J. B. S. Haldane's Rule of Succession

TL;DR

This paper explores J. B. S. Haldane's rule of succession, its historical development, variants, and philosophical implications, highlighting how these rules improve predictive intuition over Laplace's original formulation.

Contribution

It provides a comprehensive analysis of Haldane's rule, its variants, and their historical and philosophical context, enhancing understanding of Bayesian predictive rules.

Findings

Haldane's rule offers a positive probability for universal generalization.

Variants of the rule better align with intuition and common sense.

Historical analysis clarifies the development of Bayesian predictive methods.

Abstract

After Bayes, the oldest Bayesian account of enumerative induction is given by Laplace's so-called rule of succession: if all $n$ observed instances of a phenomenon to date exhibit a given character, the probability that the next instance of that phenomenon will also exhibit the character is $\frac{n+1}{n+2}$. Laplace's rule however has the apparently counterintuitive mathematical consequence that the corresponding "universal generalization" (every future observation of this type will also exhibit that character) has zero probability. In 1932, the British scientist J. B. S. Haldane proposed an alternative rule giving a universal generalization the positive probability $\frac{n+1}{n+2} \times \frac{n+3}{n+2}$. A year later Harold Jeffreys proposed essentially the same rule in the case of a finite population. A related variant rule results in a predictive probability of $\frac{n+1}{n+2} \times \frac{n+4}{n+3}$. These arguably elegant adjustments of the original Laplacean form have the advantage that they give predictions better aligned with intuition and common sense. In this paper we discuss J. B. S. Haldane's rule and its variants, placing them in their historical context, and relating them to subsequent philosophical discussions.