A few days ago I posted the following question to the list:
>One of my students is fitting a Poisson regression model, and getting BUGS to calculate the deviance at each step. She was interpreting the mean deviance in the standard GLM way, i.e. comparing it to the degrees of freedom. That doesn't seem right (the deviance in the simulations should often be bigger than it is at the MLE), but I don't know how to interpret the value, other than as a convergence diagnostic, i.e. once it has stopped dropping, we're in the neighbourhood of the mode.
>
>Is there some other interpretation of the expected deviance with respect to the posterior?
I received two responses, and several requests for a summary, so here
it is.
Stefan Van Dongen pointed out that the definition of deviance in
WinBUGS is -2*log(likelihood). To give it the interpretation my
student was using, it's necessary to subtract off -2*log(likelihood of
saturated model).
I ran a little test and confirmed that WinBUGS doesn't do this. It
behaves as documented, but I think this change would make the feature
more useful. As far as I know, in exponential families "saturated
model" is well-defined; it may not be with other distributions.
Kenneth Rice said:
>This is something of an open question! You may find a recent paper by
>David Spiegelhalter et al useful, it's available here
>
>http://www.rss.org.uk/publications/preprints.html#130302
The full reference for that paper is Spiegelhalter, D.J., Best, N.G.,
Carlin, B.P. and van der Linde, A. (2002). Bayesian measures of model
complexity and fit. JRSS B 64, 1--34.
I haven't read the whole paper in detail yet, but what it does is
suggest that the expected deviance w.r.t. the posterior can be used to
measure the complexity of a hierarchical model (an "equivalent degrees
of freedom" measure), as well as the goodness of fit.
Thanks to Stefan and Kenneth for your responses.
Duncan Murdoch
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