Continuous-time Bayesian networks (CTBNs) are graphical representations of multi-component continuous-time Markov processes as directed graphs. The edges in the network represent direct influences among components. The joint rate matrix of the multi-component process is specified by means of conditional rate matrices for each component separately. This paper addresses the situation where some of the components evolve on a time scale that is much shorter compared to the time scale of the other components. In this paper, we prove that in the limit where the separation of scales is infinite, the Markov process converges (in distribution, or weakly) to a reduced, or effective Markov process that only involves the slow components. We also demonstrate that for reasonable separation of scale (an order of magnitude) the reduced process is a good approximation of the marginal process over the slow components. We provide a simple procedure for building a reduced CTBN for this effective process, with conditional rate matrices that can be directly calculated from the original CTBN, and discuss the implications for approximate reasoning in large systems.

This paper studies a new and more general axiomatization than one presented previously for preference on likelihood gambles. Likelihood gambles describe actions in a situation where a decision maker knows multiple probabilistic models and a random sample generated from one of those models but does not know prior probability of models. This new axiom system is inspired by Jensen's axiomatization of probabilistic gambles. Our approach provides a new perspective to the role of data in decision making under ambiguity. It avoids one of the most controversial issue of Bayesian methodology namely the assumption of prior probability.

This paper is concerned with graphical criteria that can be used to solve the problem of identifying casual effects from nonexperimental data in a causal Bayesian network structure, i.e., a directed acyclic graph that represents causal relationships. We first review Pearl's work on this topic [Pearl, 1995], in which several useful graphical criteria are presented. Then we present a complete algorithm [Huang and Valtorta, 2006b] for the identifiability problem. By exploiting the completeness of this algorithm, we prove that the three basic do-calculus rules that Pearl presents are complete, in the sense that, if a causal effect is identifiable, there exists a sequence of applications of the rules of the do-calculus that transforms the causal effect formula into a formula that only includes observational quantities.

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.

We consider a Bayesian method for learning the Bayesian network structure from complete data. Recently, Koivisto and Sood (2004) presented an algorithm that for any single edge computes its marginal posterior probability in O(n 2^n) time, where n is the number of attributes; the number of parents per attribute is bounded by a constant. In this paper we show that the posterior probabilities for all the n (n - 1) potential edges can be computed in O(n 2^n) total time. This result is achieved by a forward-backward technique and fast Moebius transform algorithms, which are of independent interest. The resulting speedup by a factor of about n^2 allows us to experimentally study the statistical power of learning moderate-size networks. We report results from a simulation study that covers data sets with 20 to 10,000 records over 5 to 25 discrete attributes

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.

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.