**S**urprisingly (or not?!), I received two requests about some exercises from The Bayesian Choice, one from a group of students from McGill having difficulties solving the above, wondering about the properness of the posterior (but missing the integration of x), to whom I sent back this correction. And another one from the Czech Republic about a difficulty with the term “evaluation” by which I meant (pardon my French!) estimation.

## Archive for The Bayesian Choice

## back to the Bayesian Choice

Posted in Books, Kids, Statistics, University life with tags autoregressive model, Bayesian decision theory, Book, exercises, improper posteriors, improper prior, inverse Gamma distribution, prior predictive, The Bayesian Choice on October 17, 2018 by xi'an## about paradoxes

Posted in Books, Kids, Statistics, University life with tags bias, book review, email, Jacobian, Mark Chang, MLE, paradoxes, reparameterisation, scientific inference, The Bayesian Choice, unbiasedness on December 5, 2017 by xi'an**A**n email I received earlier today about statistical paradoxes:

I am a PhD student in biostatistics, and an avid reader of your work. I recently came across this blog post, where you review a text on statistical paradoxes, and I was struck by this section:

I found this section provocative, but I am unclear on the nature of these “paradoxes”. I reviewed my stat inference notes and came across the classic example that there is no unbiased estimator for 1/p w.r.t. a binomial distribution, but I believe you are getting at a much more general result. If it’s not too much trouble, I would sincerely appreciate it if you could point me in the direction of a reference or provide a bit more detail for these two “paradoxes”.

The text is Chang’s Paradoxes in Scientific Inference, which I indeed reviewed negatively. To answer about the bias “paradox”, it is indeed a neglected fact that, while the average of *any* transform of a sample obviously is an unbiased estimator of its mean (!), the converse does not hold, namely, an *arbitrary* transform of the model parameter θ is not necessarily enjoying an unbiased estimator. In Lehmann and Casella, Chapter 2, Section 4, this issue is (just slightly) discussed. But essentially, transforms that lead to unbiased estimators are mostly the polynomial transforms of the mean parameters… (This also somewhat connects to a recent X validated question as to why MLEs are not always unbiased. Although the simplest explanation is that the transform of the MLE is the MLE of the transform!) In exponential families, I would deem the range of transforms with unbiased estimators closely related to the collection of functions that allow for inverse Laplace transforms, although I cannot quote a specific result on this hunch.

The other “paradox” is that, if h(X) is the MLE of the model parameter θ for the observable X, the distribution of h(X) has a density different from the density of X and, hence, its maximisation in the parameter θ may differ. An example (my favourite!) is the MLE of ||a||² based on x N(a,I) which is ||x||², a poor estimate, and which (strongly) differs from the MLE of ||a||² based on ||x||², which is close to (1-p/||x||²)²||x||² and (nearly) admissible [as discussed in the Bayesian Choice].

## relativity is the keyword

Posted in Books, Statistics, University life with tags Bayes factor, model posterior probabilities, OxWaSP, relativity, Saint Giles cemetery, testing of hypotheses, The Bayesian Choice, University of Oxford on February 1, 2017 by xi'an**A**s I was teaching my introduction to Bayesian Statistics this morning, ending up with the chapter on tests of hypotheses, I found reflecting [out loud] on the relative nature of posterior quantities. Just like when I introduced the role of priors in Bayesian analysis the day before, I stressed the relativity of quantities coming out of the BBB [Big Bayesian Black Box], namely that whatever happens as a Bayesian procedure is to be understood, scaled, and relativised against the prior equivalent, i.e., that the reference measure or gauge is the prior. This is sort of obvious, clearly, but bringing the argument forward from the start avoids all sorts of misunderstanding and disagreement, in that it excludes the claims of absolute and certainty that may come with the production of a posterior distribution. It also removes the endless debate about the determination of *the* prior, by making *each* prior a reference on its own. With an additional possibility of calibration by simulation under the assumed model. Or an alternative. Again nothing new there, but I got rather excited by this presentation choice, as it seems to clarify the path to Bayesian modelling and avoid misapprehensions.

Further, the curious case of the Bayes factor (or of the posterior probability) could possibly be resolved most satisfactorily in this framework, as the [dreaded] dependence on the model prior probabilities then becomes a matter of relativity! Those posterior probabilities depend directly and almost linearly on the prior probabilities, but they should not be interpreted in an *absolute* sense as the ultimate and unique probability of the hypothesis (which anyway does not mean anything in terms of the observed experiment). In other words, this posterior probability does not need to be scaled against a U(0,1) distribution. Or against the *p*-value if anyone wishes to do so. By the end of the lecture, I was even wondering [not so loudly] whether or not this perspective was allowing for a resolution of the Lindley-Jeffreys paradox, as the resulting number could be set relative to the choice of the [arbitrary] normalising constant. Continue reading

## same data – different models – different answers

Posted in Books, Kids, Statistics, University life with tags Beaumont-en-Auge, Jeffreys priors, Laplace succession rule, non-informative priors, Normandie, Pierre Simon de Laplace, reference priors, statue, The Bayesian Choice, wikipedia on June 1, 2016 by xi'an**A**n interesting question from a reader of the Bayesian Choice came out on X validated last week. It was about Laplace’s succession rule, which I found somewhat over-used, but it was nonetheless interesting because the question was about the discrepancy of the “non-informative” answers derived from two models applied to the data: an Hypergeometric distribution in the Bayesian Choice and a Binomial on Wikipedia. The originator of the question had trouble with the difference between those two “non-informative” answers as she or he believed that there was a *single* non-informative principle that should lead to a unique answer. This does not hold, even when following a reference prior principle like Jeffreys’ invariant rule or Jaynes’ maximum entropy tenets. For instance, the Jeffreys priors associated with a Binomial and a Negative Binomial distributions differ. And even less when considering that there is no unity in reaching those reference priors. (Not even mentioning the issue of the reference dominating measure for the definition of the entropy.) This led to an informative debate, which is the point of X validated.

On a completely unrelated topic, the survey ship looking for the black boxes of the crashed EgyptAir plane is called the Laplace.

## how individualistic should statistics be?

Posted in Books, pictures, Statistics with tags ABC, All of Statistics, ancilarity, Annual Review of Statistics and Its Application, Bayesian inference, conditioning, control, foundations, frequentist inference, minimaxity, p-values, The Bayesian Choice on November 5, 2015 by xi'an**K**eli Liu and Xiao-Li Meng completed a paper on the very nature of inference, to appear in The Annual Review of Statistics and Its Application. This paper or chapter is addressing a fundamental (and foundational) question on drawing inference based a sample on a new observation. That is, in making prediction. To what extent should the characteristics of the sample used for that prediction resemble those of the future observation? In his 1921 book, *A Treatise on Probability*, Keynes thought this similarity (or individualisation) should be pushed to its extreme, which led him to somewhat conclude on the impossibility of statistics and never to return to the field again. Certainly missing the incoming possibility of comparing models and selecting variables. And not building so much on the “all models are wrong” tenet. On the contrary, classical statistics use the entire data available and the associated model to run the prediction, including Bayesian statistics, although it is less clear how to distinguish between data and control there. Liu & Meng debate about the possibility of creating controls from the data alone. Or “alone” as the model behind always plays a capital role.

“Bayes and Frequentism are two ends of the same spectrum—a spectrum defined in terms of relevance and robustness. The nominal contrast between them (…) is a red herring.”

The paper makes for an exhilarating if definitely challenging read. With a highly witty writing style. If only because the perspective is unusual, to say the least!, and requires constant mental contortions to frame the assertions into more traditional terms. For instance, I first thought that Bayesian procedures were in agreement with the ultimate conditioning approach, since it conditions on the observables and nothing else (except for the model!). Upon reflection, I am not so convinced that there is such a difference with the frequentist approach in the (specific) sense that they both take advantage of the entire dataset. Either from the predictive or from the plug-in distribution. It all boils down to how one defines “control”.

“Probability and randomness, so tightly yoked in our minds, are in fact distinct concepts (…) at the end of the day, probability is essentially a tool for bookkeeping, just like the abacus.”

Some sentences from the paper made me think of ABC, even though I am not trying to bring everything back to ABC!, as drawing controls is the nature of the ABC game. ABC draws samples or control from the prior predictive and only keeps those for which the relevant aspects (or the summary statistics) agree with those of the observed data. Which opens similar questions about the validity and precision of the resulting inference, as well as the loss of information due to the projection over the summary statistics. While ABC is not mentioned in the paper, it can be used as a benchmark to walk through it.

“In the words of Jack Kiefer, we need to distinguish those problems with `luck data’ from those with `unlucky data’.”

I liked very much recalling discussions we had with George Casella and Costas Goutis in Cornell about frequentist conditional inference, with the memory of Jack Kiefer still lingering around. However, I am not so excited about the processing of models here since, from what I understand in the paper (!), the probabilistic model behind the statistical analysis must be used to some extent in producing the control case and thus cannot be truly assessed with a critical eye. For instance, of which use is the mean square error when the model behind is unable to produce the observed data? In particular, the variability of this mean squared error is directly driven by this model. Similarly the notion of ancillaries is completely model-dependent. In the classification diagrams opposing robustness to relevance, all methods included therein are parametric. While non-parametric types of inference could provide a reference or a calibration ruler, at the very least.

Also, by continuously and maybe a wee bit heavily referring to the doctor-and-patient analogy, the paper is somewhat confusing as to which parts are analogy and which parts are methodology and to which type of statistical problem is covered by the discussion (sometimes it feels like all problems and sometimes like medical trials).

“The need to deliver individualized assessments of uncertainty are more pressing than ever.”

A final question leads us to an infinite regress: if the statistician needs to turn to individualized inference, at which level of individuality should the statistician be assessed? And who is going to provide the controls then? In any case, this challenging paper is definitely worth reading by (only mature?) statisticians to ponder about the nature of the game!