Many thanks to Andrew Millard, Durham University, for providing the following response to my enquiry concerning the imposition of constraints through post simulation MCMC sample selection. (My BUGS List submission is given below).
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Andrew's Reply
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MCMC is a method to obtain a sample from a distribution. To obtain a valid sample from the target distribution certainly requires reversibility, but once you have the sample it can be treated like any other sample from a distribution. So combining information about regions with zero probability with the validly obtained draw from a distribution where those regions had non-zero probability, one can reject samples. In effect this is a subsequent updating of knowledge using Bayes theorem to combine the MCMC posterior distribution and the constraint.
Best wishes
Andrew
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Dr. Andrew Millard [log in to unmask]
Durham University
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My Submission
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Dear BugsList,
I am uncertain about the validity of using post MCMC simulation sample selection as a mechanism for apply a constraint, one that cannot be expressed as a prior constraint. Some time ago I performed an MCMC analysis in which I rejected a few samples that gave implausible derivatives for a given function of the model parameters. On that occasion the proportion of rejected samples was small. But I had thought that this approach was not strictly valid because the resulting chain would not satisfy the reversibility criterion.
Subsequently, I have noticed several MCMC experts suggesting something along these lines. For example, in the context of imposing invertibility in AR models, Chib (1993, J. Econometrics 58:275) states (p 281) that the AR coefficient can be drawn form the untruncated posterior and retained if the roots lie outside the unit circle. In the same context Peter Congdon (Bayesian Statistical Modelling, 2nd Ed. page 251) refers to online rejection or, alternatively, subsequent selection of samples satisfying invertibility. As an aside, if one was to write custom Gibbs sampler code, as opposed to using WinBUGS, should the algorithm repeatedly sample the AR coefficient within an iteration, until the condition was met, or repeat sample the entire set of conditionals.
My questions is whether this sample rejection approach is theoretically valid or, alternatively, not strictly valid but accepted in practice.
With many thanks,
Martin King
Imaging and Biophysics,
UCL Institute of Child Health
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