[R] MCMC sampling question

[Duncan Murdoch]

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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