A Distance-Based Decision in the Credal Level

Belief function theory provides a flexible way to combine information provided by different sources. This combination is usually followed by a decision making which can be handled by a range of decision rules. Some rules help to choose the most likely hypothesis. Others allow that a decision is made on a set of hypotheses. In [6], we proposed a decision rule based on a distance measure. First, in this paper, we aim to demonstrate that our proposed decision rule is a particular case of the rule proposed in [4]. Second, we give experiments showing that our rule is able to decide on a set of hypotheses. Some experiments are handled on a set of mass functions generated randomly, others on real databases

Improving practical relevance of Bayes factors

Bayes factors are recently promoted as replacement for the heavily criticized frequentist hypothesis tests. Yet, Bayes factors are oftentimes based on the same statistical hypotheses that were employed in frequentist procedures and criticized for lacking practical relevance. To guard Bayes factors of similar shortcomings, the present dissertation attempts to elaborate on how to improve the practical relevance of Bayes factors. It appears that a formal definition of the notion of practical relevance is located within the framework of statistical decision theory. The relevance of a result naturally depends on what it is used for, and - formally speaking - such a use is a decision. Accordingly, Bayes factors were depicted and evaluated within the framework of Bayesian decision theory, in which the specification of the loss function seems to be the major obstacle to its application. Typically, information about the consequences of a decision are scarce, vague, partial, and ambiguous, prohibiting an unambiguous specification of the loss function. To deal with these specification issues, two options are discussed: First, the loss function can be simplified by employing a hypothesis-based account and, second, the required specifications can be allowed to be set-valued, i.e. imprecise, instead of precise values. In this regard, a twofold generalization of Bayes factors into the framework of decision theory and into the framework of imprecise probabilities was developed and condensed into a straightforward framework for applications. Besides, the nature of statistical hypotheses was critically evaluated, showing that - in contrast to the current conception within the literature of Bayes factors - they are merely subsets of the parameter space.Bayes-Faktoren werden neuerdings als Ersatz für die stark kritisierten frequentistischen Hypothesentests propagiert. Allerdings beruhen Bayes-Faktoren häufig auf denselben statistischen Hypothesen, die in den frequentistischen Verfahren verwendet und wegen mangelnder praktischer Relevanz kritisiert wurden. Um Bayes-Faktoren vor ähnlichen Unzulänglichkeiten zu bewahren, wird in der vorliegenden Dissertation versucht, herauszuarbeiten, wie die praktische Relevanz von Bayes-Faktoren verbessert werden kann. Es zeigt sich, dass eine formale Definition des Begriffs der praktischen Relevanz innerhalb der statistischen Entscheidungstheorie zu finden ist. Die Relevanz eines Ergebnisses hängt natürlich davon ab, wofür es verwendet wird, und eine solche Verwendung ist - formal gesehen - eine Entscheidung. Dementsprechend wurden die Bayes-Faktoren im Rahmen der Bayes'schen Entscheidungstheorie dargestellt und bewertet, wobei die Spezifikation der Verlustfunktion das größte Hindernis für ihre Anwendung zu sein scheint. Typischerweise sind die Informationen über Konsequenzen einer Entscheidung knapp, vage und mehrdeutig, was eine eindeutige und präzise Spezifikation der Verlustfunktion nahezu unmöglich macht. Um dieses Spezifikationsproblem zu lösen, werden zwei Möglichkeiten diskutiert: Erstens kann die Verlustfunktion durch die Verwendung eines hypothesenbasierten Ansatzes vereinfacht werden, und zweitens können die geforderten Spezifikationen mengenwertig, d.h. verallgemeinert, anstelle von präzisen Werten aufgefasst werden. In diesem Sinne wurde eine zweifache Verallgemeinerung der Bayes-Faktoren in die Entscheidungstheorie und in das Feld der verallgemeinerten Wahrscheinlichkeiten entwickelt und anschließend in einen anwenderfreundlichen statistischen Leitfaden verdichtet. Außerdem wurde das Wesen von statistischen Hypothesen kritisch bewertet, wobei gezeigt wurde, dass sie - im Gegensatz zur gängigen Auffassung in der Literatur über Bayes-Faktoren - lediglich Teilmengen des Parameterraums sind

Coherent frequentism

By representing the range of fair betting odds according to a pair of confidence set estimators, dual probability measures on parameter space called frequentist posteriors secure the coherence of subjective inference without any prior distribution. The closure of the set of expected losses corresponding to the dual frequentist posteriors constrains decisions without arbitrarily forcing optimization under all circumstances. This decision theory reduces to those that maximize expected utility when the pair of frequentist posteriors is induced by an exact or approximate confidence set estimator or when an automatic reduction rule is applied to the pair. In such cases, the resulting frequentist posterior is coherent in the sense that, as a probability distribution of the parameter of interest, it satisfies the axioms of the decision-theoretic and logic-theoretic systems typically cited in support of the Bayesian posterior. Unlike the p-value, the confidence level of an interval hypothesis derived from such a measure is suitable as an estimator of the indicator of hypothesis truth since it converges in sample-space probability to 1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly extended to vector parameters of interest. The derivation of upper and lower confidence levels from valid and nonconservative set estimators is formalize

Topics in inference and decision-making with partial knowledge

Two essential elements needed in the process of inference and decision-making are prior probabilities and likelihood functions. When both of these components are known accurately and precisely, the Bayesian approach provides a consistent and coherent solution to the problems of inference and decision-making. In many situations, however, either one or both of the above components may not be known, or at least may not be known precisely. This problem of partial knowledge about prior probabilities and likelihood functions is addressed. There are at least two ways to cope with this lack of precise knowledge: robust methods, and interval-valued methods. First, ways of modeling imprecision and indeterminacies in prior probabilities and likelihood functions are examined; then how imprecision in the above components carries over to the posterior probabilities is examined. Finally, the problem of decision making with imprecise posterior probabilities and the consequences of such actions are addressed. Application areas where the above problems may occur are in statistical pattern recognition problems, for example, the problem of classification of high-dimensional multispectral remote sensing image data

Confirmation, Decision, and Evidential Probability

Henry Kyburg’s theory of Evidential Probability offers a neglected tool for approaching problems in confirmation theory and decision theory. I use Evidential Probability to examine some persistent problems within these areas of the philosophy of science. Formal tools in general and probability theory in particular have great promise for conceptual analysis in confirmation theory and decision theory, but they face many challenges. In each chapter, I apply Evidential Probability to a specific issue in confirmation theory or decision theory. In Chapter 1, I challenge the notion that Bayesian probability offers the best basis for a probabilistic theory of evidence. In Chapter 2, I criticise the conventional measures of quantities of evidence that use the degree of imprecision of imprecise probabilities. In Chapter 3, I develop an alternative to orthodox utility-maximizing decision theory using Kyburg’s system. In Chapter 4, I confront the orthodox notion that Nelson Goodman’s New Riddle of Induction makes purely formal theories of induction untenable. Finally, in Chapter 5, I defend probabilistic theories of inductive reasoning against John D. Norton’s recent collection of criticisms. My aim is the development of fresh perspectives on classic problems and contemporary debates. I both defend and exemplify a formal approach to the philosophy of science. I argue that Evidential Probability has great potential for clarifying our concepts of evidence and rationality