**W**hile reading Confidence, Likelihood, Probability), by Tore Schweder and Nils Hjort, in the train from Oxford to Warwick, I came upon this unexpected property shown by Lindqvist and Taraldsen (Biometrika, 2005) that to simulate a sample **y** conditional on the realisation of a sufficient statistic, T(**y**)=t⁰, it is sufficient (!!!) to simulate the components of **y** as y=G(u,θ), with u a random variable with fixed distribution, e.g., a U(0,1), and to solve in θ the fixed point equation T(**y**)=t⁰. Assuming there exists a single solution. Brilliant (like an aurora borealis)! To borrow a simple example from the authors, take an exponential sample to be simulated given the sum statistics. As it is well-known, the conditional distribution is then a (rescaled) Beta and the proposed algorithm ends up being a standard Beta generator. For the method to work in general, T(**y**) must factorise through a function of the u’s, a so-called pivotal condition which brings us back to my post title. If this condition does not hold, the authors once again brilliantly introduce a pseudo-prior distribution on the parameter θ to make it independent from the u’s conditional on T(**y**)=t⁰. And discuss the choice of the Jeffreys prior as optimal in this setting even when this prior is improper. While the setting is necessarily one of exponential families and of sufficient conditioning statistics, I find it amazing that this property is not more well-known [at least by me!]. And wonder if there is an equivalent outside exponential families, for instance for simulating a ** t** sample conditional on the average of this sample.

## Archive for English train

## fiducial simulation

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags book review, conditional density, English train, fiducial statistics, Jeffreys prior, Monte Carlo Statistical Methods, Oxford, pseudo-random generator, simulation, Student's t distribution, sufficient statistics, University of Warwick on April 19, 2018 by xi'an## run in the parc [#1]

Posted in pictures, Running, Travel with tags Air France, English train, France, morning light, morning run, Parc de Sceaux, Paris suburbs, RER B on April 13, 2015 by xi'an## a week in Warwick

Posted in Books, Kids, Running, Statistics, University life with tags Birmingham, control variate, Coventry, English train, goose, London Midlands, Mark Girolami, Nicolas Chopin, particle MCMC, simulation model, taxi-driver, Tony O'Hagan, University of Warwick on October 19, 2014 by xi'an**T**his past week in Warwick has been quite enjoyable and profitable, from staying once again in a math house, to taking advantage of the new bike, to having several long discussions on several prospective and exciting projects, to meeting with some of the new postdocs and visitors, to attending Tony O’Hagan’s talk on “wrong models”. And then having Simo Särkkä who was visiting Warwick this week discussing his paper with me. And Chris Oates doing the same with his recent arXival with Mark Girolami and Nicolas Chopin (soon to be commented, of course!). And managing to run in dry conditions despite the heavy rains (but in pitch dark as sunrise is now quite late, with the help of a headlamp and the beauty of a countryside starry sky). I also evaluated several students’ projects, two of which led me to wonder when using RJMCMC was appropriate in comparing two models. In addition, I also eloped one evening to visit old (1977!) friends in Northern Birmingham, despite fairly dire London Midlands performances between Coventry and Birmingham New Street, the only redeeming feature being that the connecting train there was also late by one hour! (Not mentioning the weirdest taxi-driver ever on my way back, trying to get my opinion on whether or not he should have an affair… which at least kept me awake the whole trip!) Definitely looking forward my next trip there at the end of November.

## Seminar in Nottingham

Posted in Kids, pictures, Running, Statistics, Travel, University life with tags 4550 Miles from Dehli, English train, Gibbs sampler, Sherwood Forest, University of Nottingham, Ye Olde Trip to Jerusalem on March 26, 2014 by xi'an**L**ast Thursday, I gave a seminar in Nottingham, the true birthplace of the Gibbs sampler!, and I had a quite enjoyable half-day of scientific discussions in the Department of Statistics, with a fine evening tasting a local ale in the oldest (?) inn in England (Ye Olde Trip to Jerusalem) and sampling Indian dishes at 4550 Miles [plus or minus epsilon, since the genuine distance is 4200 miles) from Dehli, plus a short morning run on the very green campus. In particular, I discussed with Theo Kypraios and Simon Preston parallel ABC and their recent paper in Statistics and Computing, their use of the splitting technique of Neiswanger et al. I discussed earlier but intended here towards a better ABC approximation since (a) each term in the product could correspond to a single observation and (b) hence no summary statistic was needed and a zero tolerance could be envisioned. The paper discusses how to handle samples from terms in a product of densities, either by a Gaussian approximation or by a product of kernel estimates. And mentions connections with expectation propagation (EP), albeit not at the ABC level.

**A** minor idea that came to me during this discussion was to check whether or not a reparameterisation towards a uniform prior was a good idea: the plus of a uniform prior was that the power discussion was irrelevant, making both versions of the parallel MCMC algorithm coincide. The minus was not the computational issue since most priors are from standard families, with easily invertible cdfs, but rather why this was supposed to make a difference. When writing this on the train to Oxford, I started wondering as an ABC implementation is impervious to this reparameterisation. Indeed, simulate θ from π and pseudo-data given θ versus simulate μ from uniform and pseudo-data given T(μ) does not make a difference in the simulated pseudo-sample, hence in the distance selected θ’s, and still in one case the power does not matter while in the other case it does..!

**A**nother discussion I had during my visit led me to conclude a bit hastily that a thesis topic I had suggested to a new PhD student a few months ago had already been considered locally and earlier, although it ended up as a different, more computational than conceptual, perspective (so not all was lost for my student!). In a wider discussion around lunch, we also had an interesting foray on possible alternatives to Bayes factors and their shortcomings, which was a nice preparation to my seminar on giving up posterior probabilities for posterior error estimates. And an opportunity to mention the arXival of a proper scoring rules paper by Phil Dawid, Monica Musio and Laura Ventura, related with the one I had blogged about after the Padova workshop. And then again about a connected paper with Steve Fienberg. This lunch discussion even included some (mild) debate about Murray Aitkin’s integrated likelihood.

**A**s a completely irrelevant aside, this trip gave me the opportunity of a “pilgrimage” to Birmingham New Street train station, 38 years after “landing” for the first time in Britain! And to experience *a fresco* the multiple delays and apologies of East Midlands trains (*“we’re sorry we had to wait for this oil train in York”, “we have lost more time since B’ham”, “running a 37 minutes delay now”, “we apologize for the delay, due to trespassing”, *…), the only positive side being that delayed trains made delayed connections possible!