Model Selection in Science: The Problem of Language Variance

Abstract

ABSTRACTRecent solutions to the curve-fitting problem, described in Forster and Sober ([1994]),trade off the simplicity and fit of hypotheses by defining simplicity as the paucity ofadjustable parameters. Scott De Vito ([1997]) charges that these solutions are ‘conven-tional’ because he thinksthat the number ofadjustable parameters maychange whenthehypotheses are described differently. This he believes is exactly what is illustrated inGoodman’s new riddle of induction, otherwise known as the grue problem. However,the ‘number of adjustable parameters’ is actually a loose way of referring to a quantitythat is not language dependent. The quantity arises out of Akaike’s theorem in a waythat ensures its language invariance. 1 Introduction2 A methodological puzzle redescribed3 The curve-fitting problem redefined4 Akaike’s theorem revisited5 The grue problem as a curve-fitting problem6 Language invariance restored 1 Introduction Five years ago Elliott Sober and I published an article on the general problemof selecting from amongst a set of quantitative models (Forster and Sober[1994]). The article described a solution to the problem first worked out indetail by Akaike ([1973]). The idea has not been widely publicized evenamongst statisticians, until recently. Since then, there has been a growinginterest in the subject, especially on the part of scientists themselves.