I agree that the “Unbiased” in the title may be misleading and it was certainly not our intention to claim that we get around the MCMC bias, and we discuss this on page 6. We were trying to contrast our approach to the growing body of work that uses approximate transition kernels and therefore introduces additional bias that is very difficult to quantify. Posting the early draft on arxiv precisely had a point of encouraging open discussions like this, and to let the advantages and vulnerabilities of different approaches be subject to an open criticism: should we mess with the transition kernel? do we care about simulation or estimation? do we care about bias? do we really want to be fully Bayesian when we have a lot of data?

In terms of the second point, I was hoping that the paper had a clear attribution of the main idea to the seminal work of Rhee & Glynn. On page 2, we have the following paragraph:

“Our work builds on several breakthroughs in related areas, such as unbiased estimation for stochastic differential equations (Rhee and Glynn, 2013) and for Markov chain equilibrium expectations (Glynn and Rhee, 2014). These developments demonstrate the overarching principle that estimation is often an easier problem than simulation – a dictum we adopt and apply here in the context of Bayesian inference.”

Perhaps, the current draft has not emphasised this enough – next version will certainly rectify this.

]]>It is not clear that both of these methods can be combined while still guaranteeing a reasonable computational cost and variance.

Unless those two things are successfully combined, the use of the term “unbiased” is a bit shocking: their proposed estimator is biased. I would have expected the paper to fully explore these debiasing techniques before claiming anything. Now who’s going to do that, when this “half-baked” paper is online? With all due respect to the authors who are super inventive, nice and clever people (trying to save my career here), I bet no one is going to check that, and so the idea has been “killed in the egg”.

Another thing that shocked me (am I easily shocked!?) is that Rhee did his PhD thesis on the idea of “perfect estimation” instead of “perfect sampling”. Hence when the introduction of the paper states “In this contribution, we propose a different view on the problem. We construct a scheme that directly estimates posterior expectations with neither simulation from the full posterior nor construction of alternative approximate transition kernels – without introducing bias.” then really the view should be attributed to Rhee & Glynn.

]]>Hence I agree that the comparisons with other unbiased methods are the absolute wrong comparisons to make. People do “big data” / “big model” computations every day on data at least as large and complex as the ones in this paper using inexact (and it’s not even worth putting the scare quotes around that – just because your method is asymptotically unbiased doesn’t guarantee it’s any good in a finite computational budget) methods. I fear the situation that I ran into walking out of a recent conference session of Bayesian image analysis and hearing a person ask “why are they solving problems from the 80s with methods from the 90s?”.

I’m also slightly scared that the finite variance condition seems to require us to know more about the mixing and convergence of MCMC than we really know. But this could be me misunderstanding the condition in the paper (from my reading, to ensure it you need a non-asymptotic bound on the distance from the target). It also suggests that if the problem misbehaves (you are, for example, using a cheap but inefficient MCMC scheme for reasons of parallel scaling), the truncation time could be mammoth.

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