STAT 330 October 25, 2012

Here is the next problem set, due on Thursday, November 1.

We discussed the Errors-In-Variables case and showed how to sample this in a Gibbs sampler exactly.

We then discussed priors. I noted that if you have substantial information about the parameters of a problem you should NOT use the automatic priors we’re discussing here, you should use a prior that makes use of that information. Not to use information that you have is not a good idea.

But in many cases you do not have such information, and in these cases you can use one of these “automatic” priors with integrity.

The first example we discussed was if there is some sort of group invariance, like dice or coins (where you cannot distinguish a priori between various states, so that a prior that does not inject unknown information would be equal on all cases). Or we may think that the origin of our coordinate system should not make any difference about our results, in which case a reasonable prior would not respect translations of the origin, and would be constant. We also discussed the case where there is a scale variable, and if we think that it should not matter what units we use to describe that variable, then we should choose a prior that looks like the reciprocal of the scale variable, i.e., if s is the scale, then use prior 1/s; or transform to variable u=log(s), and in u the prior would be constant. (This is a good idea just for the improvements that sampling on u would provide).

I also discussed the situation that can present itself with some situations, where the prior generated by a left-invariant Haar prior can be different from those generated by a right-invariant Haar prior. Jim Berger strongly advocates the right-invariant choice.

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