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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. log likelihood vs. X

When running the inverse model, the likelihood vs parameter plots look quite a bit off. Additionally, one of the parameters seems to not influence the likelihood at all. log likelihood vs p1

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.

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So here are the remain pics. P3 looks exactly like P2 so leaving it out

Log likelihood vs p2 log likelihood vs p4

Again thanks for any suggestions

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