Following my earlier post about the young astronomer who feared he was running his MCMC for too long, here is an update from his visit to my office this morning. This visit proved quite an instructive visit for both of us. (Disclaimer: the picture of an observatory seen from across Brunel’s suspension bridge in Bristol is as earlier completely unrelated with the young astronomer!)
First, the reason why he thought MCMC was running too long was that the acceptance rate was plummeting down to zero, whatever the random walk scale. The reason for this behaviour is that he was actually running a standard simulated annealing algorithm, hence observing the stabilisation of the Markov chain in one of the (global) modes of the target function. In that sense, he was right that the MCMC was run for “too long”, as there was nothing to expect once the mode had been reached and the temperature turned down to zero. So the algorithm was working correctly.
Second, the astronomy problem he considers had a rather complex likelihood, for which he substituted a distance between the (discretised) observed data and (discretised) simulated data, simulated conditional on the current parameter value. Now…does this ring a bell? If not, here is a three letter clue: ABC… Indeed, the trick he had found to get around this likelihood calculation issue was to re-invent a version of ABC-MCMC! Except that the distance was re-introduced into a regular MCMC scheme as a substitute to the log-likelihood. And compared with the distance at the previous MCMC iteration. This is quite clever, even though this substitution suffers from a normalisation issue (that I already mentioned in the post about Holmes’ and Walker’s idea to turn loss functions into pseudo likelihoods. Regular ABC does not encounter this difficult, obviously. I am still bemused by this reinvention of ABC from scratch!
So we are now at a stage where my young friend will experiment with (hopefully) correct ABC steps, trying to derive the tolerance value from warmup simulations and use some of the accelerating tricks suggested by Umberto Picchini and Julie Forman to avoid simulating the characteristics of millions of stars for nothing. And we agreed to meet soon for an update. Indeed, a fairly profitable morning for both of us!
![S_nS_n^\text{T}=S_{n-1}\left(\mathbf{I}_d-\eta_n[\alpha_n-\alpha^\star]\dfrac{U_nU_n^\text{T}}{||U_n||^2}\right)S^\text{T}_{n-1}](https://s0.wp.com/latex.php?latex=S_nS_n%5E%5Ctext%7BT%7D%3DS_%7Bn-1%7D%5Cleft%28%5Cmathbf%7BI%7D_d-%5Ceta_n%5B%5Calpha_n-%5Calpha%5E%5Cstar%5D%5Cdfrac%7BU_nU_n%5E%5Ctext%7BT%7D%7D%7B%7C%7CU_n%7C%7C%5E2%7D%5Cright%29S%5E%5Ctext%7BT%7D_%7Bn-1%7D&bg=000000&fg=B0B0B0&s=0&c=20201002 "S_nS_n^\text{T}=S_{n-1}\left(\mathbf{I}_d-\eta_n[\alpha_n-\alpha^\star]\dfrac{U_nU_n^\text{T}}{||U_n||^2}\right)S^\text{T}_{n-1}")
is the current acceptance probability and 
the target acceptance rate;
is the current random noise for the proposal, 
;
is a step size sequence decaying to zero.
is proportional to the scale of the target if the latter is elliptically symmetric (but does not establish a sufficient condition for convergence). It concludes with a Law of Large Numbers for the empirical average of the 
under rather strong assumptions (on f, the target, and the matrices 
