Distributing points on the sphere as to minimize the mean square error
of the quadrature formula , where is a centered Gaussian process with covariance function . Shown is .
Le problème est de calculer où est une mesure de probabilité sur un espace et est intégrable. Si est une suite de variables aléatoires indépendantes et distribuées selon , alors on peut approximer par
qui est dit un estimateur de Monte-Carlo.
En pratique, il peut être difficile de générer . On préférera alors introduire une mesure , avec absolument continue par rapport à , de sorte que
soit une estimée de plus commode à calculer. Cette technique, dite de l’échantillonnage préférentiel, peut aussi servir à améliorer la qualité de l’estimateur par exemple en réduisant sa variance.Read More »
Sunday afternoon project. I think it is possible to get topological garanties for the reconstruction of the classification boundary using the Nash-Tognoli theorem.
Points , are distributed on some space and associated a label . It is assumed that
where is some unknown integrable function. Estimating from the data allows us to predict given .Read More »
Friday july 28 at 17:00
Rutherford Physics Building, Room 118, McGill
Next week, I’ll be talking about Bayesian learning at the Mathematical congress of the americas and at the Canadian undergraduate mathematics conference. These are somewhat challenging talks: I need to sell the idea of Bayesian statistics to a general mathematical audience (which knows nothing about it), interest them in some though problems of Bayesian nonparametrics, and then present some of our research results. This must be done in under 20 minutes.
To make the presentation more intuitive and accessible, I borrowed some language from machine learning. I’m talking about learning rather than inference, uncertain knowledge rather than subjective belief, and “asymptotic correctness” rather than consistency. These are essentially synonymous, although some authors might use them in different ways. This should not cause problems for this introductory talk.Read More »
Félix Locas presented me this problem.
Let . Show that
The series is easy to calculate. It is, for instance, the difference between the integrals of geometric series:
I explain the discretization trick that I mentioned in my previous post (Posterior consistency under possible misspecification). I think it was introduced by Walker (New approaches to Bayesian consistency (2004)).
Let be a set of densities and let be a prior on . If follows some distribution having density , then the posterior distribution of can be written as
The discretization trick is to find densities in the convex hull of (taken in the space of all densities) such that
We assume, without too much loss of generality, that our priors are discrete. When dealing with Hellinger separable density spaces, it is possible to discretize posterior distributions to study consistency (see this post about it).
Let be a prior on a countable space of probability density functions, with for all . Data follows (independently) some unknown distribution with density .
We denote by the Kullback-Leibler divergence and we let be half of the squared Hellinger distance.
The following theorem states that the posterior distribution of accumulates in Hellinger neighborhoods of , assuming the prior is root-summable (i.e. for some ) . In the well-specified case (i.e. ), the posterior accumulates in any neighborhood of . In the misspecified case, small neighborhoods of could be empty, but the posterior distribution still accumulates in sufficiently large neighborhoods (how large exactly is a function of and ).Read More »
See part 1. Most proofs are omitted; I’ll post them with the complete pdf later this week.
Recall that is is a Polish space (ie. a complete and separable metric space). It is endowed with its borel -algebra which is the smallest family of subsets of that contains its topology and that is closed under countable unions and intersections. All subsets of we consider in the following are supposed to be part of . A probability measure on is a function such that for any countable partition of we have that The set consists of all such probability measures.
Note that since is complete and separable, every probability measure is regular (and tight). It means that the measure of any can be well approximated from the measure of compact subsets of as well as from the measure of open super-sets of :
Let me review some facts. A natural metric used to compare the mass allocation of two measures is the total variation distance defined by