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, , 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.

KEYWORDS:

• Bayes factor
• normality
• evidence accumulation

2010 AMS SUBJECT CLASSIFICATION:

• 62F15

Notes

1 Such an excessive degree of evidence in favor of a general law may well constitute a world record.

2 That is, after a sufficient number of observations, the trajectories of the log Bayes factors for the different priors for are equal, only shifted by a constant. In fact, regardless of the irrational number under consideration, this constant—which corresponds to the difference in and —approaches 18.39 (for a derivation see https://osf.io/m5jas/).

3 Data were obtained using the pifast software ( numbers.computation.free.fr/Constants/PiProgram/pifast.html).

4 A frequentist statistician may object that this is a sequential design whose proper analysis demands a correction of the α-level. However, the same data may well occur in a fixed sample size design. In addition, the frequentist correction of α-levels is undefined when the digit count increases indefinitely.

5 Data were obtained using the pifast software ( numbers.computation.free.fr/Constants/PiProgram/pifast.html).

6 Data were obtained using the pifast software ( numbers.computation.free.fr/Constants/PiProgram/pifast.html).

7 Of course, in some cases this may require a very “rich” mixture, that is, a mixture prior with many components.

8 R code that allows one to explore how the results change for a different choice of a two-component Dirichlet mixture prior is available on the Open Science Framework under https://osf.io/cmn2z/.