Remarks on the Concept of Empirical Approximation

Abstract

These remarks are intended as an improved definition of Moulines’ concepts of empirical uniformity as provided in [1] p. 218. Moulines’ intention is to set forth conditions on uniformities on the class Mp of all potential full models of a given empirical theory so that its restriction to the class Mpp of all its partial models be a uniformity on Mpp. Here it is shown that Moulines’ condition (Def. 4) is too strict to be of general interest and a new definition is offered. According to this new definition, an empirical uniformity on Mp is a uniformity, U, such that every set in U contains all pairs of expansions to Mp of any single partial model. It is shown that this new definition reflects the old’s intuitive content and that empirical uniformities are precisely those obtained by expansion to Mp of arbitrary uniformities on Mpp.