Bayesian Networks: An Introduction (original) (raw)

Applied Logic Series, 2001

See the introduction to this volume for more on the distinction between logical and empirical Bayesianism. Such forms of Bayesianism are often referred to as 'objective' Bayesian positions, and confusion can arise because physical or empirical probability (frequency, propensity or chance) is often called 'objective' probability in order to distinguish it from Bayesian 'subjective' probability. In this chapter I will draw the latter distinction, using 'objective' to refer to empirical interpretations of causality and probability that are to do with objects external to an agent, and using 'subjective' to refer to interpretations of causality and probability that depend on the perspective of an agent subject. 2 If has no parents, Ô´ µ is just Ô´ µ. 3 The Bayesian network independence assumption is often called the Markov or causal Markov condition. 4 The joint distribution Ô can be determined by the direct method: Ô´ ½ AE µ É AE ½ Ô´ µ where is the state of the direct causes of which is consistent with ½ AE. Alternatively Ô may be determined by potentially more efficient propagation algorithms. See [Pearl 1988] or [Neapolitan 1990] here and for more on the formal properties of Bayesian networks. 5 See [Williamson 2000] for more on the probabilistic approach to diagnosis.

Context-specific independence in Bayesian networks

Proceedings of the Twelfth …, 1996

Bayesian networks provide a language for qualitatively representing the conditional independence properties of a distribution. This allows a natural and compact representation of the distribution, eases knowledge acquisition, and supports effective inference algorithms. It is well-known, however, that there are certain independencies that we cannot capture qualitatively within the Bayesian network structure: independencies that hold only in certain contexts, i.e., given a specific assignment of values to certain variables. In this paper, we propose a formal notion of context-specific independence (CSI), based on regularities in the conditional probability tables (CPTs) at a node. We present a technique, analogous to (and based on) d-separation, for determining when such independence holds in a given network. We then focus on a particular qualitative representation scheme---tree-structured CPTs---for capturing CSI. We suggest ways in which this representation can be used to support effective inference algorithms. In particular, we present a structural decomposition of the resulting network which can improve the performance of clustering algorithms, and an alternative algorithm based on cutset conditioning.

Markov processes in Bayesian network computation

International Journal of Electrical and Computer Engineering (IJECE), 2025

The article examines the influence of Markov processes on computations in Bayesian networks (BN), an important area of research within probabilistic graphical models. The concept of Bayesian Markov networks (BMN) is introduced, an extension of traditional Bayesian networks with the addition of a Markov constraint, according to which the probability in a node can only depend on the state of neighboring nodes. This constraint makes the model more realistic for many practical tasks, as most graphical models that reflect real-world processes possess the Markov property. The article also discusses that Bayesian networks, in the absence of evidence, actually exhibit the Markov property. However, when evidence (additional information) is introduced into the model, challenges arise that require more complex computational methods. In response, the article proposes algorithms adapted for working with Bayesian Markov networks in the presence of evidence. These algorithms are aimed at optimizing computations and reducing computational complexity. Additionally, a comparative analysis of calculations in Bayesian networks without Markov constraints and with them is conducted, highlighting the advantages and disadvantages of each approach. Special attention is paid to the practical applications of the proposed methods and their effectiveness in various scenarios.

Mutual conditional independence and its applications to model selection in Markov networks

Annals of Mathematics and Artificial Intelligence, 2020

The fundamental concepts underlying Markov networks are the conditional independence and the set of rules called Markov properties that translate conditional independence constraints into graphs. We introduce the concept of mutual conditional independence in an independent set of a Markov network, and we prove its equivalence to the Markov properties under certain regularity conditions. This extends the notion of similarity between separation in graph and conditional independence in probability to similarity between the mutual separation in graph and the mutual conditional independence in probability. Model selection in graphical models remains a challenging task due to the large search space. We show that mutual conditional independence property can be exploited to reduce the search space. We present a new forward model selection algorithm for graphical log-linear models using mutual conditional independence. We illustrate our algorithm with a real data set example. We show that for sparse models the size of the search space can be reduced from O(n 3) to O(n 2) using our proposed forward selection method rather than the classical forward selection method. We also envision that this property can be leveraged for model selection and inference in different types of graphical models.

Bayesian networks for discrete multivariate data: an algebraic approach to inference

Journal of multivariate analysis, 2003

In this paper we demonstrate how Gro¨bner bases and other algebraic techniques can be used to explore the geometry of the probability space of Bayesian networks with hidden variables. These techniques employ a parametrisation of Bayesian network by moments rather than conditional probabilities. We show that whilst Gro¨bner bases help to explain the local geometry of these spaces a complimentary analysis, modelling the positivity of probabilities, enhances and completes the geometrical picture. We report some recent geometrical results in this area and discuss a possible general methodology for the analyses of such problems.

Probabilistic Inferences in Bayesian Networks

Computing Research Repository, 2010

Bayesian network is a complete model for the variables and their relationships, it can be used to answer probabilistic queries about them. A Bayesian network can thus be considered a mechanism for automatically applying Bayes' theorem to complex problems. In the application of Bayesian networks, most of the work is related to probabilistic inferences. Any variable updating in any node of Bayesian networks might result in the evidence propagation across the Bayesian networks. This paper sums up various inference techniques in Bayesian networks and provide guidance for the algorithm calculation in probabilistic inference in Bayesian networks.

On a simple method for testing independencies in Bayesian networks

Computational Intelligence, 2017

Testing independencies is a fundamental task in reasoning with Bayesian networks (BNs). In practice, d-separation is often used for this task, since it has linear-time complexity. However, many have had difficulties understanding d-separation in BNs. An equivalent method that is easier to understand, called m-separation, transforms the problem from directed separation in BNs into classical separation in undirected graphs. Two main steps of this transformation are pruning the BN and adding undirected edges. In this paper, we propose u-separation as an even simpler method for testing independencies in a BN. Our approach also converts the problem into classical separation in an undirected graph. However, our method is based upon the novel concepts of inaugural variables and rationalization. Thereby, the primary advantage of u-separation over m-separation is that m-separation can prune unnecessarily and add superfluous edges. Our experiment results show that u-separation performs 73% fewer modifications on average than m-separation.

Special issue on PGM�� 04: Second European workshop on probabilistic graphical models 2004

2006

Probabilistic graphical models, such as Bayesian networks and Markov networks, have been around for some time by now, and have seen a remarkable rise in their popularity within the scientific community during the past decade. This community is strikingly broad and includes computer scientists, statisticians, mathematicians, physicists, and, to an increasing extent, researchers from various fields of application, such as psychology, biomedicine and finance. It is not surprising that these researchers have developed a need for having their own specialised meetings to discuss progress in their area of research.

A transformational characterization of equivalent Bayesian network structures

UAI'95, 1995

We present a simple characterization of equivalent B a yesian network structures based on local transformations. The signi cance of the characterization is twofold. First, we are able to easily prove s e v eral new invariant properties of theoretical interest for equivalent structures. Second, we use the characterization to derive an e cient algorithm that identi es all of the compelled edges in a structure. Compelled edge identication is of particular importance for learning Bayesian network structures from data because these edges indicate causal relationships when certain assumptions hold.