This paper addresses the problem of predicting the k events that are most likely to occur next, over historical real-time event streams. Existing approaches to causal prediction queries have a number of limitations. First, they exhaustively search over an acyclic causal network to find the most likely k effect events; however, data from real event streams frequently reflect cyclic causality. Second, they contain conservative assumptions intended to exclude all possible non-causal links in the causal network; it leads to the omission of many less-frequent but important causal links. We overcome these limitations by proposing a novel event precedence model and a run-time causal inference mechanism. The event precedence model constructs a first order absorbing Markov chain incrementally over event streams, where an edge between two events signifies a temporal precedence relationship between them, which is a necessary condition for causality. Then, the run-time causal inference mechanism learns causal relationships dynamically during query processing. This is done by removing some of the temporal precedence relationships that do not exhibit causality in the presence of other events in the event precedence model. This paper presents two query processing algorithms -- one performs exhaustive search on the model and the other performs a more efficient reduced search with early termination. Experiments using two real datasets (cascading blackouts in power systems and web page views) verify the effectiveness of the probabilistic top-k prediction queries and the efficiency of the algorithms. Specifically, the reduced search algorithm reduced runtime, relative to exhaustive search, by 25-80 (depending on the application) with only a small reduction in accuracy.
Stochastic sampling algorithms, while an attractive alternative to exact algorithms in very large Bayesian network models, have been observed to perform poorly in evidential reasoning with extremely unlikely evidence. To address this problem, we propose an adaptive importance sampling algorithm, AIS-BN, that shows promising convergence rates even under extreme conditions and seems to outperform the existing sampling algorithms consistently. Three sources of this performance improvement are (1) two heuristics for initialization of the importance function that are based on the theoretical properties of importance sampling in finite-dimensional integrals and the structural advantages of Bayesian networks, (2) a smooth learning method for the importance function, and (3) a dynamic weighting function for combining samples from different stages of the algorithm. We tested the performance of the AIS-BN algorithm along with two state of the art general purpose sampling algorithms, likelihood weighting (Fung & Chang, 1989; Shachter & Peot, 1989) and self-importance sampling (Shachter & Peot, 1989). We used in our tests three large real Bayesian network models available to the scientific community: the CPCS network (, 1994), the PATHFINDER network (Heckerman, Horvitz, & Nathwani, 1990), and the ANDES network (Conati, Gertner, VanLehn, & Druzdzel, 1997), with evidence as unlikely as 10-41. While the AIS-BN algorithm always performed better than the other two algorithms, in the majority of the test cases it achieved orders of magnitude improvement in precision of the results. Improvement in speed given a desired precision is even more dramatic, although we are unable to report numerical results here, as the other algorithms almost never achieved the precision reached even by the first few iterations of the AIS-BN algorithm. Since exact probabilistic inference is intractable in general for large multiply connected belief nets, approximate methods are required. A promising approach is to use heuristic search among hypotheses (instantiations of the network) to find the most probable ones, as in the TopN algorithm. Search is based on the relative probabilities of hypotheses which are efficient to compute. Given upper and lower bounds on the relative probability of partial hypotheses, it is possible to obtain bounds on the absolute probabilities of hypotheses. Best-first search aimed at reducing the maximum error progressively narrows the bounds as more hypotheses are examined. Here, qualitative probabilistic analysis is employed to obtain bounds on the relative probability of partial hypotheses for the BN20 class of networks networks and a generalization replacing the noisy OR assumption by negative synergy. The approach is illustrated by application to a very large belief network, QMR-BN, which is a reformulation of the Internist-1 system for diagnosis in internal medicine. Approximate Bayesian computation (ABC) is a popular approach to address inference problems where the likelihood function is intractable, or expensive to calculate. To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested. Most effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples (, Biometrika 86:983---990, 2009; , Technical report, 2008; , J. Roy. Soc. Interface 6:187---202, 2009) and require the careful choice of simulation parameters. In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and adaptively determines the simulation parameters. We demonstrate our algorithm on a toy example and on a birth-death-mutation model arising in epidemiology. The general problem of computing poste rior probabilities in Bayesian networks is NP hard Cooper However e cient algorithms are often possible for particular applications by exploiting problem struc tures It is well understood that the key to the materialization of such a possibil ity is to make use of conditional indepen dence and work with factorizations of joint probabilities rather than joint probabilities themselves Di erent exact approaches can be characterized in terms of their choices of factorizations We propose a new approach which adopts a straightforward way for fac torizing joint probabilities In comparison with the clique tree propagation approach our approach is very simple It allows the pruning of irrelevant variables it accommo dates changes to the knowledge base more easily it is easier to implement More importantly it can be adapted to utilize both intercausal independence and condi tional independence in one uniform frame work On the other hand clique tree prop agation is better in terms of facilitating pre computations This chapter describes a sequence of Monte Carlo methods: importance sampling, rejection sampling, the Metropolis method, and Gibbs sampling. For each method, we discuss whether the method is expected to be useful for high—dimensional problems such as arise in inference with graphical models. After the methods have been described, the terminology of Markov chain Monte Carlo methods is presented. The chapter concludes with a discussion of advanced methods, including methods for reducing random walk behaviour.  The arc reversal node reduction approach to probabilistic inference is extended to include the case of instantiated evidence by an operation called “evidence reversal.” This not only provides a technique for computing posterior joint distributions on general belief networks, but also provides insight into the methods of Pearl [1986b] and Lauritzen and Spiegelhalter [1988]. Although it is well understood that the latter two algorithms are closely related, in fact all three algorithms are identical whenever the belief network is a forest. Prediction is emerging as an essential ingredient for real-time monitoring, planning and decision support applications such as intrusion detection, e-commerce pricing and automated resource management. This paper presents a system that efficiently supports continuous prediction queries (CPQs) over streaming data using seamlessly-integrated probabilistic models. Specifically, we describe how to execute and optimize CPQs using discrete (Dynamic) Bayesian Networks as the underlying predictive model. Our primary contribution is a novel cost-based optimization framework that employs materialization, sharing, and model-specific optimization techniques to enable highly-efficient point- and range-based CPQ execution. Furthermore, we support efficient execution of top-k and threshold-based high probability queries. We characterize the behavior of our system and demonstrate significant performance gains using a prototype implementation operating on real-world network intrusion data and deployed as part of a real-time software-performance monitoring system. A new method is proposed for exploiting causal independencies in exact Bayesian network inference. A Bayesian network can be viewed as representing a factorization of a joint probability into the multiplication of a set of conditional probabilities. We present a notion of causal independence that enables one to further factorize the conditional probabilities into a combination of even smaller factors and consequently obtain a finer-grain factorization of the joint probability. The new formulation of causal independence lets us specify the conditional probability of a variable given its parents in terms of an associative and commutative operator, such as "or", "sum" or "max", on the contribution of each parent. We start with a simple algorithm VE for Bayesian network inference that, given evidence and a query variable, uses the factorization to find the posterior distribution of the query. We show how this algorithm can be extended to exploit causal independence. Empirical studies, based on the CPCS networks for medical diagnosis, show that this method is more efficient than previous methods and allows for inference in larger networks than previous algorithms.