scalable Langevin exact algorithm [armchair Read Paper]

So, Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts presented their Read Paper with discussions on the Wednesday aft! With a well-sized if virtual audience of nearly a hundred people. Here are a few notes scribbled during the Readings. And attempts at keeping the traditional structure of the meeting alive.

In their introduction, they gave the intuition of a quasi-stationary chain as the probability to be in A at time t while still alice as π(A) x exp(-λt) for a fixed killing rate λ. The concept is quite fascinating if less straightforward than stationarity! The presentation put the stress on the available recourse to an unbiased estimator of the κ rate whose initialisation scaled as O(n) but allowed a subsampling cost reduction afterwards. With a subsampling rat connected with Bayesian asymptotics, namely on how quickly the posterior concentrates. Unfortunately, this makes the practical construction harder, since n is finite and the concentration rate is unknown (although a default guess should be √n). I wondered if the link with self-avoiding random walks was more than historical.

The initialisation of the method remains a challenge in complex environments. And hence one may wonder if and how better it does when compared with SMC. Furthermore, while the motivation for using a Brownian motion stems from the practical side, this simulation does not account for the target π. This completely blind excursion sounds worse than simulating from the prior in other settings.

One early illustration for quasi stationarity was based on an hypothetical distribution of lions and wandering (Brownian) antelopes. I found that the associated concept of soft killing was not necessarily well received by …. the antelopes!

As it happens, my friend and coauthor Natesh Pillai was the first discussant! I did no not get the details of his first bimodal example. But he addressed my earlier question about how large the running time T should be. Since the computational cost should be exploding with T. He also drew a analogy with improper posteriors as to wonder about the availability of convergence assessment.

And my friend and coauthor Nicolas Chopin was the second discussant! Starting with a request to… leave the Pima Indians (model)  alone!! But also getting into a deeper assessment of the alternative use of SMCs.

scalable Langevin exact algorithm [Read Paper]

Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts (CoI: all with whom I have strong professional and personal connections!) have a Read Paper discussion happening tomorrow [under relaxed lockdown conditions in the UK, except for the absurd quatorzine on all travelers|, but still in a virtual format] that we discussed together [from our respective homes] at Paris Dauphine. And which I already discussed on this blog when it first came out.

Here are quotes I spotted during this virtual Dauphine discussion but we did not come up with enough material to build a significant discussion, although wondering at the potential for solving the O(n) bottleneck, handling doubly intractable cases like the Ising model. And noticing the nice features of the log target being estimable by unbiased estimators. And of using control variates, for once well-justified in a non-trivial environment.

“However, in practice this simple idea is unlikely to work. We can see this most clearly with the rejection sampler, as the probability of survival will decrease exponentially with t—and thus the rejection probability will often be prohibitively large.”

“This can be viewed as a rejection sampler to simulate from μ(x,t), the distribution of the Brownian motion at time  t conditional on its surviving to time t. Any realization that has been killed is ‘rejected’ and a realization that is not killed is a draw from μ(x,t). It is easy to construct an importance sampling version of this rejection sampler.”