## normality test with 10⁸ observations?

**Q**uentin Gronau and Eric-Jan Wagenmakers just arXived a rather exotic paper in that it merges experimental mathematics with Bayesian inference. The mathematical question at stake here is whether or not one of the classical irrational constants like π, e or √2 are “normal”, that is, have the same frequency for all digits in their decimal expansion. This (still) is an open problem in mathematics. Indeed, the authors do not provide a definitive answer but instead run a Bayesian testing experiment on 100 million digits on π, ending up with a Bayes factor of 2×10³¹. The figure is massive, however one must account for the number of “observations” in the sample. (Which is not a statistical sample, strictly speaking.) While I do not think the argument will convince an algebraist (as the counterargument of knowing nothing about digits after the 10⁸th one is easy to formulate!), I am also uncertain of the relevance of this huge figure, as I am unable to justify a prior on the distribution of digits if the number is not normal. Since we do not even know whether there are ~~non-~~normal numbers outside rational numbers. While the flat Dirichlet prior is a uniform prior over the simplex, to assume that all possible probability repartitions are equally possible may not appeal to a mathematician, as far as I [do not] know! Furthermore, the multinomial model imposed on (at?) the series of digit of π does not have to agree with this “data” and discrepancies may as well be due to a poor sampling model as to an inappropriate prior. The data may more agree with H⁰ than with H¹ because the sampling model in H¹ is ill-suited. The paper also considers a second prior (or posterior prior) that I do not find particularly relevant.

For all I [do not] know, the huge value of the Bayes factor may be another avatar of the Lindley-Jeffreys paradox. In the sense of my interpretation of the phenomenon as a dilution of the prior mass over an unrealistically large space. Actually, the authors mention the paradox as well (p.5) but seemingly as a criticism of a frequentist approach. The picture above has its lower bound determined by a virtual dataset that produces a χ² statistic equal to the 95% χ² quantile. Dataset that stills produces a fairly high Bayes factor. (The discussion seems to assume that the Bayes factor is a one-to-one function of the χ² statistics, which is not correct I think. I wonder if exactly 95% of the sequence of Bayes factors stays within this band. There is no theoretical reason for this to happen of course.) Hence an illustration of the Lindley-Jeffreys paradox indeed, in its first interpretation of the clash between conclusions based on both paradigms. As a conclusion, I am thus not terribly convinced that this experiment supports the use of a Bayes factor for solving this normality hypothesis. Not that I support the alternative use of the p-value of course! As a sidenote, the pdf file I downloaded from arXiv has a slight bug that interacted badly with my printer in Warwick, as shown in the picture above.

February 23, 2016 at 9:08 am

You write “Since we do not even know whether there are non-normal numbers outside rational numbers.” I think existence of non-normal numbers is not a problem: For example, if we intersperse the digits of pi with zeros to get 3.01040105090206050305… then the resulting number is not rational (since the decimal expansion is not periodic) and also not normal (since at least 50% of the digits are 0).

February 23, 2016 at 10:05 am

OK, fair enough! And do we know of any normal number that are not artificially reconstructed, i.e. do not take an infinite time to produce?