Inference in Hybrid Bayesian Networks with Nonlinear Deterministic Conditionals.

This is the peer reviewed version of the following article: Cobb, B. R. and Shenoy, P. P. (2017), Inference in Hybrid Bayesian Networks with Nonlinear Deterministic Conditionals. Int. J. Intell. Syst., 32: 1217–1246. doi:10.1002/int.21897, which has been published in final form at https://doi.org/10.1002/int.21897. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.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 math formula, math formula, math formula, and math formula. Finally, we solve two small examples of hybrid BNs containing nonlinear deterministic conditionals that arise in practice

A Decision Analysis Approach To Solving the Signaling Game

Decision analysis has traditionally been applied to choices under uncertainty involving a single decision maker. Game theory has been applied to solving games of strategic interaction between two or more players. Building upon recent work of van Binsbergen and Marx (2007. Exploring relations between decision analysis and game theory. Decision Anal. 4(1) 32–40.), this paper defines a modified decision-theoretic approach to solving games of strategic interaction between two players. Using this method, the choices of the two players are modeled with separate decision trees comprised entirely of chance nodes. Optimal policies are reflected in the probabilities in the decision trees of each player. In many cases, the optimal strategy for each player can be obtained by rolling back the opponent’s decision tree. Results are demonstrated for the multi-stage signaling game, which is difficult to model using decision nodes to represent strategies,as in the approach of van Binsbergen and Marx

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

On Transforming Belief Function Models to Probability Models

In response to reviewer comments on this paper, we have written a shorter and more focused paper: "On the Plausibility Transformation Method for Translating Belief Function Models to Probability Models," University of Kansas School of Business Working Paper No. 308, June 2004, Lawrence, KS.In this paper, we explore methods for transforming a belief function model to an equivalent probability model. We propose and define the properties of a method called the plausibility transformation method. We compare the plausibility transformation method with the pignistic transformation method. These two methods yield qualitatively different probability models. We argue that the plausibility transformation method is the correct method that maintains belief function semantics

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 (CDF’s). These methods allow inference in networks with linear deterministic variables and non-Gaussian distributions

Hybrid Influence Diagrams Using Mixtures of Truncated Exponentials

This is a short 9-pp version of a longer un-published working paper titled "Decision Making with Hybrid Influence Diagrams Using Mixtures of Truncated Exponentials," School of Business Working Paper No. 304, May 2004, Lawrence, KS.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.Partially supported by a graduate research assistantship to Barry R. Cobb from the Ronald G. Harper Professorship, and by a contract from Sparta, Inc., to Prakash P. Shenoy

Inference in Hybrid Bayesian Networks with Deterministic Variables

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