We present an application of hierarchical Bayesian estimation to robot map building. The revisiting problem occurs when a robot has to decide whether it is seeing a previously-built portion of a map, or is exploring new territory. This is a difficult decision problem, requiring the probability of being outside of the current known map. To estimate this probability, we model the structure of a "typical" environment as a hidden Markov model that generates sequences of views observed by a robot navigating through the environment. A Dirichlet prior over structural models is learned from previously explored environments. Whenever a robot explores a new environment, the posterior over the model is estimated by Dirichlet hyperparameters. Our approach is implemented and tested in the context of multi-robot map merging, a particularly difficult instance of the revisiting problem. Experiments with robot data show that the technique yields strong improvements over alternative methods.

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.

There has been great interest in identifying tractable subclasses of NP complete problems and designing efficient algorithms for these tractable classes. Constraint satisfaction and Bayesian network inference are two examples of such problems that are of great importance in AI and algorithms. In this paper we study, under the frameworks of random constraint satisfaction problems and random Bayesian networks, a typical tractable subclass characterized by the treewidth of the problems. We show that the property of having a bounded treewidth for CSPs and Bayesian network inference problem has a phase transition that occurs while the underlying structures of problems are still sparse. This implies that algorithms making use of treewidth based structural knowledge only work efficiently in a limited range of random instance.

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.

Backtracking search is a powerful algorithmic paradigm that can be used to solve many problems. It is in a certain sense the dual of variable elimination; but on many problems, e.g., SAT, it is vastly superior to variable elimination in practice. Motivated by this we investigate the application of backtracking search to the problem of Bayesian inference (Bayes). We show that natural generalizations of known techniques allow backtracking search to achieve performance guarantees similar to standard algorithms for Bayes, and that there exist problems on which backtracking can in fact do much better. We also demonstrate that these ideas can be applied to implement a Bayesian inference engine whose performance is competitive with standard algorithms. Since backtracking search can very naturally take advantage of context specific structure, the potential exists for performance superior to standard algorithms on many problems.

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.

In this paper we examine the problem of inference in Bayesian Networks with discrete random variables that have very large or even unbounded domains. For example, in a domain where we are trying to identify a person, we may have variables that have as domains, the set of all names, the set of all postal codes, or the set of all credit card numbers. We cannot just have big tables of the conditional probabilities, but need compact representations. We provide an inference algorithm, based on variable elimination, for belief networks containing both large domain and normal discrete random variables. We use intensional (i.e., in terms of procedures) and extensional (in terms of listing the elements) representations of conditional probabilities and of the intermediate factors.

WA qualitative probabilistic network models the probabilistic relationships between its variables by means of signs. Non-monotonic influences have associated an ambiguous sign. These ambiguous signs typically lead to uninformative results upon inference. A non-monotonic influence can, however, be associated with a, more informative, sign that indicates its effect in the current state of the network. To capture this effect, we introduce the concept of situational sign. Furthermore, if the network converts to a state in which all variables that provoke the non-monotonicity have been observed, a non-monotonic influence reduces to a monotonic influence. We study the persistence and propagation of situational signs upon inference and give a method to establish the sign of a reduced influence.

Most methods of exact probability propagation in Bayesian networks do not carry out the inference directly over the network, but over a secondary structure known as a junction tree or a join tree (JT). The process of obtaining a JT is usually termed {sl compilation}. As compilation is usually viewed as a whole process; each time the network is modified, a new compilation process has to be carried out. The possibility of reusing an already existing JT, in order to obtain the new one regarding only the modifications in the network has received only little attention in the literature. In this paper we present a method for incremental compilation of a Bayesian network, following the classical scheme in which triangulation plays the key role. In order to perform incremental compilation we propose to recompile only those parts of the JT which can have been affected by the networks modifications. To do so, we exploit the technique OF maximal prime subgraph decomposition in determining the minimal subgraph(s) that have to be recompiled, and thereby the minimal subtree(s) of the JT that should be replaced by new subtree(s).We focus on structural modifications : addition and deletion of links and variables.

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.