Archive for Gibbs sampling

latent variables for a hierarchical Poisson model

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , on March 11, 2021 by xi'an

Answering a question on X validated about a rather standard hierarchical Poisson model, and its posterior Gibbs simulation, where observations are (d and w being a document and a word index, resp.)

N_{w,d}\sim\mathcal P(\textstyle\sum_{1\le k\le K} \pi_{k,d}\varphi_{k,w})\qquad(1)

I found myself dragged into an extended discussion on the validation of creating independent Poisson latent variables

N_{k,w,d}\sim\mathcal P(\pi_{k,d}\varphi_{k,w})\qquad(2)

since observing their sum in (1) was preventing the latent variables (2) from being independent. And then found out that the originator of the question had asked on X validated an unanswered and much more detailed question in 2016, even though the notations differ. The question does contain the solution I proposed above, including the Multinomial distribution on the Poisson latent variables given their sum (and the true parameters). As it should be since the derivation was done in a linked 2014 paper by Gopalan, Hofman, and Blei, later published in the Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence (UAI). I am thus bemused at the question resurfacing five years later in a much simplified version, but still exhibiting the same difficulty with the conditioning principles…

freedom prior

Posted in Books, Kids, Statistics with tags , , , , , on December 9, 2020 by xi'an

Another X validated question on which I spent more time than expected. Because of the somewhat unusual parameterisation used in BDA.for the inverse χ² distribution. The interest behind the question is in the induced distribution on the parameter associated with the degrees of freedom ν of the t-distribution (question that coincided with my last modifications of my undergraduate mathematical statistics exam, involving a t sample). Whichever the prior chosen on ν, the posterior involves a nasty term

\pi(\nu)\frac{(\nu)^{n\nu/2}}{\Gamma(\nu/2)^n}{\,(v_1\cdots v_n)^{-\nu/2-1}\exp\Big\{-\nu\sigma^2}\sum_{i=1}^n1\big/2v_i\Big\}

as the Gamma function there is quickly explosive (as can be checked Stirling’s formula). Unless the prior π(ν) cancels this term, which is rather fishy as the prior would then depend on the sample size n. Even though the whole posterior is well-defined (and hence non-explosive). Rather than seeking a special prior π(ν) for computation purposes, I would thus favour a modelling restricted to integer valued ν’s as there is not much motivation in inferring about non-integer degrees of freedom.

my talk in Newcastle

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , on November 13, 2020 by xi'an

I will be talking (or rather zooming) at the statistics seminar at the University of Newcastle this afternoon on the paper Component-wise approximate Bayesian computation via Gibbs-like steps that just got accepted by Biometrika (yay!). Sadly not been there for real, as I would have definitely enjoyed reuniting with friends and visiting again this multi-layered city after discovering it for the RSS meeting of 2013, which I attended along with Jim Hobert and where I re-discussed the re-Read DIC paper. Before traveling south to Warwick to start my new appointment there. (I started with a picture of Seoul taken from the slopes of Gwanaksan about a year ago as a reminder of how much had happened or failed to happen over the past year…Writing 2019 as the year was unintentional but reflected as well on the distortion of time induced by the lockdowns!)

 

What the …?!

Posted in Books, Statistics with tags , , , , , , , , , on May 3, 2020 by xi'an

temp

too many marginals

Posted in Kids, Statistics with tags , , , , , , , on February 3, 2020 by xi'an

This week, the CEREMADE coffee room puzzle was about finding a joint distribution for (X,Y) such that (marginally) X and Y are both U(0,1), while X+Y is U(½,1+½). Beyond the peculiarity of the question, there is a larger scale problem, as to how many (if any) compatible marginals h¹(X,Y), h²(X,Y), h³(X,Y), …, need one constrains the distribution to reconstruct the joint. And wondering if any Gibbs-like scheme is available to simulate the joint.