horse ghost [jatp]

MiMo2020

On 26 and 27 March 2020, the maths department of the Université of Rouen, Normandy, France, organizes a (free) workshop on mixture distributions. With the following speakers

• • Christophe Biernacki  (Laboratoire Paul Painlevé, Univ. Lille 1 et INRIA)
  • Vincent Brault (Laboratoire Jean Kuntzmann, Univ. Grenoble Alpes)
  • Gilles Celeux  (Laboratoire de Mathématiques d’Orsay, Univ. Paris Sud et INRIA)
  • Elisabeth Gassiat  (Laboratoire de Mathématiques d’Orsay, Univ. Paris Sud)
  • Van Hà Hoang  (Laboratoire de Mathématique Raphaël Salem, Univ. Rouen Normandie)
  • Hajo Holzmann  (Philipps-University Marburg, Germany)
  • Dimitri Karlis  (Department of Statistics, Athens University of Economics and Business, Greece)
  • Trung Tin Nguyen (LMNO, Univ. Caen Normandie)
  • Andrea Rau  (Département de Génétique Animale, INRA, Jouy en Josas)
  • Pierre Vandekerkhove  (Laboratoire d’Analyse et de Mathématiques Appliquées, Univ. Paris-Est Marne-la-Vallée)
  • Cinzia Viroli  (Department of Statistical Sciences, Universita di Bologna, Italia)

Unfortunately, since this is my former department, I will not be able to attend as I am taking part into the SIAM Conference on Uncertainty Quantification (UQ20), on the very same days. In a session on likelihood-free inference.

a very quick Riddle

A very quick Riddler’s riddle last week with the question

Find the (integer) fraction with the smallest (integer) denominator strictly located between 1/2020 and 1/2019.

and the brute force resolution

for (t in (2020*2019):2021){ 
   a=ceiling(t/2020)
   if (a*2019<t) sol=c(a,t)}

leading to 2/4039 as the target. Note that

catching my train with no training

optimal choice among MCMC kernels

Last week in Siem Reap, Florian Maire [who I discovered originates from a Norman town less than 10km from my hometown!] presented an arXived joint work with Pierre Vandekerkhove at the Data Science & Finance conference in Cambodia that considers the following problem: Given a large collection of MCMC kernels, how to pick the best one and how to define what best means. Going by mixtures is a default exploration of the collection, as shown in (Tierney) 1994 for instance since this improves on both kernels (esp. when each kernel is not irreducible on its own!). This paper considers a move to local weights in the mixture, weights that are not estimated from earlier simulations, contrary to what I first understood.

As made clearer in the paper the focus is on filamentary distributions that are concentrated nearby lower-dimension sets or manifolds Since then the components of the kernel collections can be restricted to directions of these manifolds… Including an interesting case of a 2-D highly peaked target where converging means mostly simulating in x¹ and covering the target means mostly simulating in x². Exhibiting a schizophrenic tension between the two goals. Weight locally dependent means correction by Metropolis step, with cost O(n). What of Rao-Blackwellisation of these mixture weights, from weight x transition to full mixture, as in our PMC paper? Unclear to me as well [during the talk] is the use in the mixture of basic Metropolis kernels, which are not absolutely continuous, because of the Dirac mass component. But this is clarified by Section 5 in the paper. A surprising result from the paper (Corollary 1) is that the use of local weights ω(i,x) that depend on the current value of the chain does jeopardize the stationary measure π(.) of the mixture chain. Which may be due to the fact that all components of the mixture are already π-invariant. Or that the index of the kernel constitutes an auxiliary (if ancillary)  variate. (Algorithm 1 in the paper reminds me of delayed acceptance. Making me wonder if computing time should be accounted for.) A final question I briefly discussed with Florian is the extension to weights that are automatically constructed from the simulations and the target.