If the Dickey-Savage ratio does not hold…

or rather is arbitrary, then this should cause problems for methods, papers, and books that do use it, no…?! This is the question I got on Monday from an Og reader.

Actually, not necessarily: when looking for instance at O’Hagan and Forster’s Advanced Theory of Statistics 2B (2004, 7.16), the Bayes factor they construct in the model, when testing for , is correct for the same reason the “proof” of the Dickey-Savage ratio was accepted, namely the use of the “right” version of the conditional density. Similarly, when Chen, Shao and Ibrahim (2000, Section 5.10.3) introduce the ratio, and the Verdinelli-Wasserman (1996) generalisation, they implement the Monte Carlo approximation using a specific version of the full conditionals.

The plus of our perspective is therefore to give a general representation that does not involve unnatural constraints on the priors. The example below is taken from a probit example in the incoming note with Jean-Michel Marin, comparing the Dickey-Savage-like approximation to the Bayes factor with an harmonic mean version and the unbeatable Chib’s solution.

I nonetheless found [by Googling] one case where the Dickey-Savage ratio was leading to a definition problem, namely in a preprint posted by Wetzels, Grasman, and Wagenmakers,  where the authors derive the Bayes factor as the Dickey-Savage ratio under an encompassing prior constructed earlier by Klugkist et al. (2005), using a limiting argument via L’Hospital rule that seems equally contradictory with measure theoretic principles. Paradoxically, the authors mention the Borel-Kolmogorov paradox, i.e. the dependence on the conditioning σ-algebra, as a possible issue with their prior construction but, while their appendix A clearly concludes that the limit is arbitrary, they  still evacuate the issue of the choice of the version of the conditional density.