The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

TL;DR

This paper introduces a new posterior metric to analyze the continuity of conditional probabilities in transformed Gibbs measures, applicable to systems like time-evolved or noisy observations, providing a unified approach to spatial and local estimates.

Contribution

It develops a general framework using the posterior metric for deriving continuity estimates in transformed Gibbs measures without relying on a prior metric on local spaces.

Findings

Posterior metric effectively separates local and spatial continuity estimates.

Method applies to continuous spin models and time-evolved systems.

Concrete example with rotators on a sphere demonstrates practical utility.

Abstract

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the q-1 dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric.