Bayesian Statistics: An Introduction, 4th Edition

3.8 The first digit problem; invariant priors

3.8.1 A prior in search of an explanation

The problem we are going to consider in this section is not really one of statistical inference as such. What is introduced here is another argument that can sometimes be taken into account in deriving a prior distribution – that of invariance. To introduce the notion, we consider a population which appears to be invariant in a particular sense.

3.8.2 The problem

The problem we are going to consider in this section has a long history going back to Newcomb (1881). Recent references include Knuth (1969, Section 4.2.4B), Raimi (1976) and Turner (1987).

Newcomb’s basic observation, in the days where large tables of logarithms were in frequent use, was that the early pages of such tables tended to look dirtier and more worn than the later ones. This appears to suggest that numbers whose logarithms we need to find are more likely to have 1 as their first digit than 9. If you then look up a few tables of physical constants, you can get some idea as to whether this is borne out. For example, Whitaker’s Almanack (1988, p. 202) quotes the areas of 40 European countries (in square kilometres) as

28 778; 453; 83 849; 30 513; 110 912; 9251; 127 869; 43 069; 1399; 337 032; 547 026; 108 178; 248 577; 6; 131 944; 93 030; 103 000; 70 283; 301 225; 157; 2586; 316; 1; 40 844; 324 219; 312 677; 92 082; 237 500; 61; 504 782; 449 964; 41 293; 23 623; 130 439; 20 768; 78 772; 14 121; 5 571 000; 0.44; 255 804.

The ...