## conjugate priors and sufficient statistics

Posted in Statistics with tags , , , , , on March 29, 2021 by xi'an

An X validated question rekindled my interest in the connection between sufficiency and conjugacy, by asking whether or not there was an equivalence between the existence of a (finite dimension) conjugate family of priors and the existence of a fixed (in n, the sample size) dimension sufficient statistic. Outside exponential families, meaning that the support of the sampling distribution need vary with the parameter.

While the existence of a sufficient statistic T of fixed dimension d whatever the (large enough) sample size n seems to clearly imply the existence of a (finite dimension) conjugate family of priors, or rather of a family associated with each possible dominating (prior) measure,

$\mathfrak F=\{ \tilde \pi(\theta)\propto \tilde {f_n}(t_n(x_{1:n})|\theta) \pi_0(\theta)\,;\ n\in \mathbb N, x_{1:n}\in\mathfrak X^n\}$

the reverse statement is a wee bit more delicate to prove, due to the varying supports of the sampling or prior distributions. Unless some conjugate prior in the assumed family has an unrestricted support, the argument seems to limit sufficiency to a particular subset of the parameter set. I think that the result remains correct in general but could not rigorously wrap up the proof

## efficiency and the Fréchet-Darmois-Cramèr-Rao bound

Posted in Books, Kids, Statistics with tags , , , , , , , , , , , on February 4, 2019 by xi'an

Following some entries on X validated, and after grading a mathematical statistics exam involving Cramèr-Rao, or Fréchet-Darmois-Cramèr-Rao to include both French contributors pictured above, I wonder as usual at the relevance of a concept of efficiency outside [and even inside] the restricted case of unbiased estimators. The general (frequentist) version is that the variance of an estimator δ of [any transform of] θ with bias b(θ) is

I(θ)⁻¹ (1+b'(θ))²

while a Bayesian version is the van Trees inequality on the integrated squared error loss

(E(I(θ))+I(π))⁻¹

where I(θ) and I(π) are the Fisher information and the prior entropy, respectively. But this opens a whole can of worms, in my opinion since

• establishing that a given estimator is efficient requires computing both the bias and the variance of that estimator, not an easy task when considering a Bayes estimator or even the James-Stein estimator. I actually do not know if any of the estimators dominating the standard Normal mean estimator has been shown to be efficient (although there exist results for closed form expressions of the James-Stein estimator quadratic risk, including one of mine the Canadian Journal of Statistics published verbatim in 1988). Or is there a result that a Bayes estimator associated with the quadratic loss is by default efficient in either the first or second sense?
• while the initial Fréchet-Darmois-Cramèr-Rao bound is restricted to unbiased estimators (i.e., b(θ)≡0) and unable to produce efficient estimators in all settings but for the natural parameter in the setting of exponential families, moving to the general case means there exists one efficiency notion for every bias function b(θ), which makes the notion quite weak, while not necessarily producing efficient estimators anyway, the major impediment to taking this notion seriously;
• moving from the variance to the squared error loss is not more “natural” than using any [other] convex combination of variance and squared bias, creating a whole new class of optimalities (a grocery of cans of worms!);
• I never got into the van Trees inequality so cannot say much, except that the comparison between various priors is delicate since the integrated risks are against different parameter measures.

## Darmois, Koopman, and Pitman

Posted in Books, Statistics with tags , , , , , , , , on November 15, 2017 by xi'an

When [X’ed] seeking a simple proof of the Pitman-Koopman-Darmois lemma [that exponential families are the only types of distributions with constant support allowing for a fixed dimension sufficient statistic], I came across a 1962 Stanford technical report by Don Fraser containing a short proof of the result. Proof that I do not fully understand as it relies on the notion that the likelihood function itself is a minimal sufficient statistic.

## Pitman medal for Kerrie Mengersen

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on December 20, 2016 by xi'an

My friend and co-author of many years, Kerrie Mengersen, just received the 2016 Pitman Medal, which is the prize of the Statistical Society of Australia. Congratulations to Kerrie for a well-deserved reward of her massive contributions to Australian, Bayesian, computational, modelling statistics, and to data science as a whole. (In case you wonder about the picture above, she has not yet lost the medal, but is instead looking for jaguars in the Amazon.)

This medal is named after EJG Pitman, Australian probabilist and statistician, whose name is attached to an estimator, a lemma, a measure of efficiency, a test, and a measure of comparison between estimators. His estimator is the best equivariant (or invariant) estimator, which can be expressed as a Bayes estimator under the relevant right Haar measure, despite having no Bayesian motivation to start with. His lemma is the Pitman-Koopman-Darmois lemma, which states that outside exponential families, sufficient is essentially useless (except for exotic distributions like the Uniform distributions). Darmois published the result first in 1935, but in French in the Comptes Rendus de l’Académie des Sciences. And the measure of comparison is Pitman nearness or closeness, on which I wrote a paper with my friends Gene Hwang and Bill Strawderman, paper that we thought was the final paper on the measure as it was pointing out several majors deficiencies with this concept. But the literature continued to grow after that..!