For a university project, we were required to code our own Parallel Tempering Algorithm and use it to solve an Inverse Problem with 4 Parameters. Unfortunately, I'm not sure if I'm too stupid or have something completely mixed up.
The PT Algorithm should return an estimate of the Marginal Posterior Likelihood, which in theory should mean the sampler spends most of it's time at the point of highest likelihood. In the test case of using three bivariate gaussians plotting likelihood vs x/y results in the graph below displays this exactly.
Remain pics below as I can't post all of them
Any thoughts of what might be the issue? Assuming p4 can not be eliminated, can these results still be valid and how do I best evaluate the parameters of best fit? By likelihood or using other method? I already tried EM on the raw parameter data without likelihood but it fails miserably due to p4.
Thanks for any suggestions in advance. Hope the question has been asked clear enough and looking forward to your answers.