Archive for The Bayesian Choice

about paradoxes

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , on December 5, 2017 by xi'an

An 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:

“For instance, the author considers the MLE being biased to be a paradox (p.117), while omitting the much more substantial “paradox” of the non-existence of unbiased estimators of most parameters—which simply means unbiasedness is irrelevant. Or the other even more puzzling “paradox” that the secondary MLE derived from the likelihood associated with the distribution of a primary MLE may differ from the primary. (My favourite!)”

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 , , , , , , , on February 1, 2017 by xi'an

St John's College, Oxford, Feb. 23, 2012As 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

back in Oxford

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , on January 30, 2017 by xi'an

As in the previous years, I am back in Oxford (England) for my short Bayesian Statistics course in the joint Oxford-Warwick PhD programme, OxWaSP.  For some unclear reason, presumably related to the Internet connection from Oxford, I have not been able to upload my slides to Slideshare, so here the [99.9% identical] older version:

same data – different models – different answers

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on June 1, 2016 by xi'an

An 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.

off to Oxford

Posted in Kids, pictures, Travel, University life with tags , , , , , , , on January 31, 2016 by xi'an

Oxford, Feb. 23, 2012I am off to Oxford this evening for teaching once again in the Bayesian module of the OxWaSP programme. Joint PhD programme between Oxford and Warwick, supported by the EPSRC. And with around a dozen new [excellent!] PhD students every year. Here are the slides of a longer course that I will use in the coming days:

And by popular request (!) here is the heading of my Beamer file:

% Rather be using my own color
\setbeamercolor{alerted text}{fg=lightred}

how individualistic should statistics be?

Posted in Books, pictures, Statistics with tags , , , , , , , , , , , on November 5, 2015 by xi'an

keep-stats-subjectiveKeli 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.”

viemortrerbThe 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’.”

keep-calm-and-condi-tionI 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!

Think Bayes: Bayesian Statistics Made Simple

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , on October 27, 2015 by xi'an

Almost Bayes can!By some piece of luck, I came upon the book Think Bayes: Bayesian Statistics Made Simple, written by Allen B. Downey and published by Green Tea Press [which I could relate to No Starch Press, focussing on coffee!, which published Statistics Done Wrong that I reviewed a while ago] which usually publishes programming books with fun covers. The book is available on-line for free in pdf and html formats, and I went through it during a particularly exciting administrative meeting…

“Most books on Bayesian statistics use mathematical notation and present ideas in terms of mathematical concepts like calculus. This book uses Python code instead of math, and discrete approximations instead of continuous mathematics. As a result, what would be an integral in a math book becomes a summation, and most operations on probability distributions are simple loops.”

The book is most appropriately published in this collection as most of it concentrates on Python programming, with hardly any maths formula. In some sense similar to Jim Albert’s R book. Obviously, coming from maths, and having never programmed in Python, I find the approach puzzling, But just as obviously, I am aware—both from the comments on my books and from my experience on X validated—that a large group (majority?) of newcomers to the Bayesian realm find the mathematical approach to the topic a major hindrance. Hence I am quite open to this editorial choice as it is bound to include more people to think Bayes, or to think they can think Bayes.

“…in fewer than 200 pages we have made it from the basics of probability to the research frontier. I’m very happy about that.”

The choice made of operating almost exclusively through motivating examples is rather traditional in US textbooks. See e.g. Albert’s book. While it goes against my French inclination to start from theory and concepts and end up with illustrations, I can see how it operates in a programming book. But as always I fear it makes generalisations uncertain and understanding more shaky… The examples are per force simple and far from realistic statistics issues. Hence illustrates more the use of Bayesian thinking for decision making than for data analysis. To wit, those examples are about the Monty Hall problem and other TV games, some urn, dice, and coin models, blood testing, sport predictions, subway waiting times, height variability between men and women, SAT scores, cancer causality, a Geiger counter hierarchical model inspired by Jaynes, …, the exception being the final Belly Button Biodiversity dataset in the final chapter, dealing with the (exciting) unseen species problem in an equally exciting way. This may explain why the book does not cover MCMC algorithms. And why ABC is covered through a rather artificial normal example. Which also hides some of the maths computations under the carpet.

“The underlying idea of ABC is that two datasets are alike if they yield the same summary statistics. But in some cases, like the example in this chapter, it is not obvious which summary statistics to choose.¨

In conclusion, this is a very original introduction to Bayesian analysis, which I welcome for the reasons above. Of course, it is only an introduction, which should be followed by a deeper entry into the topic, and with [more] maths. In order to handle more realistic models and datasets.