Thomas Mang wrote:
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' ?
Just like others. The density at the edge of the space determines
whether you'll accept a move there, the density outside the space is
zero, so you won't.
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 ?
Sure, but remember to transform the density in a corresponding way.
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.
What does MCMC have to do with hypothesis testing? Standard hypothesis
testing has to do with the distribution of the data, not the likelihood
or posterior distribution of some parameter.
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].
What you're describing is an approach to Bayesian hypothesis testing.
I've never been convinced that Bayesian hypothesis testing is a good
approach, but some people use it.
Another way to formulate this approach is to use a prior with a point
mass at kappa = 0. You need to use a non-standard density to do
Metropolis-Hastings (I think Metropolis won't work), but MCMC is
possible. (The density needs to be evaluated as a discrete measure at 0
and a continuous measure everywhere else.)
Duncan Murdoch
cheers,
Thomas
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