Inference In Hybrid Bayesian Networks Research Articles

Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks

Hybrid Bayesian Networks (HBNs), which contain both discrete and continuous variables, arise naturally in many application areas (e.g., image understanding, data fusion, medical diagnosis, fraud detection). This paper concerns inference in an important subclass of HBNs, the conditional Gaussian (CG) networks, in which all continuous random variables have Gaussian distributions and all children of continuous random variables must be continuous. Inference in CG networks can be NP-hard even for special-case structures, such as poly-trees, where inference in discrete Bayesian networks can be performed in polynomial time. Therefore, approximate inference is required. In approximate inference, it is often necessary to trade off accuracy against solution time. This paper presents an extension to the Hybrid Message Passing inference algorithm for general CG networks and an algorithm for optimizing its accuracy given a bound on computation time. The extended algorithm uses Gaussian mixture reduction to prevent an exponential increase in the number of Gaussian mixture components. The trade-off algorithm performs pre-processing to find optimal run-time settings for the extended algorithm. Experimental results for four CG networks compare performance of the extended algorithm with existing algorithms and show the optimal settings for these CG networks.

A Review of Inference Algorithms for Hybrid Bayesian Networks

Hybrid Bayesian networks have received an increasing attention during the last years. The difference with respect to standard Bayesian networks is that they can host discrete and continuous variables simultaneously, which extends the applicability of the Bayesian network framework in general. However, this extra feature also comes at a cost: inference in these types of models is computationally more challenging and the underlying models and updating procedures may not even support closed-form solutions. In this paper we provide an overview of the main trends and principled approaches for performing inference in hybrid Bayesian networks. The methods covered in the paper are organized and discussed according to their methodological basis. We consider how the methods have been extended and adapted to also include (hybrid) dynamic Bayesian networks, and we end with an overview of established software systems supporting inference in these types of models.

Inference in Hybrid Bayesian Networks with Nonlinear Deterministic Conditionals

To enable inference in hybrid Bayesian networks (BNs) containing nonlinear deterministic conditional distributions, Cobb and Shenoy in 2005 propose approximating nonlinear deterministic functions by piecewise linear (PL) ones. In this paper, we describe a method for finding PL approximations of nonlinear functions based on a penalized mean square error (MSE) heuristic, which consists of minimizing a penalized MSE function subject to two principles, domain and symmetry. We illustrate our method for some commonly used one-dimensional and two-dimensional nonlinear deterministic functions such as W=X2, W=eX, W=X·Y, and W=X/Y. Finally, we solve two small examples of hybrid BNs containing nonlinear deterministic conditionals that arise in practice.

Inference in hybrid Bayesian networks with large discrete and continuous domains

Inference in Bayesian networks with large domain of discrete variables requires significant computational effort. In order to reduce the computational effort, current approaches often assume that discrete variables have some bounded number of values or are represented at an appropriate size of clusters. In this paper, we introduce decision-tree structured conditional probability representations that can efficiently handle a large domain of discrete and continuous variables. These representations can partition the large number of values into some reasonable number of clusters and lead to more robust parameter estimation. Very rapid computation and ability to treat both discrete and continuous variables are accomplished via modified belief propagation algorithm. Being able to compute various types of reasoning from a single Bayesian network eliminates development and maintenance issues associated with the use of distinct models for different types of reasoning. Application to real-world steel production process data is presented.

Parameter learning in hybrid Bayesian networks using prior knowledge

Mixtures of truncated basis functions have been recently proposed as a generalisation of mixtures of truncated exponentials and mixtures of polynomials for modelling univariate and conditional distributions in hybrid Bayesian networks. In this paper we analyse the problem of learning the parameters of marginal and conditional MoTBF densities when both prior knowledge and data are available. Incorporating prior knowledge provide a valuable tool for obtaining useful models, especially in domains of applications where data are costly or scarce, and prior knowledge is available from practitioners. We explore scenarios where the prior knowledge can be expressed as an MoTBF density that is afterwards combined with another MoTBF density estimated from the available data. The resulting model remains within the MoTBF class which is a convenient property from the point of view of inference in hybrid Bayesian networks. The performance of the proposed method is tested in a series of experiments carried out over synthetic and real data.

Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z=X+Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method.

Extended Shenoy–Shafer architecture for inference in hybrid bayesian networks with deterministic conditionals

The main goal of this paper is to describe an architecture for solving large general hybrid Bayesian networks (BNs) with deterministic conditionals for continuous variables using local computation. In the presence of deterministic conditionals for continuous variables, we have to deal with the non-existence of the joint density function for the continuous variables. We represent deterministic conditional distributions for continuous variables using Dirac delta functions. Using the properties of Dirac delta functions, we can deal with a large class of deterministic functions. The architecture we develop is an extension of the Shenoy–Shafer architecture for discrete BNs. We extend the definitions of potentials to include conditional probability density functions and deterministic conditionals for continuous variables. We keep track of the units of continuous potentials. Inference in hybrid BNs is then done in the same way as in discrete BNs but by using discrete and continuous potentials and the extended definitions of combination and marginalization. We describe several small examples to illustrate our architecture. In addition, we solve exactly an extended version of the crop problem that includes non-conditional linear Gaussian distributions and non-linear deterministic functions.

Inference in hybrid Bayesian networks using mixtures of polynomials

The main goal of this paper is to describe inference in hybrid Bayesian networks (BNs) using mixture of polynomials (MOP) approximations of probability density functions (PDFs). Hybrid BNs contain a mix of discrete, continuous, and conditionally deterministic random variables. The conditionals for continuous variables are typically described by conditional PDFs. A major hurdle in making inference in hybrid BNs is marginalization of continuous variables, which involves integrating combinations of conditional PDFs. In this paper, we suggest the use of MOP approximations of PDFs, which are similar in spirit to using mixtures of truncated exponentials (MTEs) approximations. MOP functions can be easily integrated, and are closed under combination and marginalization. This enables us to propagate MOP potentials in the extended Shenoy–Shafer architecture for inference in hybrid BNs that can include deterministic variables. MOP approximations have several advantages over MTE approximations of PDFs. They are easier to find, even for multi-dimensional conditional PDFs, and are applicable for a larger class of deterministic functions in hybrid BNs.

Inference in hybrid Bayesian networks

Since the 1980s, Bayesian networks (BNs) have become increasingly popular for building statistical models of complex systems. This is particularly true for boolean systems, where BNs often prove to be a more efficient modelling framework than traditional reliability techniques (like fault trees and reliability block diagrams). However, limitations in the BNs’ calculation engine have prevented BNs from becoming equally popular for domains containing mixtures of both discrete and continuous variables (the so-called hybrid domains). In this paper we focus on these difficulties, and summarize some of the last decade's research on inference in hybrid Bayesian networks. The discussions are linked to an example model for estimating human reliability.

Inference in hybrid Bayesian networks using dynamic discretization

We consider approximate inference in hybrid Bayesian Networks (BNs) and present a new iterative algorithm that efficiently combines dynamic discretization with robust propagation algorithms on junction trees. Our approach offers a significant extension to Bayesian Network theory and practice by offering a flexible way of modeling continuous nodes in BNs conditioned on complex configurations of evidence and intermixed with discrete nodes as both parents and children of continuous nodes. Our algorithm is implemented in a commercial Bayesian Network software package, AgenaRisk, which allows model construction and testing to be carried out easily. The results from the empirical trials clearly show how our software can deal effectively with different type of hybrid models containing elements of expert judgment as well as statistical inference. In particular, we show how the rapid convergence of the algorithm towards zones of high probability density, make robust inference analysis possible even in situations where, due to the lack of information in both prior and data, robust sampling becomes unfeasible.

Inference in hybrid Bayesian networks with mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated with an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the Shenoy–Shafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate an arbitrary normal PDF with any mean and a positive variance. The properties of these MTE potentials are presented, along with examples that demonstrate their use in solving hybrid Bayesian networks. Assuming that the joint density exists, MTE potentials can be used for inference in hybrid Bayesian networks that do not fit the restrictive assumptions of the conditional linear Gaussian (CLG) model, such as networks containing discrete nodes with continuous parents.