[R] MCMC sampling question
Thomas Mang
Thomas.Mang at fiwi.at
Wed Aug 12 08:46:32 CEST 2009
Hello,
Consider MCMC sampling with metropolis / metropolis hastings proposals
and a density function with a given valid parameter space. How are MCMC
proposals performed if the parameter could be located at the very
extreme of the parameter space, or even 'beyond that' ? Example to
express it and my very nontechnical 'beyond that': The von Mises
distribution is a circular distribution, describing directional trends.
It has a concentration parameter Kappa, with Kappa > 0. The lower kappa,
the flatter the distribution, and for Kappa approaching 0, it converges
into the uniform. Kappa shall be estimated [in a complex likelihood]
through MCMC, with the problem that it is possible that there truly
isn't any directional trend in the data at all, that is Kappa -> 0; the
latter would even constitute the H0.
If I log-transform Kappa to get in on the real line, will the chain then
ever fulfill convergence criteria ? The values for logged Kappa should
be on average I suppose less and less all the time. But suppose it finds
an almost flat plateau. How do I then test against the H0 - by
definition, I'll never get a Kappa = 0 exactly; so I can't compare
against that.
One idea I had: Define not only a parameter Kappa, but also one of an
indicator function, which acts as switch between a uniform and a
vonMises distribution. Call that parameter d. I could then for example
let d switch state with a 50% probability and then make usual acceptance
tests.
Is this approach realistic ? is it sound and solid or nonsense /
suboptimal? Is there a common solution to the before mentioned problem ?
[I suppose there is. Mixed effects models testing the variances of
random effects for 0 should fall into the same kind of problem].
cheers,
Thomas
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