# Can adaptive MCMC be trusted?

I am reading about adaptive MCMC (see e.g., Chapter 4 of the Handbook of Markov Chain Monte Carlo, ed. Brooks et al., 2011; and also Andrieu & Thoms, 2008).

The main result of Roberts and Rosenthal (2007) is that if the adaptation scheme satisfies the vanishing adaptation condition (plus some other technicality), adaptive MCMC is ergodic under any scheme. For example, vanishing adaptation can be easily obtained by adapting the transition operator at iteration $n$ with probability $p(n)$, with $\lim_{n \rightarrow \infty} p(n) = 0$.

This result is (a posteriori) intuitive, asymptotically. Since the amount of adaptation tends to zero, eventually it will not mess up with ergodicity. My concern is what happens with finite time.

• How do we know that adaptation is not messing up with ergodicity at a given finite time, and that a sampler is sampling from the correct distribution? If it makes sense at all, how much burn-in should one do to ensure that early adaptation is not biasing the chains?

• Do practitioners in the field trust adaptive MCMC? The reason I am asking is because I've seen many recent methods that try to build-in adaptation in other, more complex ways that are known to respect ergodicity, such as regeneration or ensemble methods (i.e., it is legit to choose a transition operator that depends on the state of other parallel chains). Alternatively, adaptation is performed only during burn-in, such as in Stan, but not at runtime. All these efforts suggest to me that adaptive MCMC as per Roberts and Rosenthal (which would be incredibly simple to implement) is not considered reliable; but perhaps there are other reasons.

• What about specific implementations, such as adaptive Metropolis-Hastings (Haario et al. 2001)?

### References

• +1 but is there finite-time guarantees even for non-adaptive MCMC? – Juho Kokkala Aug 8 '16 at 6:30
• @JuhoKokkala: probably not, but it seems that with adaptive MCMC one is adding yet another layer of possible modes of failure, which are less understood and harder to check than standard issues of convergence (which are already pretty difficult to diagnose per se). At least, that's my understanding of why practitioners (I, for one) would be wary of it. – lacerbi Aug 8 '16 at 10:19
• I think adaptation during burnin is the best way to deal with adaptation. Obviously if you have some areas of your posterior that require different tuning than others you'll have issues, but if that's the case, if you run fully adaptive MCMC you won't be allowed to adapt much because of the vanishing condition anyway... – sega_sai Nov 18 '16 at 16:28

Ergodicity and bias are about asymptotic properties of the Markov chain, they tell nothing about the behaviour and distribution of the Markov chain at a given finite time. Adaptivity has nothing to do with this issue, any MCMC algorithm may produce simulations far from the target at a given finite time.
• (+1) Thanks for the clarification. Yes, I understand that MCMC algorithms have no guarantees at a given finite time. However, in practice we do use them as if they provided good/reasonable approximation of the target distribution at a given finite time, even though in most cases there are no theoretical guarantees (AFAIK only a few cases are mathematically understood). Perhaps I should say "messing up with mixing time"? That's closer to what I meant. If you have suggestions on how to fix the language please let me know. – lacerbi Jan 5 '17 at 11:03