controlled thermodynamic integral for Bayesian model comparison
Chris Oates, Theodore Papamarkou, and Mark Girolami (all from the University of Warwick) just arXived a paper on a new form of thermodynamic integration for computing marginal likelihoods. (I had actually discussed this paper with the authors on a few occasions when visiting Warwick.) The other name of thermodynamic integration is path sampling (Gelman and Meng, 1998). In the current paper, the path goes from the prior to the posterior by a sequence of intermediary distributions using a power of the likelihood. While the path sampling technique is quite efficient a method, the authors propose to improve it through the recourse to control variates, in order to decrease the variance. The control variate is taken from Mira et al. (2013), namely a one-dimensional temperature-dependent transform of the score function. (Strictly speaking, this is an asymptotic control variate in that the mean is only asymptotically zero.) This control variate is then incorporated within the expectation inside the path sampling integral. Its arbitrary elements are then calibrated against the variance of the path sampling integral. Except for the temperature ladder where the authors use a standard geometric rate, as the approach does not account for Monte Carlo and quadrature errors. (The degree of the polynomials used in the control variates is also arbitrarily set.) Interestingly, the paper mixes a lot of recent advances, from the zero variance notion of Mira et al. (2013) to the manifold Metropolis-adjusted Langevin algorithm of Girolami and Calderhead (2011), uses as a base method pMCMC (Jasra et al., 2007). The examples processed in the paper are regression (where the controlled version truly has a zero variance!) and logistic regression (with the benchmarked Pima Indian dataset), with a counter-example of a PDE interestingly proposed in the discussion section. I quite agree with the authors that the method is difficult to envision in complex enough models. I also did not see mentions therein of the extra time involved in using this control variate idea.