spike and slab prior distributions

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Christos 2012-7-6

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hello. I am trying to compute a spike and slab prior distribution for regression coefficients to use in variable selection. Spike and slab distributions introduced in T.J Mitchell and J.J Beauchamp "Bayesian variable selection in linear regression". The definition is the following.

Pr(β_{j}=0)=h_{0j} and

Pr(β_{j}<b,β_{j} != 0)=(b+f_{j})h_{1j}, -f_{j}<b<f_{j} for f_{j} very large.

With h_{0j}+2h_{1j}f_{j}=1.

i would appreciate any suggestion. Thank you

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Answer 1

Star Strider 2012-7-7

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I assume the distributions in Figure 1 are calculated according to Eqn. 2.19 and Section 2.5, from the data they fitted to their multivariate linear model according to their procedure. After a few minutes looking through the paper (available here: http://www.isds.duke.edu/courses/Fall05/sta395/joelucas1.pdf) they also mention that ‘the posterior distribution of ß_j is a mixture of scaled and shifted t-distributions.’

So it would seem that you will have to find (or simulate) some multivariate data, scale it as they describe, and fit it according to their procedure. Then use their methods to calculate the distribution. It doesn't appear (in my brief reading of the paper, anyway) to have an analytic expression.

It's an approach to multivariate regression ('Bayesian' as compared to the more common 'Frequentist') that I wasn't aware of before (since I don't often do multivariate regression). I learned something.