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Suppose we have a Metropolis-Hastings sampler for a target distribution $f$, and we use a proposal density $Q_t$, that may depend on time $t$. By construction, $f$ is still an invariant density of the chain, at every time step (*). However, the chain is not a Markov chain anymore, at least not a time homogeneous one. I'm pretty sure there exists some theory that ensures that the process still converges to $f$, but can not find it in standard references. Where can such results be found?

* We assume that basic non degeneracy conditions hold, for instance both $f$ and all $Q_t$ are non-negative on the whole state space.

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  • $\begingroup$ Are the $Q_t$s fixed beforehand or do they depend on the history of the chain? $\endgroup$ – Juho Kokkala May 12 '17 at 6:00
  • $\begingroup$ I'm curious about both cases. But the particular $Q_t$ 'sI have depend on the history. $\endgroup$ – komark May 12 '17 at 10:09
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What you are describing is adaptive MCMC if your proposal distribution depends on the history of the chain, and is thus time dependent. There is a lot of theory about ergodicity of adaptive mcmc. There are essentially two main conditions:

  1. Diminishing Adaptation: The adaptation on the proposal distribution should diminish as a function of $t$.

  2. Containment: The time to stationarity remains bounded in probability. This is more of a technical condition and often difficult to check.

You can find a variety of references on this:

These and these set of slides go into some detail about adaptive MCMC.

This is the paper that obtains the above conditions.

Here are some examples of adaptive MCMC.

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