Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Abstract

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference -the logic of partial beliefs-can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers π, e, √ 2, and ln 2) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four constants, the evidence in favor of the general law is overwhelming. We argue that the Bayesian paradigm is particularly apt for applications in experimental mathematics, a field in which the plausibility of a general law is in need of constant revision in light of data sets whose size is increasing continually and indefinitely.

Replication Bayes factors from evidence updating

“…Specifically, the Bayes factor allows the researcher to quantify evidence to discriminate between absence of evidence (i.e., BF 01 ( d ) ≈ 1) versus evidence of absence (i.e., BF 01 ( d ) ≫ 1). The Bayes factor also allows one to monitor the evidence as the data come in (Gronau and Wagenmakers, 2017) and to design experiments in order to ensure compelling evidence. Finally, the Bayes factor can also be used to quantify replication success, a topic to which we turn next.…”

Replication Bayes Factors from Evidence Updating

“…Specifically, the Bayes factor allows the researcher to quantify evidence to discriminate between absence of evidence (i.e., BF 01 (d) ≈ 1) versus evidence of absence (i.e., BF 01 (d) 1). The Bayes factor also allows one to monitor the evidence as the data come in (Gronau and Wagenmakers, 2017) and to design experiments in order to ensure compelling evidence. Finally, the Bayes factor can also be used to quantify replication success, a topic to which we turn next.…”

Bayesian Perspectives on Mathematical Practice

“…[Marsaglia 2005] (Some claims that the digits of π fail a more subtle statistical test for randomness in [Ganz 2014] with debate in [Bailey et al 2017] and [Ganz 2017]. ) Gronau and Wagenmakers [2018] argue that this is a perfect case for appreciating the Bayesian perspective in pure mathematics. They apply a formal Bayesian analysis to the hypothesis that π is normal (in base 10), that is, that the ten digits appear equally often in its decimal expansion.…”