We investigate methods for parameter learning from incomplete data that is not missing at random. Likelihood-based methods then require the optimization of a profile likelihood that takes all possible missingness mechanisms into account. Optimzing this profile likelihood poses two main difficulties: multiple (local) maxima, and its very high-dimensional parameter space. In this paper a new method is presented for optimizing the profile likelihood that addresses the second difficulty: in the proposed AI&M (adjusting imputation and mazimization) procedure the optimization is performed by operations in the space of data completions, rather than directly in the parameter space of the profile likelihood. We apply the AI&M method to learning parameters for Bayesian networks. The method is compared against conservative inference, which takes into account each possible data completion, and against EM. The results indicate that likelihood-based inference is still feasible in the case of unknown missingness mechanisms, and that conservative inference is unnecessarily weak. On the other hand, our results also provide evidence that the EM algorithm is still quite effective when the data is not missing at random.

We present a class of inequality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network, in which some of the variables remain unmeasured. We derive bounds on causal effects that are not directly measured in randomized experiments. We derive instrumental inequality type of constraints on nonexperimental distributions. The results have applications in testing causal models with observational or experimental data.

Separable Bayesian Networks, or the Influence Model, are dynamic Bayesian Networks in which the conditional probability distribution can be separated into a function of only the marginal distribution of a node's neighbors, instead of the joint distributions. In terms of modeling, separable networks has rendered possible siginificant reduction in complexity, as the state space is only linear in the number of variables on the network, in contrast to a typical state space which is exponential. In this work, We describe the connection between an arbitrary Conditional Probability Table (CPT) and separable systems using linear algebra. We give an alternate proof on the equivalence of sufficiency and separability. We present a computational method for testing whether a given CPT is separable.

We observe that certain large-clique graph triangulations can be useful to reduce both computational and space requirements when making queries on mixed stochastic/deterministic graphical models. We demonstrate that many of these large-clique triangulations are non-minimal and are thus unattainable via the variable elimination algorithm. We introduce ancestral pairs as the basis for novel triangulation heuristics and prove that no more than the addition of edges between ancestral pairs need be considered when searching for state space optimal triangulations in such graphs. Empirical results on random and real world graphs show that the resulting triangulations that yield significant speedups are almost always non-minimal. We also give an algorithm and correctness proof for determining if a triangulation can be obtained via elimination, and we show that the decision problem associated with finding optimal state space triangulations in this mixed stochastic/deterministic setting is NP-complete.

Linear-time computational techniques have been developed for combining evidence which is available on a number of contending hypotheses. They offer a means of making the computation-intensive calculations involved more efficient in certain circumstances. Unfortunately, they restrict the orthogonal sum of evidential functions to the dichotomous structure applies only to elements and their complements. In this paper, we present a novel evidence structure in terms of a triplet and a set of algorithms for evidential reasoning. The merit of this structure is that it divides a set of evidence into three subsets, distinguishing trivial evidential elements from important ones focusing some particular elements. It avoids the deficits of the dichotomous structure in representing the preference of evidence and estimating the basic probability assignment of evidence. We have established a formalism for this structure and the general formulae for combining pieces of evidence in the form of the triplet, which have been theoretically justified.

The paper analyzes theoretically and empirically the performance of likelihood weighting (LW) on a subset of nodes in Bayesian networks. The proposed scheme requires fewer samples to converge due to reduction in sampling variance. The method exploits the structure of the network to bound the complexity of exact inference used to compute sampling distributions, similar to Gibbs cutset sampling. Yet, the extension of the previosly proposed cutset sampling principles to likelihood weighting is non-trivial due to differences in the sampling processes of Gibbs sampler and LW. We demonstrate empirically that likelihood weighting on a cutset (LWLC) is effective time-wise and has a lower rejection rate than LW when applied to networks with many deterministic probabilities. Finally, we show that the performance of likelihood weighting on a cutset can be improved further by caching computed sampling distributions and, consequently, learning 'zeros' of the target distribution.

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.

The effect of inaccuracies in the parameters of a dynamic Bayesian network can be investigated by subjecting the network to a sensitivity analysis. Having detailed the resulting sensitivity functions in our previous work, we now study the effect of parameter inaccuracies on a recommended decision in view of a threshold decision-making model. We detail the effect of varying a single and multiple parameters from a conditional probability table and present a computational procedure for establishing bounds between which assessments for these parameters can be varied without inducing a change in the recommended decision. We illustrate the various concepts involved by means of a real-life dynamic network in the field of infectious disease.

We describe an expert system, Maies, under development for analysing forensic identification problems involving DNA mixture traces using quantitative peak area information. Peak area information is represented by conditional Gaussian distributions, and inference based on exact junction tree propagation ascertains whether individuals, whose profiles have been measured, have contributed to the mixture. The system can also be used to predict DNA profiles of unknown contributors by separating the mixture into its individual components. The use of the system is illustrated with an application to a real world example. The system implements a novel MAP (maximum a posteriori) search algorithm that is briefly described.

Predicting the outcomes of future events is a challenging problem for which a variety of solution methods have been explored and attempted. We present an empirical comparison of a variety of online and offline adaptive algorithms for aggregating experts' predictions of the outcomes of five years of US National Football League games (1319 games) using expert probability elicitations obtained from an Internet contest called ProbabilitySports. We find that it is difficult to improve over simple averaging of the predictions in terms of prediction accuracy, but that there is room for improvement in quadratic loss. Somewhat surprisingly, a Bayesian estimation algorithm which estimates the variance of each expert's prediction exhibits the most consistent superior performance over simple averaging among our collection of algorithms.