Pluralism in the foundations of statistics

9-10 September 2010
Room CGU4, University of Kent, Canterbury, UK
(J2 on this map)

 

INTRODUCTION

This workshop, sponsored by the European Science Foundation, will explore various aspects of multiplicity in statistics, including methodological aspects (invoking a multiplicity of methods), evidential aspects (handling and reconciling disparate kinds of evidence, logically and psychologically) and metaphysical aspects (appealing to different interpretations of probability). The conference will also explore the extent to which such multiplicities call for unification or for pluralism. We bring together philosophers, psychologists, statisticians and economists.

ORGANISER

LOCAL ORGANISER

SPEAKERS

 

PROGRAMME

Thursday 9 September

This is joint work with Christian List and Richard Bradley. This paper introduces a generalized framework for updating in which it gives a unified characterization of the main probabilistic belief updating rules: Bayesian updating, Jeffrey updating, and Adams updating. These seemingly quite different updating rules turn out to follow from the same two underlying principles. These principles are (i) Faithfulness, whereby the updated beliefs should be consistent with the learning experience, and (ii) Conservativeness, whereby updating should preserve features on which the learning experience is silent. The only difference between the three updating rules is the kind of learning experience considered.

At the heart of the theory of Evidential probability (EP) is a solution to the reference class problem. Now, part of the controversy over EP is that this solution, along with the formal machinery of the theory itself, changed a lot over time, and that history has led some to suspect that at bottom of EP contains neither a logic nor a solution. This talk addresses how to arrest this slide by first fixing the logic and second suggesting how to triage the philosophical objections. For a start, Kyburg and Teng (2001, Uncertain Inference) provide a sounds semantics and a sketch for a completeness proof. In joint work with Terry Swift (CENTRIA, Johns Hopkins), we prove strong completeness for a very expressive fragment of EP, provide a representation theorem for this fragment within the Description Logic ALCQ, which is decidable, and provide a prototype implementation. Restricting ourselves to decidable fragments of first-order logic to formulate EP(s), we are then able to sort philosophical objections to the theory of Evidential Probability into those which we can address by altering the execution of rules within the formal language, and those which we cannot. We envision the former set of questions to engage the description logic & uncertainty in AI communities, arising from applications, whereas the latter we hope to engage philosophers of statistics. In any event, we argue that having a sound, strongly complete, and decidable version of EP addresses at least the 'no logic' charge.

Philosophers understand Objective Bayesianism as the claim that there is one particular rational assignment of degrees of belief to propositions, as opposed to the pluralist claim that in a particular situation, several assignments can be rational. In statistics, however, Objective Bayesianism usually means something else: namely the use of standardized "objective" or "reference" priors (Bernardo 1979) in data analysis, as opposed to priors that have been elicited by expert opinion, guesswork, etc. The aim of these procedures is to base the inference on the assumed statistical model and the data alone, as to avoid equivocation in scientific communication. Notwithstanding the enormous success of this project in practice, it consitutes a significant departure from orthodox Bayesian principles, and resembles Fisher's project of a logic of statistical inference. This talk has two main objectives: Historically, it elucidates the continutity from Fisher's objectivist program to modern objective Bayesianism. Systemically, it tries to place the use of reference priors in the continuum between frequentist and (proper) Bayesian methods, and to defend it against the claim that it is not more than frequentist inference dressed in Bayesian formalism.

The orthodox view in statistics has it that frequentism and Bayesianism are diametrically opposed - two totally incompatible takes on the problem of statistical inference. This paper argues to the contrary that the two approaches are complementary and need to mesh if probabilistic reasoning is to be carried out correctly.

According to probabilism, uncertain conditionals are to be reconstrcted as assertions of high conditional probability. In everyday life often encounters examples of 'conflict', in which two uncertain conditionals have contradicting consequents and both of their antecedents are instantiated or true, respectively. The often cited example of this sorty is 'Tweety', who happens to be both a bird and a penguin. We believe that Tweety is a bird then it probably can fly, and if it is a penguin then it probably cannot fly. If one reconstructs examples of 'conflict' by only one probability function, as Bayesianism requires, these examples come out as probabilistically incoherent. This seems counterintuitive. I argue that if one intends a coherent reconstruction, one has to distinguish between two probability functions, evidential probabilities (which are Bayesian), and objective probabilities, which are typically given as statistical probabilities. In the end of the talk I will present results of an experimental study on examples of 'conflict'.
The Dutch book theorem is often used to underwrite claims that a rational agent's degrees of belief can and should form a probability measure. That probability theory is the correct formal theory of uncertainty. I present two forms of the theorem, and show how each might be modified to give a different conclusion: that an agent's degrees of belief ought to be a Dempster-Shafer belief function.
A frequent charge against Bayesianism is that it is too subjective. One response to this charge is to argue for a Principle of Direct Probability (called by David Lewis the Principal Principle): subjective probabilities should match objective probabilities. A number of Dutch Book arguments have been produced in favour of this Principle. I shall argue that they all fail to establish the Principle. I conclude that subjective probabilities are, indeed, subjective.

Friday 10 September

Various scientific theories stand in a reductive relation to each other. In a recent article, we have argued that a generalized version of the Nagel-Schaffner model (GNS) is the right account of this relation. In this article, we present a Bayesian analysis of how GNS impacts on confirmation. We formalize the relation between the reducing and the reduced theory before and after the reduction using Bayesian networks, and thereby show that, post-reduction, the two theories are confirmatory of each other. We then ask when a purported reduction should be accepted on epistemic grounds. To do so, we compare the prior and posterior probabilities of the conjunction of both theories before and after the reduction and ask how well each is confirmed by the available evidence.
This paper presents a refinement of the Bayesian Information Criterion (BIC). While the original BIC selects models on the basis of complexity and fit, the so-called prior-adapted BIC allows us to choose among statistical models that differ on three scores: fit, complexity, and model size. The prior-adapted BIC can therefore accommodate comparisons among statistical models that differ only in the admissible parameter space, e.g., for choosing among models with different constraints on the parameters.
In earlier papers we have argued against epistemic foundationalism, which claims that epistemic justification must come to an end in a basic belief or ground. We showed that the foundationalist claim does not hold water if the justification is probabilistic in character – as most modern epistemologists assume it is. For a belief or proposition p-1 can have a well-defined probability, even if its justification consists of a chain such that p-1 is made probable by p-2, which in turn is made probable by p-3, and so on, ad infinitum. In the present paper we extend this result in two ways. First, we investigate what happens when the epistemic justification has the form of an infinite loop rather than an infinite linear chain. Second, we make the justification more realistic by replacing the one-dimensional chain by a two-dimensional network.
In his classical exposition of Bayesian statistics, The Foundations of Statistics (1954), L. J. Savage defended the “behavioralist outlook” against the “verbalistic outlook”: statistics deals with problems of deciding what to do rather than what to say. Savage referred to Jerzy Neyman’s 1938 paper which proposed to replace “inductive inference” with “inductive behavior”. Both of these approaches were in opposition to R. A. Fisher’s formulation of estimation and testing in statistics in traditional terms as truth-seeking methods of scientific inference. A reconciliation of inference and decision was forcefully proposed by Isaac Levi in his Gambling With Truth (1967). Levi argued against “behavioralism” that the tentative acceptance and rejection of scientific hypotheses cannot be reduced to actions that are related to practical objectives. This would reduce the role of a scientist to a practical decision-maker or a guidance counselor of a decision-maker. According to Levi’s “critical cognitivism”, science has its own theoretical objectives, defined by such “epistemic utilities” as truth, information, and explanatory power. As a development of cognitive decision theory, and in the spirit of critical scientific realism, Ilkka Niiniluoto’s Truthlikess (1987) suggests that scientific inference is defined by the attempt to maximize expected verisimilitude. This proposal allows us to interpret Bayesian point and interval estimation in terms of decisions relative to loss functions which measure the distances of a hypothesis from the truth.
The tendency (T-) theory 'A-individuals tend to be B' says that the individuals characterized by the property A 'tend to have' the property B. We will explicate the notion of T-theory and define suitable measures of the verisimilitude of T-theories. This will allow to show that T-theories can be used to describe the statistical structure of cross classified populations and that their adequacy in this task can be evaluated by measuring their verisimilitude.
The talk will present a number of examples from the judgment and decision-making literature where people have been accused of irrationality, seeming biases, and errors on the basis of task evaluations that fail to take into account key characteristics of both participants’ and experimenters’ experience. Once these are properly considered, more favourable evaluations emerge.