Archive for Rao-Blackwellisation

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Posted in Statistics with tags , , , , , , , on April 25, 2023 by xi'an

Monte Carlo swindles

Posted in Statistics with tags , , , , , , , , , on April 2, 2023 by xi'an

While reading Boos and Hugues-Olivier’s 1998 American Statistician paper on the applications of Basu’s theorem I can across the notion of Monte Carlo swindles. Where a reduced variance can be achieved without the corresponding increase in Monte Carlo budget. For instance, approximating the variance of the median statistic Μ for a Normal location family can be sped up by considering that

\text{var}(M)=\text{var}(M-\bar X)+\text{var}(\bar X)

by Basu’s theorem. However, when reading the originating 1973 paper by Gross (although the notion is presumably due to Tukey), the argument boils down to Rao-Blackwellisation (without the Rao-Blackwell theorem being mentioned). The related 1985 American Statistician paper by Johnstone and Velleman exploits a latent variable representation. It also makes the connection with the control variate approach, noticing the appeal of using the score function as a (standard) control and (unusual) swindle, since its expectation is zero. I am surprised at uncovering this notion only now… Possibly because the method only applies in special settings.

A side remark from the same 1998 paper, namely that the enticing decomposition

\mathbb E[(X/Y)^k] = \mathbb E[X^k] \big/ \mathbb E[Y^k]

when X/Y and Y are independent, should be kept out of reach from my undergraduates at all costs, as they would quickly get rid of the assumption!!!

an insufficient puzzle

Posted in Books, Kids, Statistics with tags , , , on January 12, 2022 by xi'an

A rather peculiar and challenging question on X validated,  concerning the absolute impossibility of a conditional expectation, given a non-sufficient statistic, being still a statistic (i.e., being independent on the parameter θ). Inspired from the following except from Hogg and Craig. Namely, could there exist a specific function φ(·) such that E[φ(Y¹)|Y³] does not depend on the parameter θ? I could not find a satisfactory explanation right away (and the question remains unanswered!)

After posting this entry, I thought anew that cases when the unbiased estimator φ(Y¹) is not a bijective transform of Y¹ would work as a counter-example, since Y³=φ(Y¹) is not sufficient but E[φ(Y¹) |φ(Y¹) ]=φ(Y¹) is not involving θ… And this case does not exhibit a paradox in that the variance does not decrease any further.


Posted in Statistics with tags , , , , , , , on August 18, 2021 by xi'an

Our survey paper on Rao-Blackwellisation (and the first Robert&Roberts published paper!) just appeared on-line as part of the International Statistical Review mini-issue in honour of C.R. Rao on the occasion of his 100th birthday. (With an unfortunate omission of my affiliation with Warwick!). While the papers are unfortunately beyond a paywall, except for a few weeks!, the arXiv version is still available (and presumably with less typos!).

ISBA 20.2.21

Posted in Kids, Mountains, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , on June 30, 2021 by xi'an

A second day which started earlier and more smoothly with a 100% local j-ISBA session. (Not counting the invigorating swim in Morgiou!) With talks by junior researchers from master to postdoc level, as this ISBA mirror meeting was primarily designed for them, so that they could all present their work, towards gaining more visibility for their research and facilitating more interactions with the participants. CIRM also insisted on this aspect of the workshop, which was well-attended.

I alas had to skip the poster session [and the joys of] despite skipping lunch [BND], due to organisational constraints. Then attended the Approximate Bayesian computation section, including one talk by Geoff Nicholls on confidence estimation for ABC, following upon the talk given by Kate last evening. And one by Florian Maire on learning the bound in accept-reject algorithms on the go, as in Caffo et al. (2002), which I found quite exciting and opening new possibilities, esp. if the Markov chain thus produced can be recycled towards unbiasedness without getting the constant right! For instance, by Rao-Blackwellisation, multiple mixtures, black box importance sampling, whatever. (This also reminded me of the earlier Goffinet et al. 1996.)

Followed by another Bayesian (modeling and) computation session. With my old friend Peter Müller talking about mixture inference with dependent priors (and a saturated colour scheme!), Matteo Ruggieri [who could not make it to CIRM!] on computable Bayesian inference for HMMs. Providing an impressive improvement upon particle filters for approximating the evidence. Also bringing the most realistic Chinese restaurant with conveyor belt! And Ming Yuan Zhou using optimal transport to define distance between distributions. With two different conditional distributions depending on which marginal is first fixed. And a connection with GANs (of course!).

And it was great to watch and listen to my friend Alicia Carriquiry talking on forensic statistics and the case for (or not?!) Bayes factors. And remembering Dennis Lindley. And my friend Jim Berger on frequentism versus Bayes! Consistency seems innocuous as most Bayes procedures are. Empirical coverage is another kind of consistency, isn’t it?

A remark I made when re-typing the program for CIRM is that there are surprisingly few talks about COVID-19 overall, maybe due to the program being mostly set for ISBA 2020 in Kunming. Maybe because we are more cautious than the competition…?!

And, at last, despite a higher density of boars around the CIRM facilities, no one got hurt yesterday! Unless one counts the impact of the French defeat at the Euro 2021 on the football fans here…

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