Fitting Model with MCMC?

In fitting a model with several variable I have found extremely useful a method involving the minimization of the Chi Square using a MCMC approach. In particular, I followed this tutorial http://sciencehouse.wordpress.com/2010/06/23/mcmc-and-fitting-models-to-data/

However, I cannot find around any other source. Since I will use this method for a scientific work, I would like to a) be sure that the method is theoretically correct b) cite a peer-reviewed source.

Running several simulation with complex model (6/7 parameters) showed me that the model actually works really fine, retrieving the parameter within a small error most of the time.

In particular, I am not sure about the use of $exp(-\chi^2)$ for the likelihood. Is this justified? I also tried to calculate the ratio by just dividing the oldChi2 with the newChi2, and the results were, again, satisfying. Is there any relevant difference between these two methods? Thank you

• This is nothing fancy, it's simply a Metropolis algorithm. Also, if you look closely, $\exp\left(-\chi^2 \right)$ is a Gaussian density. Aug 26, 2014 at 18:32
• I understand that, but I wonder if there is any important theoretical reason to use exp(-X^2). Could I just calculate the likelihood ratio as X^2_old/X^2new? Aug 26, 2014 at 22:35
• If you want to use the Metropolis algorithm as in the blog post, then you must use the exponential function. The Metropolis algorithm uses the ratio of densities to accept or reject proposed values. If you drop the exponential, then you are not using the ratio of densities. Aug 27, 2014 at 11:47