Exchangeable measures for subshifts

Let \Om be a Borel subset of $S^\Bbb N$ where $S$ is countable. A measure is called exchangeable on \Om, if it is supported on \Om and is invariant under every Borel automorphism of \Om which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when \Om=S^\Bbb N. We apply the ergodic theory of equivalence relations to study the case \Om\neq S^\Bbb N, and obtain versions of this theorem when \Om is a countable state Markov shift, and when \Om is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint)

Being More Realistic About Reasons: On Rationality and Reasons Perspectivism

This paper looks at whether it is possible to unify the requirements of rationality with the demands of normative reasons. It might seem impossible to do because one depends upon the agent’s perspective and the other upon features of the situation. Enter Reasons Perspectivism. Reasons perspectivists think they can show that rationality does consist in responding correctly to reasons by placing epistemic constraints on these reasons. They think that if normative reasons are subject to the right epistemic constraints, rational requirements will correspond to the demands generated by normative reasons. While this proposal is prima facie plausible, it cannot ultimately unify reasons and rationality. There is no epistemic constraint that can do what reasons perspectivists would need it to do. Some constraints are too strict. The rest are too slack. This points to a general problem with the reasons-first program. Once we recognize that the agent’s epistemic position helps determine what she should do, we have to reject the idea that the features of the agent’s situation can help determine what we should do. Either rationality crowds out reasons and their demands or the reasons will make unreasonable demands

Exchangeable random measures

Let A be a standard Borel space, and consider the space A^{\bbN^{(k)}} of A-valued arrays indexed by all size-k subsets of \bbN. This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations of \bbN. The main result is a representation theorem for such `exchangeable' random measures, obtained using the classical representation theorems for exchangeable arrays due to de Finetti, Hoover, Aldous and Kallenberg. After proving this representation, two applications of exchangeable random measures are given. The first is a short new proof of the Dovbysh-Sudakov Representation Theorem for exchangeable PSD matrices. The second is in the formulation of a natural class of limit objects for dilute mean-field spin glass models, retaining more information than just the limiting Gram-de Finetti matrix used in the study of the Sherrington-Kirkpatrick model.Comment: 24 pages. [4/23/2013:] Re-written for clarity, but no conceptual changes. [9/12/2013:] Slightly re-written to incorporate referee suggestions. [7/8/15:] Published version available at http://projecteuclid.org/euclid.aihp/143575923