Objective Bayesian Epistemology

In [Church 56, sect.07] mathematical logic or symbolic logic, or logistics, is defined as “the subject of formal logic when treated by the method of setting up a formalized language”. Among formalized languages there are purely logical (propositional and predicate) languages, languages of arithmetic and languages of set theory. Predicate languages — elementary (=first-order) as well as non-elementary (of higher orders) — are used for formal descriptions of properties of mathematical structures (first of all algebraic structures). We can use them for an axiomatic description of different classes of structures. Languages of arithmetic describe the set of natural numbers (which hardly can be described axiomatically and must anyway be regarded as existing before any axiomatic considerations). Languages of set theory have no clear semantics and are intended to formulate various axiomatic theories.

This paper presents some functional equations that have played an essential role in the characterization of utility and probability functions in decision analysis. We survey some previous results with improvements and derive several new results. We also discuss some simple methods for the solution of these equations and highlight some subtle points about their use. We show that one functional equation can determine several unknown functions within it, and that relaxing differentiability and requiring only continuity of the functions leads to generalizations of many well-known results in utility and Bayesian probability theory.

This paper offers a comparison between two decision rules for use when uncertainty is depicted by a non-trivial, convex2 set of probability func tions T. This setting for uncertainty is different from the canonical Bayesian decision theory of expected utility, which uses a singleton set, just one probability function to represent a decision maker's uncertainty. Justifications for using a non-trivial set of probabilities to depict uncer tainty date back at least a half century (Good 1952) and a foreshadowing of that idea can be found even in Keynes' (1921), where he allows that not all hypotheses may be comparable by qualitative probability in accord with, e.g., the situation where the respective intervals of probabilities for two events merely overlap with no further (joint) constraints, so that neither of the two events is more, or less, or equally probable compared with the other. Here, I will avail myself of the following simplifying assumption: Throughout, I will avoid the complexities that ensue when the decision maker's values for outcomes also are indeterminate and, in parallel with her or his uncertainty, are then depicted by a set of (cardinal) utilities. That is, for this discussion, I will contrast two decision rules when the decision maker's uncertainties, but not her/his values are indeterminate. The more familiar decision rule of the pair under discussion, T Maximin,3 requires that the decision maker ranks an option by its lower expected value, taken with respect to the convex set of probabilities, T, and then to choose an option whose lower expected value is maximum. This decision rule (as simplified by the two assumptions, above) was given a representation in terms of a binary preference relation over Anscombe Aumann horse lotteries (Gilboa and Schmeidler 1989), has been discussed

Portfolio Selection in Asset Liability Management Using Probabilistic Utility Functions

This last chapter aims to provide a concise presentation of modal logic, the logic of necessity and possibility. A modal language is a language that contains, in addition to the symbols of a propositional or predicate language, the modal operators Open image in new window and ◊, which mean respectively ‘it is necessary that’ and ‘it is possible that’.

Based upon the requirements of Bayesian object recognition theory, this paper provides the fundamental framework to determine the joint probability density function of object regions in an IR image. This probability function contains all information about the region that is required to achieve minimum probability of error recognition. The techniques advanced here are expected to be of significant use in certain rather hostile and difficult situations such as testing piping for fault conditions within operational nuclear power plants.

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to be borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment.

Reports an error in "Bridging the gap between subjective probability and probability judgments: The quantum sequential sampler" by Jiaqi Huang, Jerome R. Busemeyer, Zo Ebelt and Emmanuel M. Pothos (Psychological Review, 2025[Jul], Vol 132[4], 916-955; see record 2025-25219-001). The authors noticed a typo in Equation 17. In the matrix shown, the entry in the fourth row should be "(0 0 β+ … . .)" instead of "(0 β+ … . .)". A zero is missing from the fourth row. The online version of this article has been corrected. (The following abstract of the original article appeared in record 2025-25219-001.) One of the most important challenges in decision theory has been how to reconcile the normative expectations from Bayesian theory with the apparent fallacies that are common in probabilistic reasoning. Recently, Bayesian models have been driven by the insight that apparent fallacies are due to sampling errors or biases in estimating (Bayesian) probabilities. An alternative way to explain apparent fallacies is by invoking different probability rules, specifically the probability rules from quantum theory. Arguably, quantum cognitive models offer a more unified explanation for a large body of findings, problematic from a baseline classical perspective. This work addresses two major corresponding theoretical challenges: first, a framework is needed which incorporates both Bayesian and quantum influences, recognizing the fact that there is evidence for both in human behavior. Second, there is empirical evidence which goes beyond any current Bayesian and quantum model. We develop a model for probabilistic reasoning, seamlessly integrating both Bayesian and quantum models of reasoning and augmented by a sequential sampling process, which maps subjective probabilistic estimates to observable responses. Our model, called the Quantum Sequential Sampler, is compared to the currently leading Bayesian model, the Bayesian Sampler (J. Zhu et al., 2020) using a new experiment, producing one of the largest data sets in probabilistic reasoning to this day. The Quantum Sequential Sampler embodies several new components, which we argue offer a more theoretically accurate approach to probabilistic reasoning. Moreover, our empirical tests revealed a new, surprising systematic overestimation of probabilities. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels.

Cure rate models have been widely adopted for characterizing survival data that have long-term survivors. Under a mixture cure rate model where the population is a mixture of cured and susceptible subjects, a primary goal is to study covariate effects on the cure probability and survival function of the susceptible subjects. In this article, we propose a penalization method for estimating the mixture cure rate model where we explicitly consider the structural effects of covariates. The proposed method is more informative than the standard estimations and more flexible than the existing works on structural effects. Depending on data characteristics, we develop different penalties and corresponding computational algorithms. Simulation shows that the proposed method outperforms the alternatives by more accurately estimating parameters and identifying relevant variables. Two breast cancer datasets, one with low-dimensional clinical variables and the other with high-dimensional genetic variables, are analyzed.

Abstract. Surface flow on rilled hillslopes tends to produce sediment yields that scale nonlinearly with total hillslope length. The widespread observation lacks a single unifying theory for such a nonlinear relationship. We explore the contribution of rill network geometry to the observed yield–length scaling relationship. Relying on an idealized network geometry, we formally develop probability functions for geometric variables of contributing area and rill length. In doing so, we contribute towards a complete probabilistic foundation for the Hack distribution. Using deterministic and empirical functions, we then extend the probability theory to the hydraulic variables that are related to sediment detachment and transport. A Monte Carlo simulation samples hydraulic variables from hillslopes of different lengths to provide estimates of sediment yield. The results of this analysis demonstrate a nonlinear yield–length relationship as a result of the rill network geometry. Theory is supported by numerical modeling, wherein surface flow is routed over an idealized numerical surface and a natural surface from northern Arizona. Numerical flow routing demonstrates probability functions that resemble the theoretical ones. This work provides a unique application of the Scheidegger network to hillslope settings which, because of their finite lengths, result in unique probability functions. We have addressed sediment yields on rilled slopes and have contributed towards understanding Hack's law from a probabilistic reasoning.

The Dempster-Shafer (D-S) evidence theory has been widely used in many fields from probabilistic inference, information fusion to decision analysis due to its superiority to formulate uncertain and incomplete information under a weaker condition than the Bayesian probability theory. To support the elicitation of basic probability assignment (BPA), numerous uncertainty measures have been proposed in the recent decades, which take into account nonspecificity and structural conflict (SC) in either the entropy-alike or nonentropy form. This article is focused primarily on developing a generalized belief entropy to measure and quantify the uncertainty of BPA in a consistent and comprehensive way. The definition of the proposed belief entropy involves three components. Specifically, the first component is Shannon's definition of entropy of probability functions which can be interpreted as the measure of discord of the mass function among various focal elements. The second component is the measure of total nonspecificity which is the generalization of Dubois-Prade's definition of nonspecificity measure. Furthermore, the critical component introduced in this article is the utilization of the SC coefficient within an evidence, which fully considers the intersection between each pair of focal elements of the BPA. The characteristics and properties of the new generalized belief entropy are analyzed systematically with theoretical proofs. Finally, the proposed belief entropy is embedded to a three-tuple approach to deal with a practical multiple attribute decision-making (MADM) problem to demonstrate its validity and strengths of measuring nonspecificity and SC in uncertain decision making.

Bayesian belief networks (also known asbelief networks,causal probabilistic networks,causal nets,graphical probability networks,probabilistic cause–effect models, andprobabilistic influence diagrams) provide decision‐support for a wide range of problems involving uncertainty and probabilistic reasoning. The underlying theory of BBNs is Bayesian probability theory and the notion of propagation. Although this has been around for a long time it is only since the late 1980s that efficient algorithms and tools to implement them have been developed to enable propagation in networks with a reasonable number of variables. The recent explosion of interest in BBNs is due to these developments, which mean that for the first time realistic size problems can be solved. These developments make BBNs an excellent method for reasoning about uncertainty. BBNs have proved useful in applications such as medical diagnosis and diagnosis of mechanical failures. Within the software engineering community the interest in BBNs is motivated by the need to integrate diverse types of evidence to provide rigorous quantitative dependability/safety arguments. BBNs enable us to incorporate not just the results from testing but also other evidence about the process and product; this evidence can be both objective and subjective. The BBN forces the assessor to expose all assumptions about the impact of different forms of evidence and hence provides a visible and auditable dependability or safety argument. This article provides a very brief introduction to BBNs and explains how they can be used for assessing critical systems.

Probabilistic stit logic and its decomposition

Bayesian decision theory in its classical formulation supposes that for any rational agent and for any possible statexof the world, there is a numberP(x). which represents the agent’s judgment of the probability ofx. Similarly, the theory assumes that for any possible outcomeyof the agent’s actions, there is a numberu(y)which represents the utility or value ofyto the agent. Given these assumptions, the theory is able to define theexpected valuefor the agent of any act. Bayesian decision theory then recommends that the agent should choose, from amongst the available acts, one which has maximal expected utility.The assumption that an agent has a determinate personal probability function has often been attacked.