Archive for Roma
Last Friday I spent about 24 hours in Roma due to Clara Grazian defending her thesis there, which was awarded the highest PhD degree from both Sapienza Università di Roma and Université Paris-Dauphine. Her thesis was composed of her papers on ABC for integrated likelihood, on Jeffreys priors for mixtures (which sadly was rejected a few weeks ago), and on scoring rules à la Dawid for model choice. Clara was the first student to graduate from the joint graduate program between Sapienza and Paris-Dauphine, and I look forward the graduation of the next students!It was absolutely wonderful to be there, not only to attend the defence with Marilena Barbieri, Fabrizio Leisen, and Brunero Liseo (who was also Clara’s supervisor) and to congratulate Clara on the completion of her thesis, but also to meet [albeit much too briefly] with old friends, to enjoy great Roman food, perfect weather, my usual long run along the Tiber and twelve of its bridges in the glorious Roman morning, and “just” this unique feeling of Roma in Spring…
This morning, Clara Grazian and I arXived a paper about Jeffreys priors for mixtures. This is a part of Clara’s PhD dissertation between Roma and Paris, on which she has worked for the past year. Jeffreys priors cannot be computed analytically for mixtures, which is such a drag that it led us to devise the delayed acceptance algorithm. However, the main message from this detailed study of Jeffreys priors is that they mostly do not work for Gaussian mixture models, in that the posterior is almost invariably improper! This is a definite death knell for Jeffreys priors in this setting, meaning that alternative reference priors, like the one we advocated with Kerrie Mengersen and Mike Titterington, or the similar solution in Roeder and Wasserman, have to be used. [Disclaimer: the title has little to do with the paper, except that posterior means are off for mixtures…]
Clara Grazian and Brunero Liseo (di Roma) have just arXived a note on a method merging copulas, ABC, and empirical likelihood. The approach is rather hybrid and thus not completely Bayesian, but this must be seen as a consequence of an ill-posed problem. Indeed, as in many econometric models, the model there is not fully defined: the marginals of iid observations are represented as being from well-known parametric families (and are thus well-estimated by Bayesian tools), while the joint distribution remains uncertain and hence so does the associated copula. The approach in the paper is to proceed stepwise, i.e., to estimate correctly each marginal, well correctly enough to transform the data by an estimated cdf, and then only to estimate the copula or some aspect of it based on this transformed data. Like Spearman’s ρ. For which an empirical likelihood is computed and aggregated to a prior to make a BCel weight. (If this sounds unclear, each BEel evaluation is based on a random draw from the posterior samples, which transfers some uncertainty in the parameter evaluation into the copula domain. Thanks to Brunero and Clara for clarifying this point for me!)
At this stage of the note, there are two illustrations revolving around Spearman’s ρ. One on simulated data, with better performances than a nonparametric frequentist solution. And another one on a Garch (1,1) model for two financial time-series.
I am quite glad to see an application of our BCel approach in another domain although I feel a tiny bit uncertain about the degree of arbitrariness in the approach, from the estimated cdf transforms of the marginals to the choice of the moment equations identifying the parameter of interest like Spearman’s ρ. Especially if one uses a parametric copula which moments are equally well-known. While I see the practical gain in analysing each component separately, the object created by the estimated cdf transforms may have a very different correlation structure from the true cdf transforms. Maybe there exist consistency conditions on the estimated cdfs… Maybe other notions of orthogonality or independence could be brought into the picture to validate further the two-step solution…
As if a thumb was not enough, I lost the “new” Canon Ixus 115 H5 I bought in replacement of the (mediocre) Nikon Coolpix I lost on Ben Nevis (the title refer to the miracle mentioned in a post in February 2013, when I almost lost my (Nikon Coolpix L26) camera to the cloaca maxima, in Roma). This happened in the park on Sunday morning when I took it in my raincoat pocket to capture the serene heron standing guard at the end of the grand canal… The camera somehow fell from my pocket without me realising it (of course), presumably falling on soft ground and I only discovered it had happened five or six minutes later, when I stood next to the heron. I retraced my steps back but, even at 7:30 a Sunday morning, there was enough traffic for a runner to find it before me. (Maybe he had no gift ready for mother day!) It was not such a great camera and on its trip to Chamonix last X’mas with my daughter it had decided to host a small fungus that lived right on the lens, making zooming close to impossible. (The same thing had happened with the Nikon Coolpix the year before after falling in the snow during my X’mas ski trip.) Just a wee (bit ?) annoying… (Latest picture from the Canon Ixus to come on Sunday!)