Omiros Papaspiliopoulos (Università Bocconi): Bilinear mixed models and variational inference
Adrien Corenflos (University of Warwick): A coupling-based approach to f-divergences diagnostics for Markov chain Monte Carlo [recently discussed on the ‘Og]
Adrien Corenflos (University of Warwick) and Hai-Dang Dau (NUS) just arXived their paper on MCMC diagnostics that Adrien told me about last month, while in Warwick.
“This [f-divergence] bound is clearly suboptimal since it does not vary in t and does not take into account the mixing of the Markov chain. We present a scheme where the weights are ‘harmonized’ as the Markov chain progresses, reflecting its mixing through the notion of coupling.”
They start by opposing the classical ergodic average and embarrassingly parallel estimates obtained by N parallel chains culled of their B initial values, to couplings used in standard diagnoses. Opting for the parallel perspective, maybe rekindling the diagnostic war of the early 1990s! The evaluation tool in the paper is based on f-divergences, like the χ² divergence which naturally relates to the effective sample size when considering weighted atomic measures. When consistent, these weighted approximations produce upper bounds on the f-divergence, with exact convergence in case of independence.
In my opinion the most exciting part of the paper stands with the ability to modify these weights along MCMC iterations, since the naïve sequential importance sampling argument I also use in class keeps them constant! The trick is to (be able to) couple randomly chosen parallel chains, with the weights being averaged at each coupling event. The resulting algorithm preserves expectation (in the importance sampling sense) and consistency (in the particle sense). Furthermore, the f-divergence bound based on the weights can only decrease between iterations, which reminds me of interleaving. And exponential convergence of the weights to uniform ones (under the strong assumption of a uniformly lower bounded probability of coupling). The paper concludes with interesting remarks on perfect sampling, Rao-Blackwellisation, control variates, and backward sampling.
A long-standing gap exists between the theoretical analysis of Markov chain Monte Carlo convergence, which is often based on statistical divergences, and the diagnostics used in practice. We introduce the first general convergence diagnostics for Markov chain Monte Carlo based on any f χ² -divergence, allowing users to directly monitor, among others, the Kullback–Leibler and the divergences as well as the Hellinger and the total variation distances. Our first key contribution is a coupling-based ‘weight harmonization’ scheme that produces a direct, computable, and consistent weighting of interacting Markov chains with respect to their target distribution. The second key contribution is to show how such consistent weightings of empirical measures can be used to provide upper bounds to f -divergences in general. We prove that these bounds are guaranteed to tighten over time and converge to zero as the chains approach stationarity, providing a concrete diagnostic.
An interesting paper came out on arXiv in early December, written by Michael Brand from Monash. It is about risk-adverse Bayes estimators, which are defined as avoiding the use of loss functions (although why avoiding loss functions is not made very clear in the paper). Close to MAP estimates, they bypass the dependence of said MAPs on parameterisation by maximising instead π(θ|x)/√I(θ), which is invariant by reparameterisation if not by a change of dominating measure. This form of MAP estimate is called the Wallace-Freeman (1987) estimator [of which I never heard].
The formal definition of a risk-adverse estimator is still based on a loss function in order to produce a proper version of the probability to be “wrong” in a continuous environment. The difference between estimator and true value θ, as expressed by the loss, is enlarged by a scale factor k pushed to infinity. Meaning that differences not in the immediate neighbourhood of zero are not relevant. In the case of a countable parameter space, this is essentially producing the MAP estimator. In the continuous case, for “well-defined” and “well-behaved” loss functions and estimators and density, including an invariance to parameterisation as in my own intrinsic losses of old!, which the author calls likelihood-based loss function, mentioning f-divergences, the resulting estimator(s) is a Wallace-Freeman estimator (of which there may be several). I did not get very deep into the study of the convergence proof, which seems to borrow more from real analysis à la Rudin than from functional analysis or measure theory, but keep returning to the apparent dependence of the notion on the dominating measure, which bothers me.
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