We thus identify also a new characterization of EM fixed points, but in the semantics of EDML. Finally, we propose a hybrid EDML/EM algorithm that takes advantage of the improved empirical convergence behavior of EDML, while maintaining the monotonic improvement property of EM.

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preprint2012arXiv

New Advances in Inference by Recursive Conditioning

Recursive Conditioning (RC) was introduced recently as the first any-space algorithm for inference in Bayesian networks which can trade time for space by varying the size of its cache at the increment needed to store a floating point number. Under full caching, RC has an asymptotic time and space complexity which is comparable to mainstream algorithms based on variable elimination and clustering (exponential in the network treewidth and linear in its size). We show two main results about RC in this paper. First, we show that its actual space requirements under full caching are much more modest than those needed by mainstream methods and study the implications of this finding. Second, we show that RC can effectively deal with determinism in Bayesian networks by employing standard logical techniques, such as unit resolution, allowing a significant reduction in its time requirements in certain cases. We illustrate our results using a number of benchmark networks, including the very challenging ones that arise in genetic linkage analysis.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Node Splitting: A Scheme for Generating Upper Bounds in Bayesian Networks

We formulate in this paper the mini-bucket algorithm for approximate inference in terms of exact inference on an approximate model produced by splitting nodes in a Bayesian network. The new formulation leads to a number of theoretical and practical implications. First, we show that branchand- bound search algorithms that use minibucket bounds may operate in a drastically reduced search space. Second, we show that the proposed formulation inspires new minibucket heuristics and allows us to analyze existing heuristics from a new perspective. Finally, we show that this new formulation allows mini-bucket approximations to benefit from recent advances in exact inference, allowing one to significantly increase the reach of these approximations.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

On Bayesian Network Approximation by Edge Deletion

We consider the problem of deleting edges from a Bayesian network for the purpose of simplifying models in probabilistic inference. In particular, we propose a new method for deleting network edges, which is based on the evidence at hand. We provide some interesting bounds on the KL-divergence between original and approximate networks, which highlight the impact of given evidence on the quality of approximation and shed some light on good and bad candidates for edge deletion. We finally demonstrate empirically the promise of the proposed edge deletion technique as a basis for approximate inference.

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preprint2012arXiv

On the Robustness of Most Probable Explanations

In Bayesian networks, a Most Probable Explanation (MPE) is a complete variable instantiation with a highest probability given the current evidence. In this paper, we discuss the problem of finding robustness conditions of the MPE under single parameter changes. Specifically, we ask the question: How much change in a single network parameter can we afford to apply while keeping the MPE unchanged? We will describe a procedure, which is the first of its kind, that computes this answer for each parameter in the Bayesian network variable in time O(n exp(w)), where n is the number of network variables and w is its treewidth.

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preprint2012arXiv

Reasoning about Bayesian Network Classifiers

Bayesian network classifiers are used in many fields, and one common class of classifiers are naive Bayes classifiers. In this paper, we introduce an approach for reasoning about Bayesian network classifiers in which we explicitly convert them into Ordered Decision Diagrams (ODDs), which are then used to reason about the properties of these classifiers. Specifically, we present an algorithm for converting any naive Bayes classifier into an ODD, and we show theoretically and experimentally that this algorithm can give us an ODD that is tractable in size even given an intractable number of instances. Since ODDs are tractable representations of classifiers, our algorithm allows us to efficiently test the equivalence of two naive Bayes classifiers and characterize discrepancies between them. We also show a number of additional results including a count of distinct classifiers that can be induced by changing some CPT in a naive Bayes classifier, and the range of allowable changes to a CPT which keeps the current classifier unchanged.

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preprint2012arXiv

Sensitivity Analysis in Bayesian Networks: From Single to Multiple Parameters

Previous work on sensitivity analysis in Bayesian networks has focused on single parameters, where the goal is to understand the sensitivity of queries to single parameter changes, and to identify single parameter changes that would enforce a certain query constraint. In this paper, we expand the work to multiple parameters which may be in the CPT of a single variable, or the CPTs of multiple variables. Not only do we identify the solution space of multiple parameter changes that would be needed to enforce a query constraint, but we also show how to find the optimal solution, that is, the one which disturbs the current probability distribution the least (with respect to a specific measure of disturbance). We characterize the computational complexity of our new techniques and discuss their applications to developing and debugging Bayesian networks, and to the problem of reasoning about the value (reliability) of new information.

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preprint2012arXiv

Solving MAP Exactly using Systematic Search

MAP is the problem of finding a most probable instantiation of a set of variables in a Bayesian network given some evidence. Unlike computing posterior probabilities, or MPE (a special case of MAP), the time and space complexity of structural solutions for MAP are not only exponential in the network treewidth, but in a larger parameter known as the "constrained" treewidth. In practice, this means that computing MAP can be orders of magnitude more expensive than computing posterior probabilities or MPE. This paper introduces a new, simple upper bound on the probability of a MAP solution, which admits a tradeoff between the bound quality and the time needed to compute it. The bound is shown to be generally much tighter than those of other methods of comparable complexity. We use this proposed upper bound to develop a branch-and-bound search algorithm for solving MAP exactly. Experimental results demonstrate that the search algorithm is able to solve many problems that are far beyond the reach of any structure-based method for MAP. For example, we show that the proposed algorithm can compute MAP exactly and efficiently for some networks whose constrained treewidth is more than 40.

0 citations0 reviewsNo code linkedOpen access