Archive for multivariate probit model

21w5107 [½day 4]

Posted in Statistics with tags , , , , , , , , , , , , , , on December 3, 2021 by xi'an

Final ½ day of the 21w5107 workshop for me, as our initial plans were to stop today due to the small number of participants on site. And I had booked plane tickets early, too early. I will thus sadly miss the four afternoon talks, mea culpa! However I did attend Noiritt Chandra’s talk on Bayesian factor analysis. Which has always been a bit of a mystery to me in the sense that the number q of factors need be specified, which is a prior input one rarely controls. Here the goal is to estimate a covariance matrix with a sparse representation. And q is estimated by empirical likelihood ahead of the estimation of the matrix. The focus was on minimaxity and MCMC implementation rather than objective Bayes per se! Then, Daniele Durante spoke about analytical posteriors for probit models using unified skew-Normal priors (following a 2019 Biometrika paper). Including marginal posteriors and marginal likelihood. And for various extensions like dynamic probit models. Opening other computational issues such as simulating high dimensional truncated Normal distributions. (Potential use of delayed acceptance there?) This second talk was also drifting away from objective Bayes! In the first half of his talk, Filippo Ascolani introduced us to trees of random probability measures, each mother node being the distribution of the atoms of the children nodes. (Interestingly, Kingman is both connected to (coalescent) trees and to completely random measures.) My naïve first impression was that the distributions would get more and more degenerate as the number of levels in the tree would increase, however I am unsure this is correct as Filippo mentioned getting observations on all nodes. The talk also made me wonder at how this could be related Radford Neal’s Dirichlet trees. (Which I discovered at my first ICMS workshop about 20 years ago.) Yang Ni concluded the morning with a talk on causality that provided (to me) a very smooth (re)introduction to Bayesian causal graphs.

Even more than last time, I enormously enjoyed the workshop, its location, the fantastic staff at the hotel, and the reconnection with dear friends!, just regretting we could not be a few more. I appreciate the efforts made by on-line participants to stay connected and intervene (thanks, Ed!), but the quality of interactions is sadly of another magnitude when spending all our time together. Hopefully there will be a next time and hopefully we’ll then be back to larger size (and hopefully the location will remain the same). Hasta luego, Oaxaca!

corrected MCMC samplers for multivariate probit models

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , on May 6, 2015 by xi'an

“Moreover, IvD point out an error in Nobile’s derivation which can alter its stationary distribution. Ironically, as we shall see, the algorithms of IvD also contain an error.”

 Xiyun Jiao and David A. van Dyk arXived a paper correcting an MCMC sampler and R package MNP for the multivariate probit model, proposed by Imai and van Dyk in 2005. [Hence the abbreviation IvD in the above quote.] Earlier versions of the Gibbs sampler for the multivariate probit model by Rob McCulloch and Peter Rossi in 1994, with a Metropolis update added by Agostino Nobile, and finally an improved version developed by Imai and van Dyk in 2005. As noted in the above quote, Jiao and van Dyk have discovered two mistakes in this latest version, jeopardizing the validity of the output.

IvDykThe multivariate probit model considered here is a multinomial model where the occurrence of the k-th category is represented as the k-th component of a (multivariate) normal (correlated) vector being the largest of all components. The latent normal model being non-identifiable since invariant by either translation or scale, identifying constraints are used in the literature. This means using a covariance matrix of the form Σ/trace(Σ), where Σ is an inverse Wishart random matrix. In their 2005 implementation, relying on marginal data augmentation—which essentially means simulating the non-identifiable part repeatedly at various steps of the data augmentation algorithm—, Imai and van Dyk missed a translation term and a constraint on the simulated matrices that lead to simulations outside the rightful support, as illustrated from the above graph [snapshot from the arXived paper].

IvDyk1Since the IvD method is used in many subsequent papers, it is quite important that these mistakes are signalled and corrected. [Another snapshot above shows how much both algorithm differ!] Without much thinking about this, I [thus idly] wonder why an identifying prior is not taking the place of a hard identifying constraint, as it should solve the issue more nicely. In that it would create less constraints and more entropy (!) in exploring the augmented space, while theoretically providing a convergent approximation of the identifiable parts. I may (must!) however miss an obvious constraint preventing this implementation.

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