This paper studies decision making for Walley's partially consonant belief functions (pcb). In a pcb, the set of foci are partitioned. Within each partition, the foci are nested. The pcb class includes probability functions and possibility functions as extreme cases. Unlike earlier proposals for a decision theory with belief functions, we employ an axiomatic approach. We adopt an axiom system similar in spirit to von Neumann - Morgenstern's linear utility theory for a preference relation on pcb lotteries. We prove a representation theorem for this relation. Utility for a pcb lottery is a combination of linear utility for probabilistic lottery and binary utility for possibilistic lottery.

This paper examines a number of solution methods for decision processes with non-Markovian rewards (NMRDPs). They all exploit a temporal logic specification of the reward function to automatically translate the NMRDP into an equivalent Markov decision process (MDP) amenable to well-known MDP solution methods. They differ however in the representation of the target MDP and the class of MDP solution methods to which they are suited. As a result, they adopt different temporal logics and different translations. Unfortunately, no implementation of these methods nor experimental let alone comparative results have ever been reported. This paper is the first step towards filling this gap. We describe an integrated system for solving NMRDPs which implements these methods and several variants under a common interface; we use it to compare the various approaches and identify the problem features favoring one over the other.

Loopy and generalized belief propagation are popular algorithms for approximate inference in Markov random fields and Bayesian networks. Fixed points of these algorithms correspond to extrema of the Bethe and Kikuchi free energy. However, belief propagation does not always converge, which explains the need for approaches that explicitly minimize the Kikuchi/Bethe free energy, such as CCCP and UPS. Here we describe a class of algorithms that solves this typically nonconvex constrained minimization of the Kikuchi free energy through a sequence of convex constrained minimizations of upper bounds on the Kikuchi free energy. Intuitively one would expect tighter bounds to lead to faster algorithms, which is indeed convincingly demonstrated in our simulations. Several ideas are applied to obtain tight convex bounds that yield dramatic speed-ups over CCCP.

In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of dependency relationships in their structure. Our method effectively maps such a dense problem to a sparser one which is in some sense "closest". Exact methods can be run on the sparser problem to derive bounds on the original answer, which can be quite sharp. We present empirical results demonstrating that our method works well on the tasks of belief inference and finding the probability of the most probable explanation in belief networks, and finding the cost of the solution that violates the smallest number of constraints in constraint satisfaction problems. On one large CPCS network, for example, we were able to calculate upper and lower bounds on the conditional probability of a variable, given evidence, that were almost identical in the average case.

By elaborating on the notion of linear belief functions (Dempster 1990; Liu 1996), we propose an elementary approach to knowledge representation for expert systems using linear belief functions. We show how to use basic matrices to represent market information and financial knowledge, including complete ignorance, statistical observations, subjective speculations, distributional assumptions, linear relations, and empirical asset pricing models. We then appeal to Dempster's rule of combination to integrate the knowledge for assessing an overall belief of portfolio performance, and updating the belief by incorporating additional information. We use an example of three gold stocks to illustrate the approach.

Pearls concept OF a d - connecting path IS one OF the foundations OF the modern theory OF graphical models : the absence OF a d - connecting path IN a DAG indicates that conditional independence will hold IN ANY distribution factorising according TO that graph. IN this paper we show that IN singly - connected Gaussian DAGs it IS possible TO USE the form OF a d - connection TO obtain qualitative information about the strength OF conditional dependence.More precisely, the squared partial correlations BETWEEN two given variables, conditioned ON different subsets may be partially ordered BY examining the relationship BETWEEN the d - connecting path AND the SET OF variables conditioned upon.

In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest model able to represent the generative distribution exactly. Our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset. We show that identifying high-scoring structures is hard, even when we are given an independence oracle, an inference oracle, and/or an information oracle. Our negative results also apply to the learning of discrete-variable Bayesian networks in which each node has at most k parents, for all k > 3.

Constraint-based (CB) learning is a formalism for learning a causal network with a database D by performing a series of conditional-independence tests to infer structural information. This paper considers a new test of independence that combines ideas from Bayesian learning, Bayesian network inference, and classical hypothesis testing to produce a more reliable and robust test. The new test can be calculated in the same asymptotic time and space required for the standard tests such as the chi-squared test, but it allows the specification of a prior distribution over parameters and can be used when the database is incomplete. We prove that the test is correct, and we demonstrate empirically that, when used with a CB causal discovery algorithm with noninformative priors, it recovers structural features more reliably and it produces networks with smaller KL-Divergence, especially as the number of nodes increases or the number of records decreases. Another benefit is the dramatic reduction in the probability that a CB algorithm will stall during the search, providing a remedy for an annoying problem plaguing CB learning when the database is small.

In Non - ergodic belief networks the posterior belief OF many queries given evidence may become zero.The paper shows that WHEN belief propagation IS applied iteratively OVER arbitrary networks(the so called, iterative OR loopy belief propagation(IBP)) it IS identical TO an arc - consistency algorithm relative TO zero - belief queries(namely assessing zero posterior probabilities). This implies that zero - belief conclusions derived BY belief propagation converge AND are sound.More importantly it suggests that the inference power OF IBP IS AS strong AND AS weak, AS that OF arc - consistency.This allows the synthesis OF belief networks FOR which belief propagation IS useless ON one hand, AND focuses the investigation OF classes OF belief network FOR which belief propagation may be zero - complete.Finally, ALL the above conclusions apply also TO Generalized belief propagation algorithms that extend loopy belief propagation AND allow a crisper understanding OF their power.

We present the language {m P}{cal C}+ for probabilistic reasoning about actions, which is a generalization of the action language {cal C}+ that allows to deal with probabilistic as well as nondeterministic effects of actions. We define a formal semantics of {m P}{cal C}+ in terms of probabilistic transitions between sets of states. Using a concept of a history and its belief state, we then show how several important problems in reasoning about actions can be concisely formulated in our formalism.