The value of information to decision makers: an experimental approach using card-based decision gaming

This paper describes a novel experimental method for determining the value of different types of information to military decision makers. The experimental method used a simple scenario and a set of serials constructed from cards, each presenting a single piece of information and presented sequentially. Each of a number of pairs of players were taken through the scenario and asked to judge when they would make each of a pair of escalating responses to the situation. The data proved well suited to analysis using a probit model and is consistent with the hypothesis of a Bayesian decision mechanism with normally distributed ‘action points’. The methodology allowed the determination of weights for each of a number of different classes of information, together with estimates of the human and situational elements of variation, including estimates of the ‘prior belief’ of the different pairs of players.     

Rough sets,decision algorithms and Bayes' theorem

Rough set-based data analysis starts from a data table, called an information system. The information system contains data about objects of interest characterized in terms of some attributes. Often we distinguish in the information system condition and decision attributes. Such information system is called a decision table. The decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table. With every decision table a set of decision rules, called a decision algorithm, can be associated. It is shown that every decision algorithm reveals some well-known probabilistic properties, in particular it satisfies the total probability theorem and Bayes' theorem. These properties give a new method of drawing conclusions from data, without referring to prior and posterior probabilities, inherently associated with Bayesian reasoning.     

A Categorical Foundation for Bayesian Probability

Building on the work of Lawvere and others, we develop a categorical framework for Bayesian probability. This foundation will then allow for Bayesian representations of uncertainty to be integrated into other categorical modeling applications. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the standard setting of Polish spaces. This more general setting is advantageous, as it allows for non-trivial decision rules (Eilenberg–Moore algebras) on finite (as well as non finite) spaces. In this way, we obtain a common framework for decision theory and Bayesian probability.     

Expectation-maximization for Bayes-adaptive POMDPs

Bayes-adaptive POMDPs (BAPOMDPs) are partially observable Markov decision problems in which uncertainty in the state-transition and observation-emission probabilities can be captured by a prior distribution over the model parameters. Existing approaches to solving BAPOMDPs rely on model and trajectory sampling to guide exploration and, because of the curse of dimensionality, do not scale well when the degree of model uncertainty is large. In this paper, we begin by presenting two expectation-maximization (EM) approaches to solving BAPOMPs via finite-state controller (FSC) optimization, which at their foundation are extensions of existing EM algorithms for BAMDPs to the more general BAPOMDP setting. The first is a sampling-based EM algorithm that optimizes over a finite number of models drawn from the BAPOMDP prior, and as such is only appropriate for smaller problems with limited model uncertainty; the second approach leverages variational Bayesian methods to ensure tractability without sampling, and is most appropriate for larger domains with greater model uncertainty. Our primary novel contribution is the derivation of the constrained VB-EM algorithm, which addresses an unfavourable preference that often arises towards a certain class of policies when applying the standard VB-EM algorithm. Through an empirical study we show that the sampling-based EM algorithm is competitive with more conventional sampling-based approaches in smaller domains, and that our novel constrained VB-EM algorithm can generate quality solutions in larger domains where sampling-based approaches are no longer viable.     

Devolving command decisions in complex operations

In contemporary military endeavours, Command and Control (C2) arrangements generally aim to ensure an appropriate regulation of command-decision autonomy such that decision makers are able to act in a way that is consistent with the overall set of commanders’ intents and according to the nature of the unfolding situation. This can be a challenge, especially in situations with increasing degrees of uncertainty, ambiguity and complexity, also where individual commanders are faced with conflicting objectives. Increasingly, it seems that command decisions are being taken under conditions of internal command contention; for example, when the likely successful outcome of a tactical mission can often be at odds with the overall strategic and political aims of the campaign. The work in the paper builds on our previous research in decision making under uncertainty and conflicting objectives, where we analysed the responses of military commanders in decision experiments. We demonstrated how multi-attribute utility theory could be used to represent and understand the effects of uncertainty and conflicting objectives on a particular commander's choices. In this paper, we further develop and generalize the theory to show that the geometrical forms of expected utilities, which arise from the assumption of commander rationality, are qualitatively stable in a wide range of scenarios. This opens out into further analysis linking to Catastrophe Theory as it relates to C2 regulatory frameworks for devolving command decision freedoms. We demonstrate how an appreciation of this geometry can aid understanding of the relationship between socially complex operational environments and the prevailing C2, which can also inform selection and training of personnel, to address issues of devolving command decision-rights, as appropriate for the endeavour as a whole. The theory presented in the paper, therefore, provides a means to explore and gain insight into different approaches to regulation of C2 decision making aimed ultimately at achieving C2 agility, or at least at a conceptual language to allow its formal representation. C2 regulatory agents are discussed in terms of detailed functions for moderating command decision making, as appropriate for the degrees of uncertainty and goal contention being faced. The work also begins to address implications of any lack of experience and any differences in personality-type of the individual commanders with respect to risk-taking, open-mindedness and creativity.     

Bayesian rough set model: A further investigation

Bayesian rough set model (BRSM), as the hybrid development between rough set theory and Bayesian reasoning, can deal with many practical problems which could not be effectively handled by original rough set model. In this paper, the equivalence between two kinds of current attribute reduction models in BRSM for binary decision problems is proved. Furthermore, binary decision problems are extended to multi-decision problems in BRSM. Some monotonic measures of approximation quality for multi-decision problems are presented, with which attribute reduction models for multi-decision problems can be suitably constructed. What is more, the discernibility matrices associated with attribute reduction for binary decision and multi-decision problems are proposed, respectively. Based on them, the approaches to knowledge reduction in BRSM can be obtained which corresponds well to the original rough set methodology.     

Finding reliable solutions: event-driven probabilistic constraint programming

Real-life management decisions are usually made in uncertain environments, and decision support systems that ignore this uncertainty are unlikely to provide realistic guidance. We show that previous approaches fail to provide appropriate support for reasoning about reliability under uncertainty. We propose a new framework that addresses this issue by allowing logical dependencies between constraints. Reliability is then defined in terms of key constraints called “events”, which are related to other constraints via these dependencies. We illustrate our approach on three problems, contrast it with existing frameworks, and discuss future developments.     

Learning mixtures of polynomials of multidimensional probability densities from data using B-spline interpolation

Non-parametric density estimation is an important technique in probabilistic modeling and reasoning with uncertainty. We present a method for learning mixtures of polynomials (MoPs) approximations of one-dimensional and multidimensional probability densities from data. The method is based on basis spline interpolation, where a density is approximated as a linear combination of basis splines. We compute maximum likelihood estimators of the mixing coefficients of the linear combination. The Bayesian information criterion is used as the score function to select the order of the polynomials and the number of pieces of the MoP. The method is evaluated in two ways. First, we test the approximation fitting. We sample artificial datasets from known one-dimensional and multidimensional densities and learn MoP approximations from the datasets. The quality of the approximations is analyzed according to different criteria, and the new proposal is compared with MoPs learned with Lagrange interpolation and mixtures of truncated basis functions. Second, the proposed method is used as a non-parametric density estimation technique in Bayesian classifiers. Two of the most widely studied Bayesian classifiers, i.e., the naive Bayes and tree-augmented naive Bayes classifiers, are implemented and compared. Results on real datasets show that the non-parametric Bayesian classifiers using MoPs are comparable to the kernel density-based Bayesian classifiers. We provide a free R package implementing the proposed methods.     

A Bayesian approach for multiple criteria decision making with applications in Design for Six Sigma

Linking end-customer preferences with variables controlled at a manufacturing plant is a main idea behind popular Design for Six Sigma management techniques. Multiple criteria decision making (MCDM) approaches can be used for such purposes, but in these techniques the decision-maker's (DM) utility function, if modelled explicitly, is considered known with certainty once assessed. Here, a new algorithm is proposed to solve a MCDM problem with applications to Design for Six Sigma based on a Bayesian methodology. At a first stage, it is assumed that there are process responses that are functions of certain controllable factors or regressors. This relation is modelled based on experimental data. At a second stage, the utility function of one or more DMs or customers is described in a statistical model as a function of the process responses, based on surveys. This step considers the uncertainty in the utility function(s) explicitly. The methodology presented then maximizes the probability that the DM's or customer's utility is greater than some given lower bound with respect to the controllable factors of the first stage. Both stages are modelled with Bayesian regression techniques. The advantages of using the Bayesian approach as opposed to traditional methods are highlighted.     

Students’ emergent articulations of uncertainty while making informal statistical inferences

Research on informal statistical inference has so far paid little attention to the development of students?? expressions of uncertainty in reasoning from samples. This paper studies students?? articulations of uncertainty when engaged in informal inferential reasoning. Using data from a design experiment in Israeli Grade 5 (aged 10?C11) inquiry-based classrooms, we focus on two groups of students working with TinkerPlots on investigations with growing sample size. From our analysis, it appears that this design, especially prediction tasks, helped in promoting the students?? probabilistic language. Initially, the students oscillated between certainty-only (deterministic) and uncertainty-only (relativistic) statements. As they engaged further in their inquiries, they came to talk in more sophisticated ways with increasing awareness of what is at stake, using what can be seen as buds of probabilistic language. Attending to students?? emerging articulations of uncertainty in making judgments about patterns and trends in data may provide an opportunity to develop more sophisticated understandings of statistical inference.     

PARADIGM SHIFTS IN FISHERIES STOCK ASSESSMENT: FROM INTEGRATED ANALYSIS TO BAYESIAN ANALYSIS AND BACK AGAIN

ABSTRACT. I describe integrated analysis and Bayesian analysis, which have been two of the most influential paradigms in fisheries stock assessment during the last two decades of the twentieth century. These two paradigms have generally been considered complementary, rather than competing. However, recent advances in integrated analysis, including the special case of meta‐analysis, have made Bayesian analysis somewhat redundant. I describe how data used to create priors for use in Bayesian analysis can be integrated directly into the analyses. This provides a much more convenient way of accurately including the information and associated uncertainty into the analyses. I discuss how there is still a need to describe the uncertainty and suggest that research should focus on the most appropriate methods for doing this.     

Decision making with interval probabilities

Handling uncertainty by interval probabilities is recently receiving considerable attention by researchers. Interval probabilities are used when it is difficult to characterize the uncertainty by point-valued probabilities due to partially known information. Most of researches related to interval probabilities, such as combination, marginalization, condition, Bayesian inferences and decision, assume that interval probabilities are known. How to elicit interval probabilities from subjective judgment is a basic and important problem for the applications of interval probability theory and till now a computational challenge. In this work, the models for estimating and combining interval probabilities are proposed as linear and quadratic programming problems, which can be easily solved. The concepts including interval probabilities, interval entropy, interval expectation, interval variance, interval moment, and the decision criteria with interval probabilities are addressed. A numerical example of newsvendor problem is employed to illustrate our approach. The analysis results show that the proposed methods provide a novel and effective alternative for decision making when point-valued subjective probabilities are inapplicable due to partially known information.     

Applications of stochastic programming: Achievements and questions

When solving a decision problem under uncertainty via stochastic programming it is essential to choose or to build a suitable stochastic programming model taking into account the nature of the real-life problem, character of input data, availability of software and computer technology. Besides a brief review of history and achievements of stochastic programming, selected modeling issues concerning applications of multistage stochastic programs with recourse (the choice of the horizon, stages, methods for generating scenario trees, etc.) will be discussed.     

Bayesian decision making and military command and control

We discuss firstly the problem of military decision, in the context of the more general development of ideas in the representation of decision making. Within this context, we have considered a mathematical model—Bayesian Decision—of decision making and military command. Previous work has been extended, and applied to this problem. A distribution of belief in outcome, given that a decision is made, and a Loss function—a measure of the effect of this outcome relative to a goal—are formed. The Bayes' Decision is the decision which globally minimises the resultant bimodal (or worse) Expected Loss function. The set of all minimising decisions corresponds to the surface of an elementary Catastrophe. This allows smooth parameter changes to lead to a discontinuous change in the Bayes' decision. In future work this approach will be used to help develop a number of hypotheses concerning command processes and military headquarters structure. It will also be used to help capture such command and control processes in simulation modelling of future defence capability and force structure.     

A Bayesian paired comparison approach for relative accident probability assessment with covariate information

One of the challenges managers face when trying to understand complex, technological systems (in their efforts to mitigate system risks) is the quantification of accident probability, particularly in the case of rare events. Once this risk information has been quantified, managers and decision makers can use it to develop appropriate policies, design projects, and/or allocate resources that will mitigate risk. However, rare event risk information inherently suffers from a sparseness of accident data. Therefore, expert judgment is often elicited to develop frequency data for these high-consequence rare events. When applied appropriately, expert judgment can serve as an important (and, at times, the only) source of risk information. This paper presents a Bayesian methodology for assessing relative accident probabilities and their uncertainty using paired comparison to elicit expert judgments. The approach is illustrated using expert judgment data elicited for a risk study of the largest passenger ferry system in the US.     

Artificial intelligence and human decision making

Decision is obviously related to reasoning. One of the possible definitions of artificial intelligence (AI) refers to cognitive processes and especially to reasoning. Before making any decision, people also reason, it is therefore natural to explore the links between AI and decision making. This paper distinguishes between two aspects of decision making: diagnosis and look-ahead. It is shown that, on the one hand, AI has many relationships with diagnosis (expert systems, case-based reasoning, fuzzy set and rough set theories). On the other hand, AI has not paid enough attention to look-ahead reasoning, whose main components are uncertainty and preferences. These aspects of AI and decision making are reviewed in the paper.     

A methodology for developing Bayesian networks: An application to information technology (IT) implementation

Bayesian Networks (BNs) are probabilistic inference engines that support reasoning under uncertainty. This article presents a methodology for building an information technology (IT) implementation BN from client–server survey data. The article also demonstrates how to use the BN to predict the attainment of IT benefits, given specific implementation characteristics (e.g., application complexity) and activities (e.g., reengineering). The BN is an outcome of a machine learning process that finds the network’s structure and its associated parameters, which best fit the data. The article will be of interest to academicians who want to learn more about building BNs from real data and practitioners who are interested in IT implementation models that make probabilistic statements about certain implementation decisions.     

Comparison of the Bayesian and Randomised Decision Tree Ensembles within an Uncertainty Envelope Technique

Multiple Classifier Systems (MCSs) allow evaluation of the uncertainty of classification outcomes that is of crucial importance for safety critical applications. The uncertainty of classification is determined by a trade-off between the amount of data available for training, the classifier diversity and the required performance. The interpretability of MCSs can also give useful information for experts responsible for making reliable classifications. For this reason Decision Trees (DTs) seem to be attractive classification models for experts. The required diversity of MCSs exploiting such classification models can be achieved by using two techniques, the Bayesian model averaging and the randomised DT ensemble. Both techniques have revealed promising results when applied to real-world problems. In this paper we experimentally compare the classification uncertainty of the Bayesian model averaging with a restarting strategy and the randomised DT ensemble on a synthetic dataset and some domain problems commonly used in the machine learning community. To make the Bayesian DT averaging feasible, we use a Markov Chain Monte Carlo technique. The classification uncertainty is evaluated within an Uncertainty Envelope technique dealing with the class posterior distribution and a given confidence probability. Exploring a full posterior distribution, this technique produces realistic estimates which can be easily interpreted in statistical terms. In our experiments we found out that the Bayesian DTs are superior to the randomised DT ensembles within the Uncertainty Envelope technique.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

HRSOS: a decision support system to facilitate the privatization of military housing

The armed services must provide their personnel with acceptable housing at minimum cost within the vicinity of military installations. To achieve these housing objectives, the Department of Defense (DoD) has entered into experimental joint ventures with private developers to construct attractive housing projects on military installation property, with some of the projects reserved for military personnel. To support the analysis of the joint ventures, DoD needed a methodology that would help officials evaluate the financial feasibility and cost implications of the housing projects. A decision support system, called the Housing Revitalization Support Office System (HRSOS), was developed to provide the necessary support. This paper describes the decision support system and its underlying methodology. It outlines the process used to determine financially feasible privatized military housing initiatives, reviews the decision support system created to support the process, and describes its application at the US Department of Defense. The paper also discusses the implications for military housing management, public finance, and operations research.