Carl G. Wagner

Probability Kinematics And Commutativity

The so‐called “non‐commutativity” of probability kinematics has caused much unjustified concern. When identical learning is properly represented, namely, by identical Bayes factors rather than identical posterior probabilities, then sequential probability‐kinematical revisions behave just as they should. Our analysis is based on a variant of Field’s reformulation of probability kinematics, divested of its (inessential) physicalist gloss.

Generalized Stirling and Lah numbers

Abstract The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces, and chains. Within this theory three generalizations of Stirling numbers of the second kind, and of Lah numbers, are developed.

Consensus for belief functions and related uncertainty measures

We extend previous work of Lehrer and Wagner, and of McConway, on the consensus of probabilities, showing under axioms similar to theirs that (1) a belief function consensus of belief functions on a set with at least three members and (2) a belief function consensus of Bayesian belief functions on a set with at least four members must take the form of a weighted arithmetic mean. We observe that these results are unchanged when consensual uncertainty measures are allowed to take the form of Choquet capacities of low order monotonicity.

On the formal properties of weighted averaging as a method of aggregation

I first encountered Keith Lehrers work on consensus during the summer of 1977,l and was immediately intrigued by the possibility of developing a formal account of his model of rational group decision making. I adopted as a model for this enterprise the axiomatic method of social choice theory and was not surprised to discover that the question of how to aggregate probabilities was as complex and prob lematic as the question of how to aggregate preferences and utilities. The former issue has only recently received the kind of attention which has been directed at the latter for several decades, and is finally beginning to be as vigorously debated, as attested to by the essays in this volume. I am grateful to Barry Loe wer for organizing this forum on Rational Consensus in Science and Society. My replies to the preceding critical essays follow.

Aggregating subjective probabilities: some limitative theorems.

1 Introduction Suppose that n individuals, labeled 1,..., n 9 wish to assign consensual probabilities to a sequence of events E u ..., E k which partition some set of possibilities. Let their subjective assignments be registered in an n x k matrix P = (Pij), where py denotes the subjective probability assigned by individual / to event Ej. The question of how to aggregate the probabilities in P into a single sequence of consensual probabilities may be abstractly modeled as the problem of choosing a mapping (hereafter called a probability aggregation method, or PAM)

Generalized finite differences and Bayesian conditioning of Choquet capacities

Rotas calculus of finite differences for a locally finite poset employs a difference operator based on the Mobius function. We investigate the behavior of this operator on products and quotients and employ the lattice-of-subsets case of our results to show that r-monotonicity is preserved under Bayesian conditioning of Choquet Capacities.

Minimal covers of finite sets

We enumerate the minimal covers of a finite set S, classifying such covers by their cardinality, and also by the number of elements in S which they cover uniquely.

Generalized probability kinematics

Jeffrey conditionalization is generalized to the case in which new evidence bounds the possible revisions of a prior below by a Dempsterian lower probability. Classical probability kinematics arises within this generalization as the special case in which the evidentiary focal elements of the bounding lower probability are pairwise

The Formal Foundations of Lehrer’s Theory of Consensus

The challenge of devising rational methods for achieving group consensus has provided decision theorists with an important class of practical and theoretical problems. How should group deliberation be structured in order to enhance the exchange and thoughtful consideration of relevant information and maximize the possibility of agreement? Are there defensible ways of combining the opinions of individuals who disagree even after exhaustive discussion? Can we develop a comprehensive theory of group rationality and perhaps refine our understanding through the discovery of limitative metatheorems, which have so often been the mark of mature axiomatization?

Is Conditioning Really Incompatible with Holism

Jonathan Weisberg claims that certain probability assessments constructed by Jeffrey conditioning resist subsequent revision by a certain type of after-the-fact defeater of the reasons supporting those assessments, and that such conditioning is thus “inherently anti-holistic.” His analysis founders, however, in applying Jeffrey conditioning to a partition for which an essential rigidity condition clearly fails. Applied to an appropriate partition, Jeffrey conditioning is amenable to revision by the sort of after-the-fact defeaters considered by Weisberg in precisely the way that he demands.