Vincenzo Crupi

On Bayesian Measures of Evidential Support: Theoretical and Empirical Issues*

Epistemologists and philosophers of science have often attempted to express formally the impact of a piece of evidence on the credibility of a hypothesis. In this paper we will focus on the Bayesian approach to evidential support. We will propose a new formal treatment of the notion of degree of confirmation and we will argue that it overcomes some limitations of the currently available approaches on two grounds: (i) a theoretical analysis of the confirmation relation seen as an extension of logical deduction and (ii) an empirical comparison of competing measures in an experimental inquiry concerning inductive reasoning in a probabilistic setting.

Comparison of confirmation measures.

Alternative measures of confirmation or evidential support have been proposed to express the impact of ascertaining one event on the credibility of another. We report an experiment that compares the adequacy of several such measures as descriptions of confirmation judgment in a probabilistic context.

Probability, confirmation, and the conjunction fallacy

The conjunction fallacy has been a key topic in debates on the rationality of human reasoning and its limitations. Despite extensive inquiry, however, the attempt to provide a satisfactory account of the phenomenon has proved challenging. Here we elaborate the suggestion (first discussed by Sides, Osherson, Bonini, & Viale, 2002) that in standard conjunction problems the fallacious probability judgements observed experimentally are typically guided by sound assessments of confirmation relations, meant in terms of contemporary Bayesian confirmation theory. Our main formal result is a confirmation-theoretic account of the conjunction fallacy, which is proven robust (i.e., not depending on various alternative ways of measuring degrees of confirmation). The proposed analysis is shown distinct from contentions that the conjunction effect is in fact not a fallacy, and is compared with major competing explanations of the phenomenon, including earlier references to a confirmation-theoretic account.

A Second Look at the Logic of Explanatory Power (with Two Novel Representation Theorems)

We discuss the probabilistic analysis of explanatory power and prove a representation theorem for posterior ratio measures recently advocated by Schupbach and Sprenger. We then prove a representation theorem for an alternative class of measures that rely on the notion of relative probability distance. We end up endorsing the latter, as relative distance measures share the properties of posterior ratio measures that are genuinely appealing, while overcoming a feature that we consider undesirable. They also yield a telling result concerning formal accounts of explanatory power versus inductive confirmation, thereby bridging our discussion to a so-called no-miracle argument.

Confirmation as partial entailment: A representation theorem in inductive logic

Abstract The most prominent research program in inductive logic – here just labeled The Program, for simplicity – relies on probability theory as its main building block and aims at a proper generalization of deductive-logical relations by a theory of partial entailment. We prove a representation theorem by which a class of ordinally equivalent measures of inductive support or confirmation is singled out as providing a uniquely coherent way to work out these two major sources of inspiration of The Program.

New Axioms for Probability and Likelihood Ratio Measures

Probability ratio and likelihood ratio measures of inductive support and related notions have appeared as theoretical tools for probabilistic approaches in the philosophy of science, the psychology of reasoning, and artificial intelligence. In an effort of conceptual clarification, several authors have pursued axiomatic foundations for these two families of measures. Such results have been criticized, however, as relying on unduly demanding or poorly motivated mathematical assumptions. We provide two novel theorems showing that probability ratio and likelihood ratio measures can be axiomatized in a way that overcomes these difficulties. 1 Introduction 2 Axioms for Probability Ratio Measures 3 Axioms for Likelihood Ratio Measures 4 Discussion 1 Introduction 2 Axioms for Probability Ratio Measures 3 Axioms for Likelihood Ratio Measures 4 Discussion

Towards a Grammar of Bayesian Confirmation

The foundations of a detailed grammar of Bayesian confirmation are presented as a theoretical tool for the formal analysis of reasoning in epistemology and philosophy of science. After a discussion of core intuitions grounding the measurement of confirmation in probabilistic terms, a number of basic, derived and structural properties of Bayesian incremental confirmation are defined, distinguished and investigated in their logical relationships. Illustrations are provided that a thorough development of this line of research would yield an appropriate general framework of inquiry for several analyses and debates surrounding confirmation and Bayesian confirmation in particular.

Determinants of confirmation

Epistemologists often suppose that the extent to which evidencee confirms hypothesisH depends on probabilities involvinge andH, and nothing more. We show experimentally that human reasoners sometimes violate this assumption.

Irrelevant Conjunction: Statement and Solution of a New Paradox*

The so‐called problem of irrelevant conjunction has been seen as a serious challenge for theories of confirmation. It involves the consequences of conjoining irrelevant statements to a hypothesis that is confirmed by some piece of evidence. Following Hawthorne and Fitelson, we reconstruct the problem with reference to Bayesian confirmation theory. Then we extend it to the case of conjoining irrelevant statements to a hypothesis that is disconfirmed by some piece of evidence. As a consequence, we obtain and formally present a novel and more troublesome problem of irrelevant conjunction. We conclude by indicating a possible solution based on a measure‐sensitive approach and by critically discussing a major alternative way to address the problem.

Updating: Learning versus supposing

Bayesian orthodoxy posits a tight relationship between conditional probability and updating. Namely, the probability of an event A after learning B should equal the conditional probability of A given B prior to learning B. We examine whether ordinary judgment conforms to the orthodox view. In three experiments we found substantial differences between the conditional probability of an event A supposing an event B compared to the probability of A after having learned B. Specifically, supposing B appears to have less impact on the credibility of A than learning that B is true.