# 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?

• hard to say without more information – Taylor Jan 29 '18 at 20:41

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 $-6.0$.