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 and cold (-18C in some southern suburbs!): 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.)
Archive for Chib’s approximation
A+B+C=F… mess
Posted in Travel with tags Chib's approximation, Paris, RER, train on January 9, 2010 by xi'anOn resolving the Savage-Dickey paradox
Posted in R, Statistics with tags Bayes factor, Bayesian model choice, bridge sampling, Chib's approximation, Dickey-Savage ratio, importance sampling, Monte Carlo methods, R program on November 16, 2009 by xi'anAfter a long month of polishing our explanations and adding a full-scale comparison with the solution of Verdinelli and Wasserman (1995). we have now completed (and rearXived) our rewriting of the paper on the Savage-Dickey paradox. And we eventually made the submission to the Annals of Statistics, with the hope that the mix of measure-theory and computational techniques therein would appeal to the journal. In this extended version, we show that the Verdinelli and Wasserman (1995) representation
![B_{01} = \dfrac{\pi_1(\theta_0|x)}{\pi(\theta_0)} \, \mathbb{E}^{\pi_1(\psi|\theta_0,x)}[\pi_0(\psi)/\pi_1(\psi|\theta_0)]](https://s0.wp.com/latex.php?latex=B_%7B01%7D+%3D+%5Cdfrac%7B%5Cpi_1%28%5Ctheta_0%7Cx%29%7D%7B%5Cpi%28%5Ctheta_0%29%7D+%5C%2C+%5Cmathbb%7BE%7D%5E%7B%5Cpi_1%28%5Cpsi%7C%5Ctheta_0%2Cx%29%7D%5B%5Cpi_0%28%5Cpsi%29%2F%5Cpi_1%28%5Cpsi%7C%5Ctheta_0%29%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
only depends on a specific version of the conditional density
where 
We also found that a quick-and-dirty sufficient condition for the above Verdinelli and Wasserman (1995) representation to apply is that Bayes’ theorem
holds in 
![B_{01} = \dfrac{\tilde\pi_1(\theta_0|x)}{\pi(\theta_0)} \, \mathbb{E}^{\pi_1(\theta,\psi|x)}[\pi_0(\psi)/\pi_1(\psi|\theta_0)]](https://s0.wp.com/latex.php?latex=B_%7B01%7D+%3D+%5Cdfrac%7B%5Ctilde%5Cpi_1%28%5Ctheta_0%7Cx%29%7D%7B%5Cpi%28%5Ctheta_0%29%7D+%5C%2C+%5Cmathbb%7BE%7D%5E%7B%5Cpi_1%28%5Ctheta%2C%5Cpsi%7Cx%29%7D%5B%5Cpi_0%28%5Cpsi%29%2F%5Cpi_1%28%5Cpsi%7C%5Ctheta_0%29%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
has the same unbiasedness and variability as the Verdinelli and Wasserman (1995) alternative on the Pima Indian benchmark used in our survey of Bayes factor approximation methods. The theoretical comparison between both Monte Carlo solutions remains an open question. The R objects needed to produce the boxplots above are available both in Dauphine and Montpellier.
