This seems to relate to Kennedy and Kull (1985) in that you have essentially an unbiased estimator of the acceptance probability, Gaussianity being irrelevant there….

]]>In my case, evaluation of my likelihood involves solving a PDE by a completely different Markov Chain. Therefore, my acceptance probability comes with a very special error: Instead of getting the true α, I get α + N, where N is Gaussian. This gives me a confidence interval around my estimate of α. Note that I can make Variance(N) arbitrarily small (with computational effort). So, rather than being content with an initial estimate of α, I reduce Variance(N) until this confidence interval is small enough.

So what I’m looking for (doubtful that it’s out there) is work where acceptance is done with some confidence…

The case I deal with is rather special, but there is a great deal of work done (in the “inverse problems” community) where evaluation of the likelihood involves solving a PDE, and is therefore done with some error.

]]>The approximation of ABC (Approximative Bayesian Computation) is in the target, not in the acceptance probability or the importance weight. As stressed by Wikinson (2008) and Fearnhead and Prangle (2010), ABC is an exact simulation method wrt to an approximate target distribution. It remains that it brings an approximation effect into the Bayesian inference that is hard to evaluate.

]]>Indeed, it is… Provided you care about the quality of the approximation.

I took a look at Kennedy & Kull (1985) as I could not get hold of the second paper and it appears like a primitive version of Andrieu & Roberts (2009) in that the essential input of the paper is the idea to replace the acceptance probability with an unbiased estimator of the acceptance probability (their formula (6)), which is hard as you wrote, because it mixes unbiasedness with boundary constraints, since those probability estimates must remain between 0 and 1, something that sounds impossible if the density ratio is not bounded away from zero and infinity…

]]>Well, it’s hard…

]]>What you described is really close to ‘domain decomposition methods’, where you split the spatial domain into chunks. If you do it correctly, the interaction terms only appear in one block and the art is then to approximate this block correctly. It’s usually used as a parallel preconditioner for CG-like methods for large iterative systems (although it’s also pretty much how you distribute a Cholesky factorisation).

]]>Thank you, Radford. I will take a look at those. Before even looking at them, let me ask how comes they did not have more of an impact in the field?

]]>Ok, you certainly know much better about INLA. My restatement is then that INLA would not solve it much better! I would think a “sparsification” of the matrix leading to independent blocks could provide a feasible approximation running INLA… Evaluating the impact of this sparsification is obviously complicated but if this is the only solution….

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