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 {\mathcal N}(\mu,\sigma^2) model, when testing for \sigma=\sigma_0, 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.

Variations of three approximations to the Bayes factor

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.

4 Responses to “If the Dickey-Savage ratio does not hold…”

  1. […] lemma, Lebesgue’s dominated convergence theorem, matrix algebra, Euler’s formula, the Borel-Kolmogorov paradox, Taylor expansions (I dislike the use of HOT for “higher order terms” in math […]

  2. […] A+B+C=F… mess In the past days, travelling has been hampered by the snowy weather in and around Paris and I used my bike every time I could as the most reliable mean of transportation. Today, I resorted to the local train to reach Paris-Dauphine but the RER B line was incredibly slow, presumably because of the snow: I then tried to switch to the RER C line in Saint-Michel but found it was not operating at all when arriving on the platform, because of a fire in one suburban station!  I thus went back on the B line for one stop to use the RER A line instead, only to find on the platform that a technical problem on a train had stuckd all following trains in stations… I ended up in the metro instead with a few thousand other stuck passengers. And an half an hour delay. I had similar difficulties on the way back as the C line was still stuck on the B line but how could I complain when considering the fate of my colleague Jean-Michel Marin who came to work with me over these two days?! He got his bike stolen yesterday morning then his train was delayed close to two hours due to snow and cold and the major storm hitting the South-West this afternoon means he will be delayed at least four hours on his way back… (I am glad he came, despite those hardships, because we clarified a puzzling issue about Chib’s approximation.) […]

  3. […] A+B+C=F… mess In the past days, travelling has been hampered by the snowy weather in and around Paris and I used my bike every time I could as the most reliable mean of transportation. Today, I resorted to the local train to reach Paris-Dauphine but the RER B line was incredibly slow, presumably because of the snow: I then tried to switch to the RER C line in Saint-Michel but found it was not operating at all when arriving on the platform, because of a fire in one suburban station!,I thus went back on the B line for one stop to use the RER A line instead, only to find on the platform that a technical problem on a train had stuckd all following trains in stations… I ended up in the metro instead with a few thousand other stuck passengers. And an half an hour delay. I had similar difficulties on the way back as the C line was still stuck on the B line but how could I complain when considering the fate of my colleague Jean-Michel Marin who came to work with me over these two days?! He got his bike stolen yesterday morning then his train was delayed close to two hours due to snow and cold and the major storm hitting the South-West this afternoon means he will be delayed at least four hours on his way back… (I am glad he came, despite those hardships, because we clarified a puzzling issue about Chib’s approximation.) […]

  4. […] of the correctng factor close to 1/1.17, and very stable, which means the picture in the previous post is next to being correct! Still, there was a clear mistake in the original representation I should […]

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