I am trying to use MCMC to obtain the posterior probabilities of the free parameters of a model. I have tried first to leave two free parameters for my model and I was able to estimate the posterior probability for these free parameters using EMCEE. The reason that I am confident about the results is that I have tested them on simulated data and the peaks of posteriors were very close to true values.
Now, I have set four free parameters for this model but the posteriors that I have obtained is basically similar to my priors. I have chosen uniform priors for two new parameters that I left free. The posteriors for these two free parameters got uniform distributions with some spikes.
I have to mention that the characteristic property of the data is that it is very noisy. I am wondering whether I have messed up with the new likelihood that I set up for four parameters or whether MCMC can not constrain the free parameters given the property of the data (being noisy) or I need to use special type of resampling to obtain the proper posterior?
Update: Here is a file contain the information of all points have been sampled by parallel tempering method. I have used 20 temperatures, 500 walkers and for 200 times iterations. I flatted the whole resampled data for different temperatures.
In the following I computed the likelihood for the gridded values of two new parameters fixing two other parameters. The maximum value of likelihood is in agreement with the true value in the mock data. However when I leave all four parameters free and use MCMC to get the posteriors for them it fails.
I have another question: The quantity that I measure with the model is the orientation of each object with respect to the center for the two given free parameters of the model. However, each object has an intrinsic orientation which is the main source of error in this measurement. At the moment, I use the rms value of all the object's orientations as $\sigma$ in the likelihood but how could I include the effect of each individual intrinsic orientation of object in the likelihood in order to get a better un-biased estimator?