Archive for Doeblin’s condition

adaptive independent Metropolis-Hastings

Posted in Statistics with tags , , , , , , on May 8, 2018 by xi'an

When rereading this paper by Halden et al. (2009), I was reminded of the earlier and somewhat under-appreciated Gåsemyr (2003). But I find the convergence results therein rather counter-intuitive in that they seem to justify adaptive independent proposals with no strong requirement. Besides the massive Doeblin condition:

“The Doeblin condition essentially requires that all the proposal distribution [sic] has uniformly heavier tails than the target distribution.”

Even when the adaptation is based on an history vector made of rejected values and non-replicated accepted values. Actually  convergence of this sequence of adaptive proposals kernels is established under a concentration of the Doeblin constants a¹,a²,… towards one, in the sense that

E[(1-a¹)(1-a²)…]=0.

The reason may be that, with chains satisfying a Doeblin condition, there is a probability to reach stationarity at each step. Equal to a¹, a², … And hence to ignore adaptivity since each kernel keep the target π invariant. So in the end this is not so astounding. (The paper also reminded me of Wolfgang [or Vincent] Doeblin‘s short and tragic life.)

approximations of Markov Chains [another garden of forking paths]

Posted in Books, Mountains, pictures, Statistics, University life with tags , , , , , , , , , , on March 15, 2016 by xi'an

On the Sétaz cabin ride, Valloire, Dec. 23, 2011James Johndrow and co-authors from Duke wrote a paper on approximate MCMC that was arXived last August and that I missed. David Dunson‘s talk at MCMski made me aware of it. The paper studies the impact of replacing a valid kernel with a close approximation. Which is a central issue for many usages of MCMC in complex models, as exemplified by the large number of talks on that topic at MCMski.

“All of our bounds improve with the MCMC sample path length at the expected rate in t.”

A major constraint in the paper is Doeblin’s condition, which implies uniform geometric ergodicity. Not only it is a constraint on the Markov kernel but it is also one for the Markov operator in that it may prove impossible to… prove. The second constraint is that the approximate Markov kernel is close enough to the original, which sounds reasonable. Even though one can always worry that the total variation norm is too weak a norm to mean much. For instance, I presume with some confidence that this does not prevent the approximate Markov kernel from not being ergodic, e.g., not irreducible, not absolutely continuous wrt the target, null recurrent or transient. Actually, the assumption is stronger in that there exists a collection of approximations for all small enough values ε of the total variation distance. (Small enough meaning ε is much smaller than the complement α to 1 of the one step distance between the Markov kernel and the target. With poor kernels, the approximation must thus be very good.) This is less realistic than assuming the availability of one single approximation associated with an existing but undetermined distance ε. (For instance, the three examples of Section 3 in the paper show the existence of approximations achieving a certain distance ε, without providing a constructive determination of such approximations.) Under those assumptions, the average of the sequence of Markov moves according to the approximate kernel converges to the target in total variation (and in expectation for bounded functions). With sharp bounds on those distances. I am still a bit worried at the absence of conditions for the approximation to be ergodic.

“…for relatively short path lengths, there should exist a range of values for which aMCMC offers better performance in the compminimax sense.”

The paper also includes computational cost into the picture. Introducing the notion of compminimax error, which is the smallest (total variation) distance among all approximations at a given computational budget. Quite an interesting, innovative, and relevant notion that may however end up being too formal for practical use. And that does not include the time required to construct and calibrate the approximations.