Log posterior probability in MCMC is decreasing I am running a Markov chain Monte Carlo simulation (of phylogenetic trees using Beast2) and checking the traces, I notice that the log of the posterior probability has been generally decreasing by 1000 log units over the last 400000 samples.
I would expect the posterior to move upwards in general until it approaches a (local) maximum and then to jump around in that region, only rarely leaving it for lower-probability areas and probably returning soon.
Is something wrong with my expectation; or do I need to check my setup (in particular my operators – but they are standard Beast and should be tested to be reasonably symmetric); or should I just wait much longer because it's likely that the chain will move somewhere else again and increase in probability?
 A: Your expectation that the posterior probability should generally increase while the Markov chain has not converged yet is false. This can be seen in the following simple example.
Let your chain sample a random variable (prior and likelihood do not matter here, so I give just the posterior) which is in state 0 with a probability of $\frac{1}{4}$ and in each of 300 different other states with probability $\frac{1}{400}$. Then you will see the chain (which in this case is might even be IID) will be in $log(p) \approx -6.0$ states for most of the time, and when it's not, it is very likely that you will observe a decrease from $-1.4$ to $-6.0$.
So if you have many states each having a probability density of those 1000 log units lower, and only relatively few at the higher log posterior level, such a decrease in posterior probability is not by itself a concern – you mention that you sample a tree space, and as you will know, tree spaces are huge, so don't take this as an alarm signal all by itself.
