MCMC Sampling and Boundary Penalty

[aliakhavan89]

Is there a way to avoid sampling from parameter boundaries, apart from manually defining a penalty term in the payoff function? I am fine to get a uniform distribution if MCMC cannot recover a proper posterior distribution. By the way, I have not defined a probability distribution for the prior.

Screenshot 2024-07-16 at 3.53.14 PM.png (80.27 KiB) Viewed 10389 times

[tomfid]

I guess the likelihood was also uniform over this experiment?

Reflecting off the boundaries is tricky in multidimensions, but we could probably improve on this behavior. In the short run I don't have a good solution though.

A couple possibilities would be to use the logistic transform on the parameter, or give it a Beta prior. For the latter, you can ignore the gamma terms and use x^(a-1)*(1-x)^(b-1) with a=b>>1.

It might be possible to suggest other options if we knew what the param represents.

[aliakhavan89]

Thank you! It is a normal log-likelihood. I have attached the toy model here. These are great suggestions. I think defining a prior and including it (mean and standard deviation) in the payoff as the initial condition would resolve the issue. But I guess I should keep the barrier penalty.

[tomfid]

What param are we looking at in the histogram?

[tomfid]

Oops - missing data.vdfx.

[tomfid]

Anyway ... I can replicate this with a simpler example, so no need. Will explore a better boundary reflection approach.

[tomfid]

Misspoke earlier though. For the Beta coefficients, a=b>>1 gives a centered Normalish distribution around 0.5. If you want something flattish, but avoiding extremes, you'd want 1<a=b<2.

[aliakhavan89]

Hi Tom, Sorry for not uploading the data. I have attached the whole folder here. I did a quick test with the boundary penalty, and it gave a me a very good result, though I added an additional parameter (Exp Sigma, which is not shown in the graph). I'll try to implement your suggestions. Thanks a lot!

Screenshot 2024-07-18 at 2.45.40 PM.png (371.21 KiB) Viewed 10293 times

[aliakhavan89]

I borrowed the boundary penalty formulation from a template that normalizes the estimated parameters, but I didn't properly adjust the equation based on the parameter boundaries I have in the model. So, I got it to work by accident. I'll share a revised version, though now I have some doubts about the template.

[tomfid]

I modified the hard bounds constraint behavior so the concentration of mass at the bound no longer happens, at least in a case like this with a uniform likelihood and (implicitly) flat prior over a unit hypercube.

It probably still makes sense to apply some kind of prior within the range. We plan to make this easier, so that you can specify priors within the .voc list, though more complex situations (hierarchy) may still require equations.

[tomfid]

This experiment does suggest another interesting case: if all the likelihoods and/or priors have a beta distribution with small a,b then all the density should be concentrated at the extremes of the range (or worse, the corners of the hypercube if it's multidimensional). In theory the algorithm can handle this, but I'm not sure how true it is in practice. It might make sense to apply some kind of rank transformation to normalize things.