## minimaxity of a Bayes estimator

Posted in Books, Kids, Statistics, University life with tags , , , , , on February 2, 2015 by xi'an

Today, while in Warwick, I spotted on Cross Validated a question involving “minimax” in the title and hence could not help but look at it! The way I first understood the question (and immediately replied to it) was to check whether or not the standard Normal average—reduced to the single Normal observation by sufficiency considerations—is a minimax estimator of the normal mean under an interval zero-one loss defined by

$\mathcal{L}(\mu,\hat{\mu})=\mathbb{I}_{|\mu-\hat\mu|>L}=\begin{cases}1 &\text{if }|\mu-\hat\mu|>L\\ 0&\text{if }|\mu-\hat{\mu}|\le L\\ \end{cases}$

where L is a positive tolerance bound. I had not seen this problem before, even though it sounds quite standard. In this setting, the identity estimator, i.e., the normal observation x, is indeed minimax as (a) it is a generalised Bayes estimator—Bayes estimators under this loss are given by the centre of an equal posterior interval—for this loss function under the constant prior and (b) it can be shown to be a limit of proper Bayes estimators and its Bayes risk is also the limit of the corresponding Bayes risks. (This is a most traditional way of establishing minimaxity for a generalised Bayes estimator.) However, this was not the question asked on the forum, as the book by Zacks it referred to stated that the standard Normal average maximised the minimal coverage, which amounts to the maximal risk under the above loss. With the strange inversion of parameter and estimator in the minimax risk:

$\sup_\mu\inf_{\hat\mu} R(\mu,\hat{\mu})\text{ instead of } \sup_\mu\inf_{\hat\mu} R(\mu,\hat{\mu})$

which makes the first bound equal to 0 by equating estimator and mean μ. Note however that I cannot access the whole book and hence may miss some restriction or other subtlety that would explain for this unusual definition. (As an aside, note that Cross Validated has a protection against serial upvoting, So voting up or down at once a large chunk of my answers on that site does not impact my “reputation”!)

Posted in Kids, pictures, Statistics, University life with tags , , , , , , on January 19, 2015 by xi'an

Now my grading is over, I can reflect on the unexpected difficulties in the mathematical statistics exam. I knew that the first question in the multiple choice exercise, borrowed from Cross Validation, was going to  be quasi-impossible and indeed only one student out of 118 managed to find the right solution. More surprisingly, most students did not manage to solve the (absence of) MLE when observing that n unobserved exponential Exp(λ) were larger than a fixed bound δ. I was also amazed that they did poorly on a N(0,σ²) setup, failing to see that

$\mathbb{E}[\mathbb{I}(X_1\le -1)] = \Phi(-1/\sigma)$

and determine an unbiased estimator that can be improved by Rao-Blackwellisation. No student reached the conditioning part. And a rather frequent mistake more understandable due to the limited exposure they had to Bayesian statistics: many confused parameter λ with observation x in the prior, writing

$\pi(\lambda|x) \propto \lambda \exp\{-\lambda x\} \times x^{a-1} \exp\{-bx\}$

$\pi(\lambda|x) \propto \lambda \exp\{-\lambda x\} \times \lambda^{a-1} \exp\{-b\lambda\}$

hence could not derive a proper posterior.

## which parameters are U-estimable?

Posted in Books, Kids, Statistics, University life with tags , , , , , , , on January 13, 2015 by xi'an

Today (01/06) was a double epiphany in that I realised that one of my long-time beliefs about unbiased estimators did not hold. Indeed, when checking on Cross Validated, I found this question: For which distributions is there a closed-form unbiased estimator for the standard deviation? And the presentation includes the normal case for which indeed there exists an unbiased estimator of σ, namely

$\frac{\Gamma(\{n-1\}/{2})}{\Gamma({n}/{2})}2^{-1/2}\sqrt{\sum_{k=1}^n(x_i-\bar{x})^2}$

which derives directly from the chi-square distribution of the sum of squares divided by σ². When thinking further about it, if a posteriori!, it is now fairly obvious given that σ is a scale parameter. Better, any power of σ can be similarly estimated in a unbiased manner, since

$\left\{\sum_{k=1}^n(x_i-\bar{x})^2\right\}^\alpha \propto\sigma^\alpha\,.$

And this property extends to all location-scale models.

So how on Earth was I so convinced that there was no unbiased estimator of σ?! I think it stems from reading too quickly a result in, I think, Lehmann and Casella, result due to Peter Bickel and Erich Lehmann that states that, for a convex family of distributions F, there exists an unbiased estimator of a functional q(F) (for a sample size n large enough) if and only if q(αF+(1-α)G) is a polynomial in 0α1. Because of this, I had this

impression that only polynomials of the natural parameters of exponential families can be estimated by unbiased estimators… Note that Bickel’s and Lehmann’s theorem does not apply to the problem here because the collection of Gaussian distributions is not convex (a mixture of Gaussians is not a Gaussian).

This leaves open the question as to which transforms of the parameter(s) are unbiasedly estimable (or U-estimable) for a given parametric family, like the normal N(μ,σ²). I checked in Lehmann’s first edition earlier today and could not find an answer, besides the definition of U-estimability. Not only the question is interesting per se but the answer could come to correct my long-going impression that unbiasedness is a rare event, i.e., that the collection of transforms of the model parameter that are U-estimable is a very small subset of the whole collection of transforms.

Posted in Kids, pictures, Statistics, University life with tags , , , , on January 11, 2015 by xi'an

## 10 Little’s simple ideas

Posted in Books, Statistics, University life with tags , , , , , , , , on July 17, 2013 by xi'an

“I still feel that too much of academic statistics values complex mathematics over elegant simplicity — it is necessary for a research paper to be complicated in order to be published.” Roderick Little, JASA, p.359

Roderick Little wrote his Fisher lecture, recently published in JASA, around ten simple ideas for statistics. Its title is “In praise of simplicity not mathematistry! Ten simple powerful ideas for the statistical scientist”. While this title is rather antagonistic, blaming mathematical statistics for the rise of mathematistry in the field (a term borrowed from Fisher, who also invented the adjective ‘Bayesian’), the paper focus on those 10 ideas and very little on why there is (would be) too much mathematics in statistics:

1. Make outcomes univariate
2. Bayes rule, for inference under an assumed model
3. Calibrated Bayes, to keep inference honest
4. Embrace well-designed simulation experiments
5. Distinguish the model/estimand, the principles of estimation, and computational methods
6. Parsimony — seek a good simple model, not the “right” model
7. Model the Inclusion/Assignment and try to make it ignorable
8. Consider dropping parts of the likelihood to reduce the modeling part
9. Potential outcomes and principal stratification for causal inferenc
10. Statistics is basically a missing data problem

“The mathematics of problems with infinite parameters is interesting, but with finite sample sizes, I would rather have a parametric model. “Mathematistry” may eschew parametric models because the asymptotic theory is too simple, but they often work well in practice.” Roderick Little, JASA, p.365

Both those rules and the illustrations that abund in the paper are reflecting upon Little’s research focus and obviously apply to his model in a fairly coherent way. However, while a mostly parametric model user myself, I fear the rejection of non-parametric techniques is far too radical. It is more and more my convinction that we cannot handle the full complexity of a realistic structure in a standard Bayesian manner and that we have to give up on the coherence and completeness goals at some point… Using non-parametrics and/or machine learning on some bits and pieces then makes sense, even though it hurts elegance and simplicity.

“However, fully Bayes inference requires detailed probability modeling, which is often a daunting task. It seems worth sacrifycing some Bayesian inferential purity if the task can be simplified.” Roderick Little, JASA, p.366

I will not discuss those ideas in detail, as some of them make complete sense to me (like Bayesian statistics laying its assumptions in the open) and others remain obscure (e.g., causality) or with limited applicability. It is overall a commendable Fisher lecture that focus on methodology and the practice of statistical science, rather than on theory. I however do not see the reason why maths should be blamed for this state of the field. Nor why mathematical statistics journals like AoS would carry some responsibility in the lack of further applicability in other fields.  Students of statistics do need a strong background in mathematics and I fear we are losing ground in this respect, at least judging by the growing difficulty in finding measure theory courses abroad for our exchange undergradutes from Paris-Dauphine. (I also find the model misspecification aspects mostly missing from this list.)

## mathematical statistics books with Bayesian chapters [incomplete book reviews]

Posted in Books, Statistics, University life with tags , , , , , , , , on July 9, 2013 by xi'an

I received (in the same box) two mathematical statistics books from CRC Press, Understanding Advanced Statistical Methods by Westfall and Henning, and Statistical Theory A Concise Introduction by Abramovich and Ritov. For review in CHANCE. While they are both decent books for teaching mathematical statistics at undergraduate borderline graduate level, I do not find enough of a novelty in them to proceed to a full review. (Given more time, I could have changed my mind about the first one.) Instead, I concentrate here on their processing of the Bayesian paradigm, which takes a wee bit more than a chapter in either of them. (And this can be done over a single métro trip!) The important following disclaimer applies: comparing both books is highly unfair in that it is only because I received them together. They do not necessarily aim at the same audience. And I did not read the whole of either of them.

First, the concise Statistical Theory  covers the topic in a fairly traditional way. It starts with a warning about the philosophical nature of priors and posteriors, which reflect beliefs rather than frequency limits (just like likelihoods, no?!). It then introduces priors with the criticism that priors are difficult to build and assess. The two classes of priors analysed in this chapter are unsurprisingly conjugate priors (which hyperparameters have to be determined or chosen or estimated in the empirical Bayes heresy [my words!, not the authors’]) and “noninformative (objective) priors”.  The criticism of the flat priors is also traditional and leads to the  group invariant (Haar) measures, then to Jeffreys non-informative priors (with the apparent belief that Jeffreys only handled the univariate case). Point estimation is reduced to posterior expectations, confidence intervals to HPD regions, and testing to posterior probability ratios (with a warning about improper priors). Bayes rules make a reappearance in the following decision-theory chapter, as providers of both admissible and minimax estimators. This is it, as Bayesian techniques are not mentioned in the final “Linear Models” chapter. As a newcomer to statistics, I think I would be as bemused about Bayesian statistics as when I got my 15mn entry as a student, because here was a method that seemed to have a load of history, an inner coherence, and it was mentioned as an oddity in an otherwise purely non-Bayesian course. What good could this do to the understanding of the students?! So I would advise against getting this “token Bayesian” chapter in the book

“You are not ignorant! Prior information is what you know prior to collecting the data.” Understanding Advanced Statistical Methods (p.345)

Second, Understanding Advanced Statistical Methods offers a more intuitive entry, by justifying prior distributions as summaries of prior information. And observations as a mean to increase your knowledge about the parameter. The Bayesian chapter uses a toy but very clear survey examplew to illustrate the passage from prior to posterior distributions. And to discuss the distinction between informative and noninformative priors. (I like the “Ugly Rule of Thumb” insert, as it gives a guideline without getting too comfy about it… E.g., using a 90% credible interval is good enough on p.354.) Conjugate priors are mentioned as a result of past computational limitations and simulation is hailed as a highly natural tool for analysing posterior distributions. Yay! A small section discusses the purpose of vague priors without getting much into details and suggests to avoid improper priors by using “distributions with extremely large variance”, a concept we dismissed in Bayesian Core! For how large is “extremely large”?!

“You may end up being surprised to learn in later chapters (..) that, with classical methods, you simply cannot perform the types of analyses shown in this section (…) And that’s the answer to the question, “What good is Bayes?””Understanding Advanced Statistical Methods (p.345)