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Let's say I'm trying to sample from a posterior distribution, $p(\theta|x)$, which I suspect is very "bumpy" (10s to 1000s of small modes). Moreover, the evaluation of the posterior kernel $K(\theta|x) \propto p(\theta|x)$ is very expensive to evaluate (and not conjugate) and the data set is large enough such that I would like to avoid any technique based on parallel chains.

However, I would like to avoid being stuck in any one mode for too long during the sampling process, so I employ a standard annealed MCMC technique (such as Metroplis-Hastings with an annealed acceptance probability) but I never cool the temperature parameter. The distribution I'm sampling from is thus "flattened", and easier to sample from in the case of many modes, but the stationary distribution for the Markov chain has clearly changed from the one defined by $K(\theta|x)$.

Does anyone have any references about what happens to the stationary distribution of the Markov chain in this case?

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    $\begingroup$ It will be the stationary distribution corresponding to the acceptance distribution that goes with what you actually accept $\endgroup$
    – Glen_b
    Jan 29, 2014 at 23:29
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    $\begingroup$ Right, I was hoping I could say a little more than that though. I suppose if you accepted everything (e.g. T = infinity), the stationary distribution is just the proposal distribution. Then I guess the most you could say is that a constant T controls the mixture between the density defined by $K(\theta|X)$ and the proposal. $\endgroup$ Jan 30, 2014 at 22:23

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There are two things in mind here. One is "annealing". This concept is not used to sample from a distribution, but to find miniums of functions, see stimulated annealing. When you fix the $\beta$, then you are using Metropolis-Hastings algorithm, used to draw samples from a distribution.

As you correctly mention, the distribution changed. And the reason is that you are... using another distribution...

Let's recap: you want to draw samples from $P(x)$. To that, you employed a MCMC with an acceptance

$$A(x\rightarrow x') = \min\left(1, \frac{P(x')}{P(x)}\frac{g(x'\rightarrow x)}{g(x\rightarrow x')}\right)$$

where $g(x\rightarrow x')$ the conditional of proposing $x'$ given you were at $x$.

If you use $P(x)=K(x)$, you sample from your posterior, if you use $P(x)=e^{-\beta E}$, you will sample from other distribution.

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So your problem is to sample from a posterior $p(\theta|x)$. If annealing this distribution, then by definition, you sample from a distribution proportional to $p(\theta|x)^{\frac{1}{T}}$ which is definitely not what you want to sample from. However, there is the possibility of sampling from this annealed distribution, and weighting the samples. However, if $T$ is very large, i.e. the annealed distribution is very much flatter than the original $p(\theta|x)$, then the weights will suffer from much degeneracy - i.e. most of your samples will have weights very close zero, with the occasional few samples being further from zero. Perhaps you might consider using Parallel Tempering or Wang Landau? Both Parallel tempering and Wang Landau allow you to sample using multiple temperatures.

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