Best way of using MAP/ML point when fitting about dynesty

[lalit-pathak]

So if we define a ball/ellipsoid/cube in the posterior space around the ML point, How do we exactly set the boundaries? In the case of small boundaries, we could get railings in the posteriors, right? In that case, we would either need to repeat the exercise for various boundaries or make an effective fisher matrix covering some given volume of the posterior.

[segasai]

My thinking was that if we define an ellipsoid around the MAP value and then maybe sample from the prior that has 99% of the volume inside the ellipsoid and 1% outside. This way the majority of the sampling will be focused on the ellipsoid, but if there is substatial posterior volume outside, it'll likely still be captured. But this is a vague idea, I am not sure it's implementable.

Specifically if x is parameter within the unit Cube then the posterior is just
$1/Z* L(x) $
, but now if we adopt the prior
$\pi(x)$
so with the volume requirement given above and then we'd sample the posterior of the form
$\pi(x) * (1/z * 1/\pi(x) * L(x))$
This is technically the same posterior as before, but the sampling will mostly avoid low L regions.

The problem is I'm not sure there is a parameter transformation implementing this kind of prior.

[mvsoom]

One way of using the MAP approximation is to fit a MVN to it (Laplace approximation) and use that as a proposal distribution, to be incorporated into the prior, much like the expressions in your previous reply. Here is a fine short paper exploring this idea: https://arxiv.org/pdf/2212.01760.pdf.

[lalit-pathak]

@mvsoom Thanks for posting this nice paper here. I will look into it.