posterior probability of modes

[Radek Poleski]

Dear All,

I'm working on a problem in which the posterior has 4 separate modes, let's call them A, B, C, and D. The modes A and B are close enough that the walkers can move from one to another (they have overlapping 3-sigma contours). Same is for C and D, but not A and C, B and C, A and D, or B and D. Is it possible to get posterior probability of these modes from EMCEE? As I understand, I can define the border between A and B, count posterior samples in each of them n(A) and n(B), and take posterior probability p(A)/p(B) = n(A)/n(B). But what about comparing A+B vs C+D? Can I start the walkers from uniform distributions, so that each can fall in either of the modes, and then take [p(A)+p(B)] / [p(C)+p(D)] = [n(A)+n(B)] / [n(C)+n(D)]? Since the walkers are not able to move from modes A or B to C or D, then these probabilities are set after a small number of steps. Hence, I should run many walkers with a small number of steps to get the posterior probability of the modes. Is it a legitimate approach?

Kind regards,

Radek

[Dan Foreman-Mackey]

Hi Radek,

emcee isn't going to be the right tool for this job unfortunately. You can't rely on the right numbers of walkers ending up in the relevant modes as a way to estimate their relative mass. You could potentially define a custom move to swap modes and then use ensemble moves for local exploration, but that might be more of a research problem. I'd recommend something like nested sampling here either hoping that it can move between modes or by separately sampling in the two regimes and combining the results weighted by the estimated evidence.

Best,

Dan

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Dan Foreman-Mackey

Associate Research Scientist

Flatiron Institute

https://dfm.io

[Radek Poleski]

Hi,

Ok, it's not possible to use EMCEE for these 4 modes, but what about simpler problems. Imagine I have only 2 modes and the walkers are able to move from one to another or back. Can I calculate their relative posterior probability using EMCEE?

Cheers,

Radek

[Dan Foreman-Mackey]

Yes - if all the walkers are sufficiently moving between modes then the relative number of samples in each mode give you the relative masses. 

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