Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have been used for problems ranging from tracking planes and missiles to predicting the economy. However, HMMs and KFMs are limited in their “expressive power”. Dynamic Bayesian Networks (DBNs) generalize HMMs by allowing the state space to be represented in factored form, instead of as a single discrete random variable. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. In this thesis, I will discuss how to represent many different kinds of models as DBNs, how to perform exact and approximate inference in DBNs, and how to learn DBN models from sequential data. In particular, the main novel technical contributions of this thesis are as follows: a way of representing Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3), where T is the length of the sequence; an exact smoothing algorithm that takes O(log T) space instead of O(T); a simple way of using the junction tree algorithm for online inference in DBNs; new complexity bounds on exact online inference in DBNs; a new deterministic approximate inference algorithm called factored frontier; an analysis of the relationship between the BK algorithm and loopy belief propagation; a way of applying Rao-Blackwellised particle filtering to DBNs in general, and the SLAM (simultaneous localization and mapping) problem in particular; a way of extending the structural EM algorithm to DBNs; and a variety of different applications of DBNs. However, perhaps the main value of the thesis is its catholic presentation of the field of sequential data modelling.

We show how to use a variational approximation to the logistic function to perform approximate inference in Bayesian networks containing discrete nodes with continuous parents. Essentially, we convert the logistic function to a Gaussian, which facilitates exact inference, and then iteratively adjust the variational parameters to improve the quality of the approximation. We demonstrate experimentally that this approximation is much faster than sampling, but comparable in accuracy. We also introduce a simple new technique for handling evidence, which allows us to handle arbitrary distributionson observed nodes, as well as achieving a significant speedup in networks with discrete variables of large cardinality. 1

The paper presents an efficient computational method for performing sensitivity analysis in discrete Bayesian networks. The method exploits the structure of conditional probabilities of a target node given the evidence. First, the set of parameters which are relevant to the calculation of the conditional probabilities of the target node is identified. Next, this set is reduced by removing those combinations of the parameters which either contradict the available evidence or are incompatible. Finally, using the canonical components associated with the resulting subset of parameters, the desired conditional probabilities are obtained. In this way, an important saving in the calculations is achieved. The proposed method can also be used to compute exact upper and lower bounds for the conditional probabilities, hence a sensitivity analysis can be easily performed. Examples are used to illustrate the proposed methodology.

We survey the literature on methods for inference and learning in Bayesian Networks composed of discrete and continuous nodes, in which the continuous nodes have a multivariate Gaussian distribution, whose mean and variance depends on the values of the discrete nodes. We also briefly consider hybrid Dynamic Bayesian Networks, an extension of switching Kalman filters. This report is meant to summarize what is known at a sufficient level of detail to enable someone to implement the algorithms, but without dwelling on formalities.

The assessments obtained for the various conditional probabilities of a Bayesian belief network inevitably are inaccurate. The inaccuracies involved influence the reliability of the network's output. By subjecting the belief network to a sensitivity analysis with respect to its conditional probabilities, the reliability of the output can be investigated. Unfortunately, straightforward sensitivity analysis of a Bayesian belief network is highly time-consuming. In this paper, we show that, by qualitative considerations, several analyses can be identified as being uninformative as the conditional probabilities under study cannot affect the network's output. In addition, we show that the analyses that are informative comply with simple mathematical functions; more specifically, we show that the network's output can be expressed as a quotient of two functions that are linear in a conditional probability under study. These properties allow for considerably reducing the computational burden of se...

This paper presents a general and efficient framework for probabilistic inference and learning from arbitrary uncertain information. It exploits the calculation properties of finite mixture models, conjugate families and factorization. Both the joint probability density of the variables and the likelihood function of the (objective or subjective) observation are approximated by a special mixture model, in such a way that any desired conditional distribution can be directly obtained without numerical integration. We have developed an extended version of the expectation maximization (EM) algorithm to estimate the parameters of mixture models from uncertain training examples (indirect observations). As a consequence, any piece of exact or uncertain information about both input and output values is consistently handled in the inference and learning stages. This ability, extremely useful in certain situations, is not found in most alternative methods. The proposed framework is formally just...

Bayesian network is a compact representation for probabilistic models and inference. They have been used successfully for multisensor fusion and situation assessment. It is well known that, in general, the inference algorithms to compute the exact posterior probability of the target state are either computationally infeasible for dense networks or impossible for mixed discretecontinuous networks. In those cases, one approach is to compute the approximate results using simulation methods. This paper proposes efficient inference methods for those cases. The goal is not to compute the exact or approximate posterior probability of the target state, but to identify the top (most likely) ones in an efficient manner. The approach is to use intelligent simulation techniques where previous samples will be used to guide the future sampling strategy. By focusing the sampling on the "important" space, we are able to sort out the top candidates quickly. Simulation results are included to demonstrate the performances of the algorithms.

In this paper we show how Bayesian network models can be used to perform a sensitivity analysis using symbolic, as opposed to numeric, computations. An example of damage assessment of concrete structures of buildings is used for illustrative purposes. Initially, normal or Gaussian Bayesian network models are described together with an algorithm for numerical propagation of uncertainty in an incremental form. Next, the algorithm is implemented symbolically, in Mathematica code, and applied to answer some queries related to the damage assessment of concrete structures of buildings. Finally, the conditional means and variances of the of nodes given the evidence are shown to be rational functions of the parameters, thus, discovering its parametric structure, which can be efficiently used in sensitivity analysis.

In this paper we deal with the problem of interpreting data coming from a dynamic system by using Causal Probabilistic Networks, a probabilistic graphical model particularly appealing in Intelligent Data Analysis. We discuss the different approaches presented in the literature, outlining their pros and cons through a simple training example. Then, we present a new method for reconstructing the state of the dynamic system, based on Markov Chain Monte Carlo algorithms, called Dynamic Probabilistic Network smoothing (DPN-smoothing). Finally, we present an example of the application of DPN-smoothing in the field of signal deconvolution.

The relation between a probability computed from a Bayesian network and the parameters in the network is a simple mathematical function. Any prior probability can be expressed as a multilinear function in the network's parameters; any posterior probability is a quotient of such functions. These functions serve to yield insight into the robustness of a Bayesian network. To this end, it suffices to establish the coefficients in the functions. In general, the investigation into the relation between a probability computed from the network and the network's parameters is called a sensitivity analysis. In the past, various methods for sensitivity analysis have been suggested, most of them being computationally very expensive. In this paper, a new method is presented, that is both elegant and efficient. The method builds upon a junction tree representation of a Bayesian network for its computational architecture. Basically, the method amounts to the propagation of vectors of coeffi...