The prediction of posttraumatic stress disorder (PTSD) has gained a lot of interest in clinical studies. Identifying patients with a high risk of PTSD can guide mental healthcare workers when making treatment decisions. The main goal of this paper is to propose several Bayesian network (BN) models to assess the probability that a veteran has PTSD when first visiting a U.S. Department of Veteran Affairs (VA) facility seeking medical care. The current practice is to use a five-question test called PC-PTSD-5. We aim to use the PC-PTSD-5 test, which is currently administered to most incoming new patients, and demographic information, military service history, and medical history. We construct a Bayes information criterion score-based BN, a group L2-regularized BN (GL2-regularized BN), and a naïve Bayes BN to assess the probability that a patient has PTSD. The GL2-regularized BN is a new method for constructing a BN motivated by some of the challenges of analyzing this data set. A secondary goal is to identify which features are important in predicting PTSD. We discover that the following features help compute the probability of PTSD: PC-PTSD-5, service-connected flag, combat flag, agent orange flag, military sexual trauma flag, traumatic brain injury, and age. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0174 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0174 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The primary goal is to define conditional belief functions in the Dempster-Shafer theory. We do so similarly to probability theory's notion of conditional probability tables. Conditional belief functions are necessary for constructing directed graphical belief function models in the same sense as conditional probability tables are necessary for constructing Bayesian networks. We provide examples of conditional belief functions, including those obtained by Smets' conditional embedding. Besides defining conditional belief functions, we state and prove a few basic properties of conditionals. In the belief-function literature, conditionals are defined starting from a joint belief function. Conditionals are then defined using the removal operator, an inverse of Dempster's combination operator. When such conditionals are well-defined belief functions, we show that our definition is equivalent to these definitions.

In 2018, Jiroušek and Shenoy proposed a definition of entropy for Dempster-Shafer (D-S) belief functions called decomposable entropy (d-entropy). This paper provides an algorithm for computing the d-entropy of directed graphical D-S belief function models. We illustrate the algorithm using Almond's Captain's Problem example. For belief function undirected graphical models, assuming that the set of belief functions in the model is non-informative, the belief functions are distinct. We illustrate this using Haenni-Lehmann's Communication Network problem. As the joint belief function for this model is quasi-consonant, it follows from a property of d-entropy that the d-entropy of this model is zero, and no algorithm is required. For a class of undirected graphical models, we provide an algorithm for computing the d-entropy of such models. Finally, the d-entropy coincides with Shannon's entropy for the probability mass function of a single random variable and for a large multi-dimensional probability distribution expressed as a directed acyclic graph model called a Bayesian network. We illustrate this using Lauritzen-Spiegelhalter's Chest Clinic example represented as a belief-function directed graphical model.

How do you make inferences from a Bayesian network (BN) model with missing information? For example, we may not have priors for some variables or may not have conditionals for some states of the parent variables. It is well-known that the Dempster-Shafer (D-S) belief function theory is a generalization of probability theory. So, a solution is to embed an incomplete BN model in a D-S belief function model, omit the missing data, and then make inferences from the belief function model. We will demonstrate this using an implementation of a local computation algorithm for D-S belief function models called the “Belief function machine.” One advantage of this approach is that we get interval estimates of the probabilities of interest. Using Laplacian (equally likely) or maximum entropy priors or conditionals for missing data in a BN may lead to point estimates for the probabilities of interest, masking the uncertainty in these estimates. Bayesian reasoning is unable to reason from an incomplete model. A Bayesian sensitivity analysis of the missing parameters is not a substitute for a belief-function analysis.

Realistic decision-making often occurs with insufficient time to gather all possible evidence before a decision must be rendered, requiring efficient processes for prioritizing between candidate action sequences. The proposed Judicial Evidential Reasoning framework encodes decision-maker questions as rigorously testable hypotheses and proposes actions to resolve the hypotheses in the face of ambiguous, incomplete, and uncertain evidence. Dempster-Shafer theory is applied to model hypothesis knowledge and quantify ambiguity, and an equal-effort heuristic is proposed time-efficiency and impartiality to combat confirmation bias. This work includes derivation of the generalized formulation, computational tractability considerations for improved performance, several illustrative examples, and sample application to a space situational awareness sensor network tasking scenario. The results show strong hypothesis resolution and robustness to fixation due to poor prior evidence.

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 , , , and . Finally, we solve two small examples of hybrid BNs containing nonlinear deterministic conditionals that arise in practice.

In this paper, we discuss some practical issues that arise in solving hybrid Bayesian networks that include deterministic conditionals for continuous variables. We show how exact inference can become intractable even for small networks due to the difficulty in handling deterministic conditionals (for continuous variables). We propose some strategies for carrying out the inference task using mixtures of polynomials (MOPs) and mixtures of truncated exponentials. MOPs can be defined on hypercubes or hyperrhombuses. We compare these two methods. A key strategy is to reapproximate large potentials with potentials consisting of fewer pieces and lower degrees/number of terms. We discuss several methods for reapproximating potentials. We illustrate our methods in a practical application consisting of solving a stochastic program evaluation and review technique (PERT) network.

In this paper we analyze the use of hybrid Bayesian networks in domains that include deterministic conditionals for continuous variables. We show how exact inference can become infeasible even for small networks, due to the difficulty in handling functional relationships. We compare two strategies for carrying out the inference task, using mixtures of polynomials (MOPs) and mixtures of truncated exponentials (MTEs).

This paper proposes a linear belief function (LBF) approach to evaluate portfolio performance. By drawing on the notion of LBFs, an elementary approach to knowledge representation in expert systems is proposed. It is shown how to use basic matrices to represent market information and financial knowledge, including complete ignorance, statistical observations, subjective speculations, distributional assumptions, linear relations, and empirical asset-pricing models. The authors then appeal to Dempster's rule of combination to integrate the knowledge for assessing the overall belief of portfolio performance and updating the belief by incorporating additional evidence. An example of three gold stocks is used to illustrate the approach.

We describe a simple method for propagating belief functions in AND-trees. We exploit the properties of AND-trees to make our method simpler than the general method discussed by Shenoy and Shafer, and Dempster and Kong. We illustrate our method for aggregation of evidence in a financial audit. © 1995 John Wiley & Sons, Inc.

Valuation networks have been proposed as graphical representations of valuation-based systems. The axiomatic framework of valuation-based systems is able to capture many uncertainty calculi including probability theory, Dempster-Shafer's belief-function theory, Spohn's epistemic belief theory, and Zadeh's possibility theory. In this paper, we show how valuation networks encode conditional independence relations. For the probabilistic case, the class of probability models encoded by valuation networks includes undirected graph models, directed acyclic graph models, directed balloon graph models, and recursive causal graph models.

This paper has three main results. First, we present a new computational technique for checking for inconsistencies in valuation-based systems. This technique is different from the implicit enumeration method of Davis-Putnam and its variants. Our technique uses the divide-and-conquer method of dynamic programming. The computational complexity of this technique depends on the sizes of the valuations and on the graphical structure of the valuation-based system. Second, if a valuation-based system is consistent, we describe a method for generating a model for the system, i.e., an assignment of values for each variable that is consistent with each valuation in the system. Third, if a valuation-based system is inconsistent, we describe a method for isolating a minimal inconsistent set of valuations. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.