We characterize probabilities in Bayesian networks in terms of algebraic expressions called quasi-probabilities. These are arrived at by casting Bayesian networks as noisy AND-OR-NOT networks, and viewing the subnetworks that lead to a node as arguments for or against a node. Quasi-probabilities are in a sense the "natural" algebra of Bayesian networks: we can easily compute the marginal quasi-probability of any node recursively, in a compact form; and we can obtain the joint quasi-probability of any set of nodes by multiplying their marginals (using an idempotent product operator). Quasi-probabilities are easily manipulated to improve the efficiency of probabilistic inference. They also turn out to be representable as square-wave pulse trains, and joint and marginal distributions can be computed by multiplication and complementation of pulse trains.

For many real-life Bayesian networks, common knowledge dictates that the output established for the main variable of interest increases with higher values for the observable variables. We define two concepts of monotonicity to capture this type of knowledge. We say that a network is isotone in distribution if the probability distribution computed for the output variable given specific observations is stochastically dominated by any such distribution given higher-ordered observations; a network is isotone in mode if a probability distribution given higher observations has a higher mode. We show that establishing whether a network exhibits any of these properties of monotonicity is coNPPP-complete in general, and remains coNP-complete for polytrees. We present an approximate algorithm for deciding whether a network is monotone in distribution and illustrate its application to a real-life network in oncology.

Generalized belief propagation (GBP) has proven to be a promising technique for approximate inference tasks in AI and machine learning. However, the choice of a good set of clusters to be used in GBP has remained more of an art then a science until this day. This paper proposes a sequential approach to adding new clusters of nodes and their interactions (i.e. "regions") to the approximation. We first review and analyze the recently introduced region graphs and find that three kinds of operations ("split", "merge" and "death") leave the free energy and (under some conditions) the fixed points of GBP invariant. This leads to the notion of "weakly irreducible" regions as the natural candidates to be added to the approximation. Computational complexity of the GBP algorithm is controlled by restricting attention to regions with small "region-width". Combining the above with an efficient (i.e. local in the graph) measure to predict the improved accuracy of GBP leads to the sequential "region pursuit" algorithm for adding new regions bottom-up to the region graph. Experiments show that this algorithm can indeed perform close to optimally.

Maximum a Posteriori assignment (MAP) is the problem of finding the most probable instantiation of a set of variables given the partial evidence on the other variables in a Bayesian network. MAP has been shown to be a NP-hard problem [22], even for constrained networks, such as polytrees [18]. Hence, previous approaches often fail to yield any results for MAP problems in large complex Bayesian networks. To address this problem, we propose AnnealedMAP algorithm, a simulated annealing-based MAP algorithm. The AnnealedMAP algorithm simulates a non-homogeneous Markov chain whose invariant function is a probability density that concentrates itself on the modes of the target density. We tested this algorithm on several real Bayesian networks. The results show that, while maintaining good quality of the MAP solutions, the AnnealedMAP algorithm is also able to solve many problems that are beyond the reach of previous approaches.

Although many real-world stochastic planning problems are more naturally formulated by hybrid models with both discrete and continuous variables, current state-of-the-art methods cannot adequately address these problems. We present the first framework that can exploit problem structure for modeling and solving hybrid problems efficiently. We formulate these problems as hybrid Markov decision processes (MDPs with continuous and discrete state and action variables), which we assume can be represented in a factored way using a hybrid dynamic Bayesian network (hybrid DBN). This formulation also allows us to apply our methods to collaborative multiagent settings. We present a new linear program approximation method that exploits the structure of the hybrid MDP and lets us compute approximate value functions more efficiently. In particular, we describe a new factored discretization of continuous variables that avoids the exponential blow-up of traditional approaches. We provide theoretical bounds on the quality of such an approximation and on its scale-up potential. We support our theoretical arguments with experiments on a set of control problems with up to 28-dimensional continuous state space and 22-dimensional action space.

As real-world Bayesian networks continue to grow larger and more complex, it is important to investigate the possibilities for improving the performance of existing algorithms of probabilistic inference. Motivated by examples, we investigate the dependency of the performance of Lazy propagation on the message computation algorithm. We show how Symbolic Probabilistic Inference (SPI) and Arc-Reversal (AR) can be used for computation of clique to clique messages in the addition to the traditional use of Variable Elimination (VE). In addition, the paper presents the results of an empirical evaluation of the performance of Lazy propagation using VE, SPI, and AR as the message computation algorithm. The results of the empirical evaluation show that for most networks, the performance of inference did not depend on the choice of message computation algorithm, but for some randomly generated networks the choice had an impact on both space and time performance. In the cases where the choice had an impact, AR produced the best results.

Based on a recent development in the area of error control coding, we introduce the notion of convolutional factor graphs (CFGs) as a new class of probabilistic graphical models. In this context, the conventional factor graphs are referred to as multiplicative factor graphs (MFGs). This paper shows that CFGs are natural models for probability functions when summation of independent latent random variables is involved. In particular, CFGs capture a large class of linear models, where the linearity is in the sense that the observed variables are obtained as a linear ransformation of the latent variables taking arbitrary distributions. We use Gaussian models and independent factor models as examples to emonstrate the use of CFGs. The requirement of a linear transformation between latent variables (with certain independence restriction) and the bserved variables, to an extent, limits the modelling flexibility of CFGs. This structural restriction however provides a powerful analytic tool to the framework of CFGs; that is, upon taking the Fourier transform of the function represented by the CFG, the resulting function is represented by a FG with identical structure. This Fourier transform duality allows inference problems on a CFG to be solved on the corresponding dual MFG.

We introduce a probabilistic formalism subsuming Markov random fields of bounded tree width and probabilistic context free grammars. Our models are based on a representation of Boolean formulas that we call case-factor diagrams (CFDs). CFDs are similar to binary decision diagrams (BDDs) but are concise for circuits of bounded tree width (unlike BDDs) and can concisely represent the set of parse trees over a given string under a given context free grammar (also unlike BDDs). A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give an insideoutside algorithm for simultaneously computing the marginal of each Boolean variable, and a Viterbi algorithm for finding the mininum cost variable assignment. Both algorithms run in time proportional to the size of the CFD.

A key prerequisite to optimal reasoning under uncertainty in intelligent systems is to start with good class probability estimates. This paper improves on the current best probability estimation trees (Bagged-PETs) and also presents a new ensemble-based algorithm (MOB-ESP). Comparisons are made using several benchmark datasets and multiple metrics. These experiments show that MOB-ESP outputs significantly more accurate class probabilities than either the baseline B-PETs algorithm or the enhanced version presented here (EB-PETs). These results are based on metrics closely associated with the average accuracy of the predictions. MOB-ESP also provides much better probability rankings than B-PETs. The paper further suggests how these estimation techniques can be applied in concert with a broader category of classifiers.

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.