On June 24th the Royal Statistical Society hosted, virtually, a discussion meeting of the paper Quasi-stationary Monte Carlo and the ScaLE Algorithm, by Pollock, Fearnhead, Johansen and Roberts.

This is one of a number of recent papers that is looking at alternatives to MCMC for Bayesian Statistics, motivated in part by the challenges of running MCMC for big data. The key idea of the paper is to try and draw samples from the posterior distribution by simulating a Markov process with killing. Such processes will eventually die out, and there long-term behaviour is often described by its quasi-stationary distribution. That is we consider the distribution of the process at time t conditional on death not occurring until after t; and then the quasi-stationary distribution is the limit of this distribution (if it exists) as t tends to infinity. (Interestingly, quasi-stationary distributions also arise as distributions of interest in other statistical situations; for example in extremes.)

The idea behind this is that it opens up new algorithms for sampling from a posterior. Though this does not come without some challenges -- not the least simulating from the quasi-stationary distribution is much harder than simulating from the stationary distribution in MCMC. The algorithm presented in the paper is based on constructing a killed Brownian motion, for which it is straightforward to derive the killing rate so that the killed Brownian motion has the posterior distribution as its quasi-stationary distribution. However simulating killed Brownian motion is non-trivial, and requires the use of ideas from exact simulation of diffusions. Simulating from the quasi-stationary distribution is then achieved by embedding the forward simulation of the killed Brownian motion within a Sequential Monte Carlo Algorithm. The paper shows that the resulting algorithm, called ScaLE, can have good properties in terms of how it scales with the number of data points -- as we can simulate the killed Brownian motion exactly whilst only using subsamples of data at each iteration.

The main contribution of the paper is to open up the potential for new algorithms for Bayesian computation. The ScaLE algorithm is just one possibility -- and there is scope for interesting extensions such as using alternatives to SMC for simulating from the quasi-stationary distribution; or for using alternatives to killed Brownian motion (which would avoid the overhead of using exact simulation methods).

The Discussion meeting gives an opportunity for short contributions linked to the paper and the related areas of scalable computational methods, and alternatives to MCMC.

You will find full details of the meeting and can register from https://rss.org.uk/training-events/events/key-events/discussion-papers/. You can also download the PDF preprint of the paper. There is opportunity to contribute to the discussion of the paper either as part of the meeting or by submitting written contributions.

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