Hybrid Bayesian networks with linear deterministic variables

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Hybrid Bayesian networks with linear deterministic variables

Proceedings Article•

Hybrid Bayesian networks with linear deterministic variables

Barry R. Cobb¹, Prakash P. Shenoy²•Institutions (2)

Virginia Military Institute¹, University of Kansas²

26 Jul 2005-pp 136-144

TL;DR: In this paper, operations required for performing inference with conditionally deterministic variables in hybrid Bayesian networks are developed and these methods allow inference in networks with Deterministic variables where continuous variables may be non-Gaussian, and their density functions can be approximated by mixtures of truncated exponentials.

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Abstract: When a hybrid Bayesian network has conditionally deterministic variables with continuous parents, the joint density function for the continuous variables does not exist. Conditional linear Gaussian distributions can handle such cases when the continuous variables have a multi-variate normal distribution and the discrete variables do not have continuous parents. In this paper, operations required for performing inference with conditionally deterministic variables in hybrid Bayesian networks are developed. These methods allow inference in networks with deterministic variables where continuous variables may be non-Gaussian, and their density functions can be approximated by mixtures of truncated exponentials. There are no constraints on the placement of continuous and discrete nodes in the network.

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Journal Article•DOI•

Inference in hybrid Bayesian networks using mixtures of polynomials

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Prakash P. Shenoy¹, James C. West¹•Institutions (1)

University of Kansas¹

01 Jul 2011-International Journal of Approximate Reasoning

TL;DR: 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), which are similar in spirit to using mixtures of truncated exponentials (MTEs) approxims.

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126 citations

Cites background or methods from "Hybrid Bayesian networks with linea..."

...Cobb and Shenoy [2006] and Cobb et al. [2006] propose using a non-linear optimization technique for finding MTE approximations for several commonly used one-dimensional distributions. Cobb and Shenoy [2005a, b] extend this approach to BNs with linear and non-linear deterministic variables. In the latter case, they approximate non-linear deterministic functions by piecewise linear ones. Rumi and Salmeron [2007] describe approximate probability propagation with MTE approximations that have only two exponential terms in each piece. Romero et al. [2006] describe learning MTE potentials from data, and Langseth et al....
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...Cobb and Shenoy [2005b] propose approximating non-linear deterministic functions by piecewise linear deterministic functions, and then using MTEs....
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...Cobb and Shenoy [2006] and Cobb et al. [2006] propose using a non-linear optimization technique for finding MTE approximations for several commonly used one-dimensional distributions....
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...Cobb and Shenoy [2006] and Cobb et al. [2006] propose using a non-linear optimization technique for finding MTE approximations for several commonly used one-dimensional distributions. Cobb and Shenoy [2005a, b] extend this approach to BNs with linear and non-linear deterministic variables. In the latter case, they approximate non-linear deterministic functions by piecewise linear ones. Rumi and Salmeron [2007] describe approximate probability propagation with MTE approximations that have only two exponential terms in each piece....
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...Cobb and Shenoy [2006] and Cobb et al. [2006] propose using a non-linear optimization technique for finding MTE approximations for several commonly used one-dimensional distributions. Cobb and Shenoy [2005a, b] extend this approach to BNs with linear and non-linear deterministic variables. In the latter case, they approximate non-linear deterministic functions by piecewise linear ones. Rumi and Salmeron [2007] describe approximate probability propagation with MTE approximations that have only two exponential terms in each piece. Romero et al. [2006] describe learning MTE potentials from data, and Langseth et al. [2010] investigate the use of MTE approximations where the coefficients are restricted to integers....
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Inference in hybrid Bayesian networks wih mixtures of truncated exponenials

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Barry R. Cobb, Prakash P. Shenoy

01 Apr 2006

TL;DR: MTE potentials are presented that approximate an arbitrary normal PDF with any mean and a positive variance, and can be used for inference in hybrid Bayesian networks that do not fit the restrictive assumptions of the conditional linear Gaussian model.

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Abstract: 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.

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80 citations

Cites methods from "Hybrid Bayesian networks with linea..."

...MTE methods can be used when continuous variables have non-Gaussian distributions [5] and the deterministic functions are non-linear [4]....
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Journal Article•DOI•

Inference in hybrid Bayesian networks with mixtures of truncated exponentials

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Barry R. Cobb¹, Prakash P. Shenoy²•Institutions (2)

Virginia Military Institute¹, University of Kansas²

01 Apr 2006-International Journal of Approximate Reasoning

TL;DR: In this paper, a mixture of truncated exponentials (MTE) potentials is proposed to approximate an arbitrary normal PDF with any mean and a positive variance, which can be used for inference in hybrid Bayesian networks.

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74 citations

Proceedings Article•

Inference in hybrid Bayesian networks using mixtures of Gaussians

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Prakash P. Shenoy¹•Institutions (1)

University of Kansas¹

13 Jul 2006

TL;DR: In this article, the authors describe a method for exact inference in general hybrid Bayesian networks (BNs) with a mixture of discrete and continuous chance variables, which consists of approximating general hybrid BNs by a mix of Gaussians (MoG) BNs.

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Abstract: The main goal of this paper is to describe a method for exact inference in general hybrid Bayesian networks (BNs) (with a mixture of discrete and continuous chance variables). Our method consists of approximating general hybrid Bayesian networks by a mixture of Gaussians (MoG) BNs. There exists a fast algorithm by Lauritzen-Jensen (LJ) for making exact inferences in MoG Bayesian networks, and there exists a commercial implementation of this algorithm. However, this algorithm can only be used for MoG BNs. Some limitations of such networks are as follows. All continuous chance variables must have conditional linear Gaussian distributions, and discrete chance nodes cannot have continuous parents. The methods described in this paper will enable us to use the LJ algorithm for a bigger class of hybrid Bayesian networks. This includes networks with continuous chance nodes with non-Gaussian distributions, networks with no restrictions on the topology of discrete and continuous variables, networks with conditionally deterministic variables that are a nonlinear function of their continuous parents, and networks with continuous chance variables whose variances are functions of their parents.

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38 citations

Journal Article•DOI•

A review of representation issues and modeling challenges with influence diagrams

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Concha Bielza¹, Manuel Herrera Gómez², Prakash P. Shenoy³•Institutions (3)

Technical University of Madrid¹, University of Granada², University of Kansas³

01 Jun 2011-Omega-international Journal of Management Science

TL;DR: This paper looks at the representation of asymmetric decision problems including conditional distribution trees, sequential decision diagrams, and sequential valuation networks and the use of continuous chance and decision variables, including continuous conditionally deterministic variables.

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Abstract: Since their introduction in the mid 1970s, influence diagrams have become a de facto standard for representing Bayesian decision problems. The need to represent complex problems has led to extensions of the influence diagram methodology designed to increase the ability to represent complex problems. In this paper, we review the representation issues and modeling challenges associated with influence diagrams. In particular, we look at the representation of asymmetric decision problems including conditional distribution trees, sequential decision diagrams, and sequential valuation networks. We also examine the issue of representing the sequence of decision and chance variables, and how it is done in unconstrained influence diagrams, sequential valuation networks, and sequential influence diagrams. We also discuss the use of continuous chance and decision variables, including continuous conditionally deterministic variables. Finally, we discuss some of the modeling challenges faced in representing decision problems in practice and some software that is currently available.

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38 citations

Cites background or methods from "Hybrid Bayesian networks with linea..."

...For non-linear deterministic functions, Cobb and Shenoy [15] suggest approximating a non-linear deterministic function by a piecewise linear function, and then using the technique proposed in [14]....
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...The class of MTE functions is also closed under transformations required by linear deterministic functions [14], but not for non-linear deterministic functions....
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References

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Axioms for Probability and Belief-Function Propagation.

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Prakash P. Shenoy¹, Glenn Shafer¹•Institutions (1)

University of Kansas¹

01 Jan 2008

TL;DR: This paper describes an abstract framework and axioms under which exact local computation of marginals is possible and shows how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.

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Abstract: In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.

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579 citations

"Hybrid Bayesian networks with linea..." refers methods in this paper

...Cobb and Shenoy [2005a] demonstrate that the operations used for propagation satisfy the Shenoy-Shafer axioms [Shenoy and Shafer 1990] for local propagation and maintain density potentials that are within the class of MTE density potentials....
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Book Chapter•DOI•

Axioms for probability and belief-function propagation

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Prakash P. Shenoy¹, Glenn Shafer¹•Institutions (1)

University of Kansas¹

01 Jun 1990

TL;DR: In this article, an abstract framework and axioms under which exact local computation of marginals is possible are presented. But the primitive objects of the framework are variables and valuations.

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521 citations

Journal Article•DOI•

Stable local computation with conditional Gaussian distributions

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Steffen L. Lauritzen¹, Frank Jensen¹•Institutions (1)

Aalborg University¹

01 Apr 2001-Statistics and Computing

TL;DR: A propagation scheme for Bayesian networks with conditional Gaussian distributions that does not have the numerical weaknesses of the scheme derived in Lauritzen and Spiegelhalter is described.

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Abstract: This article describes a propagation scheme for Bayesian networks with conditional Gaussian distributions that does not have the numerical weaknesses of the scheme derived in Lauritzen (Journal of the American Statistical Association 87: 1098–1108, 1992). The propagation architecture is that of Lauritzen and Spiegelhalter (Journal of the Royal Statistical Society, Series B 50: 157– 224, 1988). In addition to the means and variances provided by the previous algorithm, the new propagation scheme yields full local marginal distributions. The new scheme also handles linear deterministic relationships between continuous variables in the network specification. The computations involved in the new propagation scheme are simpler than those in the previous scheme and the method has been implemented in the most recent version of the HUGIN software.

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285 citations

"Hybrid Bayesian networks with linea..." refers background in this paper

...Conditional linear Gaussian (CLG) models [Lauritzen and Jensen 2001] can handle such cases when continuous variables have a multi-variate Gaussian distribution and discrete nodes do not have continuous parents....
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Journal Article•DOI•

LAZY propagation: a junction tree inference algorithm based on lazy evaluation

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Anders L. Madsen¹, Finn Verner Jensen¹•Institutions (1)

Aalborg University¹

01 Sep 1999-Artificial Intelligence

TL;DR: A junction tree based inference architecture exploiting the structure of the original Bayesian network and independence relations induced by evidence to improve the efficiency of inference is presented.

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240 citations

"Hybrid Bayesian networks with linea..." refers methods in this paper

...According to the LAZY propagation scheme [Madsen and Jensen 1999] the parts of the mixed potentials are not combined, but rather maintained as a decomposed set of potentials during combination....
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Book Chapter•DOI•

Mixtures of Truncated Exponentials in Hybrid Bayesian Networks

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Serafín Moral¹, Rafael Rumí², Antonio Salmerón²•Institutions (2)

University of Granada¹, University of Almería²

19 Sep 2001

TL;DR: The properties of the MTE distribution are studied and it is shown how exact probability propagation can be carried out by means of a local computation algorithm.

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Abstract: In this paper we propose the use of mixtures of truncated exponential (MTE) distributions in hybrid Bayesian networks. We study the properties of the MTE distribution and show how exact probability propagation can be carried out by means of a local computation algorithm. One feature of this model is that no restriction is made about the order among the variables either discrete or continuous. Computations are performed over a representation of probabilistic potentials based on probability trees, expanded to allow discrete and continuous variables simultaneously. Finally, a Markov chain Monte Carlo algorithm is described with the aim of dealing with complex networks.

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235 citations