Barry R. Cobb

On the plausibility transformation method for translating belief function models to probability models

In this paper, we propose the plausibility transformation method for translating Dempster-Shafer (D-S) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to probability models. We argue that the plausibility transformation method produces probability models that are consistent with D-S semantics of belief function models, and that, in some examples, the pignistic transformation method produces results that appear to be inconsistent with Dempsters rule of combination.

A Comparison of Bayesian and Belief Function Reasoning

The goal of this paper is to compare the similarities and differences between Bayesian and belief function reasoning. Our main conclusion is that although there are obvious differences in semantics, representations, and the rules for combining and marginalizing representations, there are many similarities. We claim that the two calculi have roughly the same expressive power. Each calculus has its own semantics that allow us to construct models suited for these semantics. Once we have a model in either calculus, one can transform it to the other by means of a suitable transformation.

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.

Approximating probability density functions in hybrid Bayesian networks with mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by 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 standard PDF’s and applications of these potentials for solving inference problems in hybrid Bayesian networks. These approximations will extend the types of inference problems that can be modelled with Bayesian networks, as demonstrated using three examples.

A Comparison of Methods for Transforming Belief Function Models to Probability Models

Recently, we proposed a new method called the plausibility transformation method to convert a belief function model to an equivalent probability model. In this paper, we compare the plausibility transformation method with the pignistic transformation method. The two transformation methods yield qualitatively different probability models. We argue that the plausibility transformation method is the correct method for translating a belief function model to an equivalent probability model that maintains belief function semantics.

REAL OPTIONS VOLATILITY ESTIMATION WITH CORRELATED INPUTS

Real Options Analysis (ROA) provides a framework for valuing reactive and proactive managerial flexibility in investment decisions. Estimating the volatility parameter for a real options model is challenging because there are typically no historical returns for the underlying asset and no current market prices. A previously developed method of using simulation to estimate the volatility parameter for a real investment is demonstrated. The effects of serial price correlation and price-demand cross-correlation on volatility parameters developed with this method are explained. Finally, managerial implications of these findings are discussed.

Decision making with hybrid influence diagrams using mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for representing continuous chance variables in influence diagrams. Also, MTE potentials can be used to approximate utility functions. This paper introduces MTE influence diagrams, which can represent decision problems without restrictions on the relationships between continuous and discrete chance variables, without limitations on the distributions of continuous chance variables, and without limitations on the nature of the utility functions. In MTE influence diagrams, all probability distributions and the joint utility function (or its multiplicative factors) are represented by MTE potentials and decision nodes are assumed to have discrete state spaces. MTE influence diagrams are solved by variable elimination using a fusion algorithm.

Nonlinear deterministic relationships in bayesian networks

In a Bayesian network with continuous variables containing a variable(s) that is a conditionally deterministic function of its continuous parents, the joint density function does not exist. Conditional linear Gaussian distributions can handle such cases when the deterministic function is linear and the continuous variables have a multi-variate normal distribution. In this paper, operations required for performing inference with nonlinear conditionally deterministic variables are developed. We perform inference in networks with nonlinear deterministic variables and non-Gaussian continuous variables by using piecewise linear approximations to nonlinear functions and modeling probability distributions with mixtures of truncated exponentials (MTE) potentials.

Bayesian Network Models with Discrete and Continuous Variables

Bayesian networks are powerful tools for handling problems which are specified through a multivariate probability distribution. A broad background of theory and methods have been developed for the case in which all the variables are discrete. However, situations in which continuous and discrete variables coexist in the same problem are common in practice. In such cases, usually the continuous variables are discretized and therefore all the existing methods for discrete variables can be applied, but the price to pay is that the obtained model is just an approximation. In this chapter we study two frameworks where continuous and discrete variables can be handled simultaneously without using discretization. These models are based on the CG and MTE distributions.

Operations for inference in continuous Bayesian networks with linear deterministic variables

An important class of continuous Bayesian networks are those that have linear conditionally deterministic variables (a variable that is a linear deterministic function of its parents). In this case, the joint density function for the variables in the network does not exist. Conditional linear Gaussian (CLG) distributions can handle such cases when all variables are normally distributed. In this paper, we develop operations required for performing inference with linear conditionally deterministic variables in continuous Bayesian networks using relationships derived from joint cumulative distribution functions. These methods allow inference in networks with linear deterministic variables and non-Gaussian distributions.