Computing evidence

Posted in Books, R, Statistics with tags , , , , , , , , , , on November 29, 2010 by xi'an

The book Random effects and latent variable model selection, edited by David Dunson in 2008 as a Springer Lecture Note. contains several chapters dealing with evidence approximation in mixed effect models. (Incidentally, I would be interested in the story behind the  Lecture Note as I found no explanation in the backcover or in the preface. Some chapters but not all refer to a SAMSI workshop on model uncertainty…) The final chapter written by Joyee Ghosh and David Dunson (similar to a corresponding paper in JCGS) contains in particular the interesting identity that the Bayes factor opposing model h to model h-1 can be unbiasedly approximated by (the average of the terms)

$\dfrac{f(x|\theta_{i,h},\mathfrak{M}=h-1)}{f(x|\theta_{i,h},\mathfrak{M}=h)}$

when

• $\mathfrak{M}$ is the model index,
• the $\theta_{i,h}$‘s are simulated from the posterior under model h,
• the model $\mathfrak{M}=h-1$ only considers the h-1 first components of $\theta_{i,h}$,
• the prior under model h-1 is the projection of the prior under model h. (Note that this marginalisation is not the projection used in Bayesian Core.)