Rao-Blackwellisation in the MCMC era

A few months ago, as indicated on this blog, I was contacted by ISR editors to write a piece on Rao-Blackwellisation, towards a special issue celebrating Calyampudi Radhakrishna Rao’s 100th birthday. Gareth Roberts and I came up with this survey, now on arXiv, discussing different aspects of Monte Carlo and Markov Chain Monte Carlo that pertained to Rao-Blackwellisation, one way or another. As I discussed the topic with several friends over the Fall, it appeared that the difficulty was more in setting the boundaries. Than in finding connections. In a way anything conditioning or demarginalising or resorting to auxiliary variates is a form of Rao-Blackwellisation. When re-reading the JASA Gelfand and Smith 1990 paper where I first saw the link between the Rao-Blackwell theorem and simulation, I realised my memory of it had drifted from the original, since the authors proposed there an approximation of the marginal based on replicas rather than the original Markov chain. Being much closer to Tanner and Wong (1987) than I thought. It is only later that the true notion took shape. [Since the current version is still a draft, any comment or suggestion would be most welcomed!]

Bangalore workshop [ಬೆಂಗಳೂರು ಕಾರ್ಯಾಗಾರ]

Second day at the Indo-French Centre for Applied Mathematics and the workshop. Maybe not the most exciting day in terms of talks (as I missed the first two plenary sessions by (a) oversleeping and (b) running across the campus!). However I had a neat talk with another conference participant that led to [what I think are] interesting questions… (And a very good meal in a local restaurant as the guest house had not booked me for dinner!)

To wit: given a target like

the simulation of λ can be demarginalised into the simulation of

where z is a latent (and artificial) variable. This means a Gibbs sampler simulating λ given z and z given λ can produce an outcome from the target (*). Interestingly, another completion is to consider that the z_i‘s are U(0,y_i) and to see the quantity

as an unbiased estimator of the target. What’s quite intriguing is that the quantity remains the same but with different motivations: (a) demarginalisation versus unbiasedness and (b) z_i ∼ Exp(λ) versus z_i ∼ U(0,y_i). The stationary is the same, as shown by the graph below, the core distributions are [formally] the same, … but the reasoning deeply differs.

Obviously, since unbiased estimators of the likelihood can be justified by auxiliary variable arguments, this is not in fine a big surprise. Still, I had not thought of the analogy between demarginalisation and unbiased likelihood estimation previously.