Archive for defensive mixture

approximation of Bayes Factors via mixing

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on December 21, 2020 by xi'an

A [new version of a] paper by Chenguang Dai and Jun S. Liu got my attention when it appeared on arXiv yesterday. Due to its title which reminded me of a solution to the normalising constant approximation that we proposed in the 2010 nested sampling evaluation paper we wrote with Nicolas. Recovering bridge sampling—mentioned by Dai and Liu as an alternative to their approach rather than an early version—by a type of Charlie Geyer (1990-1994) trick. (The attached slides are taken from my MCMC graduate course, with a section on the approximation of Bayesian normalising constants I first wrote for a short course at Jim Berger’s 70th anniversary conference, in San Antonio.)

A difference with the current paper is that the authors “form a mixture distribution with an adjustable mixing parameter tuned through the Wang-Landau algorithm.” While we chose it by hand to achieve sampling from both components. The weight is updated by a simple (binary) Wang-Landau version, where the partition is determined by which component is simulated, ie by the mixture indicator auxiliary variable. Towards using both components on an even basis (à la Wang-Landau) and stabilising the resulting evaluation of the normalising constant. More generally, the strategy applies to a sequence of surrogate densities, which are chosen by variational approximations in the paper.

approximating evidence with missing data

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on December 23, 2015 by xi'an

University of Warwick, May 31 2010Panayiota Touloupou (Warwick), Naif Alzahrani, Peter Neal, Simon Spencer (Warwick) and Trevelyan McKinley arXived a paper yesterday on Model comparison with missing data using MCMC and importance sampling, where they proposed an importance sampling strategy based on an early MCMC run to approximate the marginal likelihood a.k.a. the evidence. Another instance of estimating a constant. It is thus similar to our Frontier paper with Jean-Michel, as well as to the recent Pima Indian survey of James and Nicolas. The authors give the difficulty to calibrate reversible jump MCMC as the starting point to their research. The importance sampler they use is the natural choice of a Gaussian or t distribution centred at some estimate of θ and with covariance matrix associated with Fisher’s information. Or derived from the warmup MCMC run. The comparison between the different approximations to the evidence are done first over longitudinal epidemiological models. Involving 11 parameters in the example processed therein. The competitors to the 9 versions of importance samplers investigated in the paper are the raw harmonic mean [rather than our HPD truncated version], Chib’s, path sampling and RJMCMC [which does not make much sense when comparing two models]. But neither bridge sampling, nor nested sampling. Without any surprise (!) harmonic means do not converge to the right value, but more surprisingly Chib’s method happens to be less accurate than most importance solutions studied therein. It may be due to the fact that Chib’s approximation requires three MCMC runs and hence is quite costly. The fact that the mixture (or defensive) importance sampling [with 5% weight on the prior] did best begs for a comparison with bridge sampling, no? The difficulty with such study is obviously that the results only apply in the setting of the simulation, hence that e.g. another mixture importance sampler or Chib’s solution would behave differently in another model. In particular, it is hard to judge of the impact of the dimensions of the parameter and of the missing data.