Archive for Statistical Science

the first Bayesian

Posted in Statistics with tags , , , , , , , on February 20, 2018 by xi'an

In the first issue of Statistical Science for this year (2018), Stephen Stiegler pursues the origins of Bayesianism as attributable to Richard Price, main author of Bayes’ Essay. (This incidentally relates to an earlier ‘Og piece on that notion!) Steve points out the considerable inputs of Price on this Essay, even though the mathematical advance is very likely to be entirely Bayes’. It may however well be Price who initiated Bayes’ reflections on the matter, towards producing a counter-argument to Hume’s “On Miracles”.

“Price’s caution in addressing the probabilities of hypotheses suggested by data is rare in early literature.”

A section of the paper is about Price’s approach data-determined hypotheses and to the fact that considering such hypotheses cannot easily fit within a Bayesian framework. As stated by Price, “it would be improbable as infinite to one”. Which is a nice way to address the infinite mass prior.

 

amazing appendix

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , on February 13, 2018 by xi'an

In the first appendix of the 1995 Statistical Science paper of Besag, Green, Higdon and Mengersen, on MCMC, “Bayesian Computation and Stochastic Systems”, stands a fairly neat result I was not aware of (and which Arnaud Doucet, with his unrivalled knowledge of the literature!, pointed out to me in Oxford, avoiding me the tedium to try to prove it afresco!). I remember well reading a version of the paper in Fort Collins, Colorado, in 1993 (I think!) but nothing about this result.

It goes as follows: when running a Metropolis-within-Gibbs sampler for component x¹ of a collection of variates x¹,x²,…, thus aiming at simulating from the full conditional of x¹ given x⁻¹ by making a proposal q(x|x¹,x⁻¹), it is perfectly acceptable to use a proposal that depends on a parameter α (no surprise so far!) and to generate this parameter α anew at each iteration (still unsurprising as α can be taken as an auxiliary variable) and to have the distribution of this parameter α depending on the other variates x²,…, i.e., x⁻¹. This is the surprising part, as adding α as an auxiliary variable was messing up the update of x⁻¹. But the proof as found in the 1995 paper [page 35] does not require to consider α as such as it establishes global balance directly. (Or maybe still detailed balance when writing the whole Gibbs sampler as a cycle of Metropolis steps.) Terrific! And a whiff mysterious..!

Bayes, reproducibility and the Quest for Truth

Posted in Books, Statistics, University life with tags , , , , , on April 27, 2017 by xi'an

Don Fraser, Mylène Bédard, and three coauthors have written a paper with the above dramatic title in Statistical Science about the reproducibility of Bayesian inference in the framework of what they call a mathematical prior. Connecting with the earlier quick-and-dirty tag attributed by Don to Bayesian credible intervals.

“We provide simple (…) counter-examples to general claims that Bayes can offer accuracy for statistical inference. To obtain this accuracy with Bayes, more effort is required compared to recent likelihood methods (…) [and] accuracy beyond first order is routinely not available (…) An alternative is to view default Bayes as an exploratory technique and then ask does it do as it overtly claims? Is it reproducible as understood in contemporary science? (…) No one has answers although speculative claims abound.” (p. 1)

The early stages of the paper questions the nature of a prior distribution in terms of objectivity and reproducibility, which strikes me as a return to older debates on the nature of probability. And of a dubious insistence on the reality of a prior when the said reality is customarily and implicitly assumed for the sampling distribution. While we “can certainly ask how [a posterior] quantile relates to the true value of the parameter”, I see no compelling reason why the associated quantile should be endowed with a frequentist coverage meaning, i.e., be more than a normative indication of the deviation from the true value. (Assuming there is such a parameter.) To consider that the credible interval of interest can be “objectively” assessed by simulation experiments evaluating its coverage is thus doomed from the start (since there is not reason for the nominal coverage) and situated on the wrong plane since it stems from the hypothetical frequentist model for a range of parameter values. Instead I find simulations from (generating) models useful in a general ABC sense, namely by producing realisations from the predictive one can assess at which degree of roughness the data is compatible with the formal construct. To bind reproducibility to the frequentist framework thus sounds wrong [to me] as being model-based. In other words, I do not find the definition of reproducibility used in the paper to be objective (literally bouncing back from Gelman and Hennig Read Paper)

At several points in the paper, the legal consequences of using a subjective prior are evoked as legally binding and implicitly as dangerous. With the example of the L’Aquila expert trial. I have trouble seeing the relevance of this entry as an adverse lawyer is as entitled to attack the expert on her or his sampling model. More fundamentally, I feel quite uneasy about bringing this type of argument into the debate!

nonparametric Bayesian clay for robust decision bricks

Posted in Statistics with tags , , , , , , on January 30, 2017 by xi'an

Just received an email today that our discussion with Judith of Chris Holmes and James Watson’s paper was now published as Statistical Science 2016, Vol. 31, No. 4, 506-510… While it is almost identical to the arXiv version, it can be read on-line.

ISBA 2016 [#3]

Posted in pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , on June 16, 2016 by xi'an

Among the sessions I attended yesterday, I really liked the one on robustness and model mispecification. Especially the talk by Steve McEachern on Bayesian inference based on insufficient statistics, with a striking graph of the degradation of the Bayes factor as the prior variance increases. I sadly had no time to grab a picture of the graph, which compared this poor performance against a stable rendering when using a proper summary statistic. It clearly relates to our work on ABC model choice, as well as to my worries about the Bayes factor, so this explains why I am quite excited about this notion of restricted inference. In this session, Chris Holmes also summarised his two recent papers on loss-based inference, which I discussed here in a few posts, including the Statistical Science discussion Judith and I wrote recently. I also went to the j-ISBA [section] session which was sadly under-attended, maybe due to too many parallel sessions, maybe due to the lack of unifying statistical theme.

comments on Watson and Holmes

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , , on April 1, 2016 by xi'an


“The world is full of obvious things which nobody by any chance ever observes.” The Hound of the Baskervilles

In connection with the incoming publication of James Watson’s and Chris Holmes’ Approximating models and robust decisions in Statistical Science, Judith Rousseau and I wrote a discussion on the paper that has been arXived yesterday.

“Overall, we consider that the calibration of the Kullback-Leibler divergence remains an open problem.” (p.18)

While the paper connects with earlier ones by Chris and coauthors, and possibly despite the overall critical tone of the comments!, I really appreciate the renewed interest in robustness advocated in this paper. I was going to write Bayesian robustness but to differ from the perspective adopted in the 90’s where robustness was mostly about the prior, I would say this is rather a Bayesian approach to model robustness from a decisional perspective. With definitive innovations like considering the impact of posterior uncertainty over the decision space, uncertainty being defined e.g. in terms of Kullback-Leibler neighbourhoods. Or with a Dirichlet process distribution on the posterior. This may step out of the standard Bayesian approach but it remains of definite interest! (And note that this discussion of ours [reluctantly!] refrained from capitalising on the names of the authors to build easy puns linked with the most Bayesian of all detectives!)

Harold Jeffreys’ default Bayes factor [for psychologists]

Posted in Books, Statistics, University life with tags , , , , , , on January 16, 2015 by xi'an

“One of Jeffreys’ goals was to create default Bayes factors by using prior distributions that obeyed a series of general desiderata.”

The paper Harold Jeffreys’s default Bayes factor hypothesis tests: explanation, extension, and application in Psychology by Alexander Ly, Josine Verhagen, and Eric-Jan Wagenmakers is both a survey and a reinterpretation cum explanation of Harold Jeffreys‘ views on testing. At about the same time, I received a copy from Alexander and a copy from the journal it had been submitted to! This work starts with a short historical entry on Jeffreys’ work and career, which includes four of his principles, quoted verbatim from the paper:

  1. “scientific progress depends primarily on induction”;
  2. “in order to formalize induction one requires a logic of partial belief” [enters the Bayesian paradigm];
  3. “scientific hypotheses can be assigned prior plausibility in accordance with their complexity” [a.k.a., Occam’s razor];
  4. “classical “Fisherian” p-values are inadequate for the purpose of hypothesis testing”.

“The choice of π(σ)  therefore irrelevant for the Bayes factor as long as we use the same weighting function in both models”

A very relevant point made by the authors is that Jeffreys only considered embedded or nested hypotheses, a fact that allows for having common parameters between models and hence some form of reference prior. Even though (a) I dislike the notion of “common” parameters and (b) I do not think it is entirely legit (I was going to write proper!) from a mathematical viewpoint to use the same (improper) prior on both sides, as discussed in our Statistical Science paper. And in our most recent alternative proposal. The most delicate issue however is to derive a reference prior on the parameter of interest, which is fixed under the null and unknown under the alternative. Hence preventing the use of improper priors. Jeffreys tried to calibrate the corresponding prior by imposing asymptotic consistency under the alternative. And exact indeterminacy under “completely uninformative” data. Unfortunately, this is not a well-defined notion. In the normal example, the authors recall and follow the proposal of Jeffreys to use an improper prior π(σ)∝1/σ on the nuisance parameter and argue in his defence the quote above. I find this argument quite weak because suddenly the prior on σ becomes a weighting function... A notion foreign to the Bayesian cosmology. If we use an improper prior for π(σ), the marginal likelihood on the data is no longer a probability density and I do not buy the argument that one should use the same measure with the same constant both on σ alone [for the nested hypothesis] and on the σ part of (μ,σ) [for the nesting hypothesis]. We are considering two spaces with different dimensions and hence orthogonal measures. This quote thus sounds more like wishful thinking than like a justification. Similarly, the assumption of independence between δ=μ/σ and σ does not make sense for σ-finite measures. Note that the authors later point out that (a) the posterior on σ varies between models despite using the same data [which shows that the parameter σ is far from common to both models!] and (b) the [testing] Cauchy prior on δ is only useful for the testing part and should be replaced with another [estimation] prior when the model has been selected. Which may end up as a backfiring argument about this default choice.

“Each updated weighting function should be interpreted as a posterior in estimating σ within their own context, the model.”

The re-derivation of Jeffreys’ conclusion that a Cauchy prior should be used on δ=μ/σ makes it clear that this choice only proceeds from an imperative of fat tails in the prior, without solving the calibration of the Cauchy scale. (Given the now-available modern computing tools, it would be nice to see the impact of this scale γ on the numerical value of the Bayes factor.) And maybe it also proceeds from a “hidden agenda” to achieve a Bayes factor that solely depends on the t statistic. Although this does not sound like a compelling reason to me, since the t statistic is not sufficient in this setting.

In a differently interesting way, the authors mention the Savage-Dickey ratio (p.16) as a way to represent the Bayes factor for nested models, without necessarily perceiving the mathematical difficulty with this ratio that we pointed out a few years ago. For instance, in the psychology example processed in the paper, the test is between δ=0 and δ≥0; however, if I set π(δ=0)=0 under the alternative prior, which should not matter [from a measure-theoretic perspective where the density is uniquely defined almost everywhere], the Savage-Dickey representation of the Bayes factor returns zero, instead of 9.18!

“In general, the fact that different priors result in different Bayes factors should not come as a surprise.”

The second example detailed in the paper is the test for a zero Gaussian correlation. This is a sort of “ideal case” in that the parameter of interest is between -1 and 1, hence makes the choice of a uniform U(-1,1) easy or easier to argue. Furthermore, the setting is also “ideal” in that the Bayes factor simplifies down into a marginal over the sample correlation only, under the usual Jeffreys priors on means and variances. So we have a second case where the frequentist statistic behind the frequentist test[ing procedure] is also the single (and insufficient) part of the data used in the Bayesian test[ing procedure]. Once again, we are in a setting where Bayesian and frequentist answers are in one-to-one correspondence (at least for a fixed sample size). And where the Bayes factor allows for a closed form through hypergeometric functions. Even in the one-sided case. (This is a result obtained by the authors, not by Jeffreys who, as the proper physicist he was, obtained approximations that are remarkably accurate!)

“The fact that the Bayes factor is independent of the intention with which the data have been collected is of considerable practical importance.”

The authors have a side argument in this section in favour of the Bayes factor against the p-value, namely that the “Bayes factor does not depend on the sampling plan” (p.29), but I find this fairly weak (or tongue in cheek) as the Bayes factor does depend on the sampling distribution imposed on top of the data. It appears that the argument is mostly used to defend sequential testing.

“The Bayes factor (…) balances the tension between parsimony and goodness of fit, (…) against overfitting the data.”

In fine, I liked very much this re-reading of Jeffreys’ approach to testing, maybe the more because I now think we should get away from it! I am not certain it will help in convincing psychologists to adopt Bayes factors for assessing their experiments as it may instead frighten them away. And it does not bring an answer to the vexing issue of the relevance of point null hypotheses. But it constitutes a lucid and innovative of the major advance represented by Jeffreys’ formalisation of Bayesian testing.