Abstract

This paper is concerned with developing a model of inductive inference of patterns in sets of numbers. In philosophy, this is known as the curve fitting problem, when the sets of numbers are sets of pairs of real numbers. In computer science, this is the standard way of framing the problem of inductive inference, when the sets of numbers are ordered sets of integers. A solution to the curve fitting problem should contribute to the understanding of scientific inference, since much of scientific inference consists of induction of patterns in numerical data. It is proposed that a pattern or theory about a set of numbers must be both accurate and simple. The problem with this proposal is that there is no good definition of simplicity. It is suggested here that simplicity is stability. Stability is the ability to resist damage due to random accidents. This heuristic definition leads to two precise definitions. The first precise definition is a measure of the stability of directed graphs. This is useful for measuring the stability of algorithms, since algorithms are naturally described by directed graphs . The second precise definition is a measure of the stability of linear regression equations. This provides a solution to the curve fitting problem for linear regression equations