One of the most useful sensitivity analysis techniques of decision analysis is the computation of value of information (or clairvoyance), the difference in value obtained by changing the decisions by which some of the uncertainties are observed. In this paper, some simple but powerful extensions to previous algorithms are introduced which allow an efficient value of information calculation on the rooted cluster tree (or strong junction tree) used to solve the original decision problem.

Graphical models based on conditional independence support concise encodings of the subjective belief of a single agent. A natural question is whether the consensus belief of a group of agents can be represented with equal parsimony. We prove, under relatively mild assumptions, that even if everyone agrees on a common graph topology, no method of combining beliefs can maintain that structure. Even weaker conditions rule out local aggregation within conditional probability tables. On a more positive note, we show that if probabilities are combined with the logarithmic opinion pool (LogOP), then commonly held Markov independencies are maintained. This suggests a straightforward procedure for constructing a consensus Markov network. We describe an algorithm for computing the LogOP with time complexity comparable to that of exact Bayesian inference.

In this paper, we analyze the relationship between probability and Spohn's theory for representation of uncertain beliefs. Using the intuitive idea that the more probable a proposition is, the more believable it is, we study transformations from probability to Sphonian disbelief and vice-versa. The transformations described in this paper are different from those described in the literature. In particular, the former satisfies the principles of ordinal congruence while the latter does not. Such transformations between probability and Spohn's calculi can contribute to (1) a clarification of the semantics of nonprobabilistic degree of uncertain belief, and (2) to a construction of a decision theory for such calculi. In practice, the transformations will allow a meaningful combination of more than one calculus in different stages of using an expert system such as knowledge acquisition, inference, and interpretation of results.

Dynamic Bayesian networks provide a compact and natural representation for complex dynamic systems. However, in many cases, there is no expert available from whom a model can be elicited. Learning provides an alternative approach for constructing models of dynamic systems. In this paper, we address some of the crucial computational aspects of learning the structure of dynamic systems, particularly those where some relevant variables are partially observed or even entirely unknown. Our approach is based on the Structural Expectation Maximization (SEM) algorithm. The main computational cost of the SEM algorithm is the gathering of expected sufficient statistics. We propose a novel approximation scheme that allows these sufficient statistics to be computed efficiently. We also investigate the fundamental problem of discovering the existence of hidden variables without exhaustive and expensive search. Our approach is based on the observation that, in dynamic systems, ignoring a hidden variable typically results in a violation of the Markov property. Thus, our algorithm searches for such violations in the data, and introduces hidden variables to explain them. We provide empirical results showing that the algorithm is able to learn the dynamics of complex systems in a computationally tractable way.

Structure and parameters in a Bayesian network uniquely specify the probability distribution of the modeled domain. The locality of both structure and probabilistic information are the great benefits of Bayesian networks and require the modeler to only specify local information. On the other hand this locality of information might prevent the modeler - and even more any other person - from obtaining a general overview of the important relationships within the domain. The goal of the work presented in this paper is to provide an "alternative" view on the knowledge encoded in a Bayesian network which might sometimes be very helpful for providing insights into the underlying domain. The basic idea is to calculate a mixture approximation to the probability distribution represented by the Bayesian network. The mixture component densities can be thought of as representing typical scenarios implied by the Bayesian model, providing intuition about the basic relationships. As an additional benefit, performing inference in the approximate model is very simple and intuitive and can provide additional insights. The computational complexity for the calculation of the mixture approximations criticaly depends on the measure which defines the distance between the probability distribution represented by the Bayesian network and the approximate distribution. Both the KL-divergence and the backward KL-divergence lead to inefficient algorithms. Incidentally, the latter is used in recent work on mixtures of mean field solutions to which the work presented here is closely related. We show, however, that using a mean squared error cost function leads to update equations which can be solved using the junction tree algorithm. We conclude that the mean squared error cost function can be used for Bayesian networks in which inference based on the junction tree is tractable. For large networks, however, one may have to rely on mean field approximations.

Decision-making problems in uncertain or stochastic domains are often formulated as Markov decision processes (MDPs). Policy iteration (PI) is a popular algorithm for searching over policy-space, the size of which is exponential in the number of states. We are interested in bounds on the complexity of PI that do not depend on the value of the discount factor. In this paper we prove the first such non-trivial, worst-case, upper bounds on the number of iterations required by PI to converge to the optimal policy. Our analysis also sheds new light on the manner in which PI progresses through the space of policies.

We present a hybrid constraint-based/Bayesian algorithm for learning causal networks in the presence of sparse data. The algorithm searches the space of equivalence classes of models (essential graphs) using a heuristic based on conventional constraint-based techniques. Each essential graph is then converted into a directed acyclic graph and scored using a Bayesian scoring metric. Two variants of the algorithm are developed and tested using data from randomly generated networks of sizes from 15 to 45 nodes with data sizes ranging from 250 to 2000 records. Both variations are compared to, and found to consistently outperform two variations of greedy search with restarts.

In this article we propose a qualitative (ordinal) counterpart for the Partially Observable Markov Decision Processes model (POMDP) in which the uncertainty, as well as the preferences of the agent, are modeled by possibility distributions. This qualitative counterpart of the POMDP model relies on a possibilistic theory of decision under uncertainty, recently developed. One advantage of such a qualitative framework is its ability to escape from the classical obstacle of stochastic POMDPs, in which even with a finite state space, the obtained belief state space of the POMDP is infinite. Instead, in the possibilistic framework even if exponentially larger than the state space, the belief state space remains finite.

The paper is a second in a series of two papers evaluating the power of a new scheme that generates search heuristics mechanically. The heuristics are extracted from an approximation scheme called mini-bucket elimination that was recently introduced. The first paper introduced the idea and evaluated it within Branch-and-Bound search. In the current paper the idea is further extended and evaluated within Best-First search. The resulting algorithms are compared on coding and medical diagnosis problems, using varying strength of the mini-bucket heuristics. Our results demonstrate an effective search scheme that permits controlled tradeoff between preprocessing (for heuristic generation) and search. Best-first search is shown to outperform Branch-and-Bound, when supplied with good heuristics, and sufficient memory space.

Next-generation sequencing technologies provide a revolutionary tool for generating gene expression data. Starting with a fixed RNA sample, they construct a library of millions of differentially abundant short sequence tags or "reads", which constitute a fundamentally discrete measure of the level of gene expression. A common limitation in experiments using these technologies is the low number or even absence of biological replicates, which complicates the statistical analysis of digital gene expression data. Analysis of this type of data has often been based on modified tests originally devised for analysing microarrays; both these and even de novo methods for the analysis of RNA-seq data are plagued by the common problem of low replication. We propose a novel, non-parametric Bayesian approach for the analysis of digital gene expression data. We begin with a hierarchical model for modelling over-dispersed count data and a blocked Gibbs sampling algorithm for inferring the posterior distribution of model parameters conditional on these counts. The algorithm compensates for the problem of low numbers of biological replicates by clustering together genes with tag counts that are likely sampled from a common distribution and using this augmented sample for estimating the parameters of this distribution. The number of clusters is not decided a priori, but it is inferred along with the remaining model parameters. We demonstrate the ability of this approach to model biological data with high fidelity by applying the algorithm on a public dataset obtained from cancerous and non-cancerous neural tissues.