[digitalzombie]

I think you create a markov chain and from there see if the chain end up in a steady state somewhere.

Bayesian use MCMC to indirectly get the joint distribution without actually doing integrating by sampling the distribution and getting the average from the chain. You start anywhere reasonable or guess where to start from the chain so you usually burn the first 10-15% of the chain and get the average of the chain.

You can have multiple chains in the simulation each chain represent a parameter estimate.

edit:

The reason why Bayesian use MCMC is because integration is hard and each model have their own different integration problem. If you choose to model salmon migration you may have your own take on the model and in the end you have this nasty integration for that take of your model. If you change your model you have a new integration problem... You can try to integrate it or you can just MCMC it and by pass integration. Instead of integrating to get the PDF you can just sample from the joint distribution and estimate it (the parameters of the distributions) from the sample.