**T**oday, Ewan Cameron arXived a paper that generalises our Robert and Marin (2010) paper on the measure theoretic difficulties (or impossibilities) of the Savage-Dickey ratio and on the possible resolutions. (A paper of mine’s I like very much despite it having neither impact nor quotes, whatsoever! Until this paper.) I met Ewan last year when he was completing a PhD with Tony Pettitt at QUT in astrostatistics, but he ~~also worked~~ did not work on this transdimensional ABC algorithm with application to worm invasion in Northern Alberta (arXive I reviewed last week)… Ewan also runs a blog called Another astrostatistics blog, full of goodies, incl. the one where he announces he moves to… zoology in Oxford! Anyway, this note extends our paper and a mathematically valid Savage-Dickey ratio representation to the case when the posterior distributions have no density against the Lebesgue measure. For instance for Dirichlet processes or Gaussian processes priors. Using generic Radon-Nykodim derivatives instead. The example is somewhat artificial, superimposing a Dirichlet process prior onto the Old faithful benchmark. But this is an interesting entry, worth mentioning, into the computation of Bayes factors. And the elusive nature of the Savage-Dickey ratio representation.

## Archive for Dickey-Savage ratio

## generalised Savage-Dickey ratio

Posted in Statistics, University life with tags astrostatistics, Australia, Brisbane, Dickey-Savage ratio, Old Faithful geyser, QUT, University of Oxford, Yellowstone national park, zoology on November 11, 2013 by xi'an## paradoxes in scientific inference

Posted in Books, Statistics, University life with tags Allan Birnbaum, book cover, book reviews, CHANCE, Deborah Mayo, Dickey-Savage ratio, Error and Inference, Fermat, Likelihood Principle, p-values, paradoxes, scientific inference, Shigeo Fukuda, Simpson's paradox on November 23, 2012 by xi'an**T**his CRC Press book was sent to me for review in CHANCE: *Paradoxes in Scientific Inference* is written by Mark Chang, vice-president of AMAG Pharmaceuticals. The topic of scientific paradoxes is one of my primary interests and I have learned a lot by looking at Lindley-Jeffreys and Savage-Dickey paradoxes. However, I did not find a renewed sense of excitement when reading the book. The very first (and maybe the best!) paradox with *Paradoxes in Scientific Inference* is that it is a book from the future! Indeed, its copyright year is 2013 (!), although I got it a few months ago. (Not mentioning here the cover mimicking Escher’s “paradoxical” pictures with dices. A sculpture due to Shigeo Fukuda and apparently not quoted in the book. As I do not want to get into another dice cover polemic, I will abstain from further comments!)

**N**ow, getting into a deeper level of criticism (!), I find the book very uneven and overall quite disappointing. (Even missing in its statistical foundations.) Esp. given my initial level of excitement about the topic!

**F**irst, there is a tendency to turn *everything* into a paradox: obviously, when writing a book about paradoxes, everything looks like a paradox! This means bringing into the picture every paradox known to man and then some, i.e., things that are either un-paradoxical (e.g., Gödel’s incompleteness result) or uninteresting in a scientific book (e.g., the birthday paradox, which may be surprising but is far from a paradox!). Fermat’s theorem is also quoted as a paradox, even though there is nothing in the text indicating in which sense it is a paradox. (Or is it because it is simple to express, hard to prove?!) Similarly, Brownian motion is considered a paradox, as “*reconcil[ing] the paradox between two of the greatest theories of physics (…): thermodynamics and the kinetic theory of gases*” (p.51) 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!)

“*When the null hypothesis is rejected, the p-value is the probability of the type I error.*” *Paradoxes in Scientific Inference* (p.105)

“*The p-value is the conditional probability given H _{0}.” Paradoxes in Scientific Inference (p.106)*

**S**econd, the depth of the statistical analysis in the book is often found missing. For instance, Simpson’s paradox is not analysed from a statistical perspective, only reported as a fact. Sticking to statistics, take for instance the discussion of Lindley’s paradox. The author seems to think that the problem is with the different conclusions produced by the frequentist, likelihood, and Bayesian analyses (p.122). This is completely wrong: Lindley’s (or Lindley-Jeffreys‘s) paradox is about the lack of significance of Bayes factors based on improper priors. Similarly, when the likelihood ratio test is introduced, the reference threshold is given as equal to 1 and no mention is later made of compensating for different degrees of freedom/against over-fitting. The discussion about *p*-values is equally garbled, witness the above quote which (a) conditions upon the rejection and (b) ignores the dependence of the *p*-value on a realized random variable. Continue reading

## Bayesian model selection

Posted in Books, R, Statistics with tags Bayes factors, Bayesian model choice, book review, Chib's approximation, Dickey-Savage ratio, hypothesis testing, MCMC algorithms, model choice, particle filters, posterior predictive on December 8, 2010 by xi'an**L**ast week, I received a box of books from the ** International Statistical Review**, for reviewing them. I thus grabbed the one whose title was most appealing to me, namely

**by Tomohiro Ando. I am indeed interested in both the nature of testing hypotheses or more accurately of assessing models, as discussed in both my talk at the Seminar of philosophy of mathematics at Université Paris Diderot a few days ago and the post on Murray Aitkin’s alternative, and the computational aspects of the resulting Bayesian procedures, including evidence, the Savage-Dickey paradox, nested sampling, harmonic mean estimators, and more…**

*Bayesian Model Selection and Statistical Modeling***A**fter reading through the book, I am alas rather disappointed. What I consider to be innovative or at least “novel” parts with comparison with existing books (like Chen, Shao and Ibrahim, 2000, which remains a reference on this topic) is based on papers written by the author over the past five years and it is mostly a sort of asymptotic Bayes analysis that I do not see as particularly Bayesian, because involving the “true” distribution of the data. The coverage of the existing literature on Bayesian model choice is often incomplete and sometimes misses the point, as discussed below. This is especially true for the computational aspects that are generally mistreated or at least not treated in a way from which a newcomer to the field would benefit. The author often takes complex econometric examples for illustration, which is nice; however, he does not pursue the details far enough for the reader to be able to replicate the study without further reading. (An example is given by the coverage of stochastic volatility in Section 4.5.1, pages 83-84.) The few exercises at the end of each chapter are rather unhelpful, often sounding rather like notes than true problems (an extreme case is Exercise 6 pages 196-197 which introduces the Metropolis-Hastings algorithm within the exercise (although it has already been defined on pages 66-67) and then asks to derive the marginal likelihood estimator. Another such exercise on page 164-165 introduces the theory of DNA microarrays and gene expression in ten lines (which are later repeated verbatim on page 227), then asks to identify marker genes responsible for a certain trait.) The overall feeling after reading this book is thus that the contribution to the field of ** Bayesian Model Selection and Statistical Modeling** is too limited and disorganised for the book to be recommended as “

*helping you choose the right Bayesian model*” (backcover).

## Statistical Inference

Posted in Books, Statistics, University life with tags Bayes factors, Bayesian Analysis, Bayesian model choice, Dickey-Savage ratio, harmonic mean estimator, joint posterior, likelihood ratio, MCMC, mixture estimation, Pitman nearness on November 16, 2010 by xi'an**F**ollowing the publication of several papers on the topic of integrated evidence (about competing models), Murray Aitkin has now published a book entitled * Statistical Inference* and I have now finished reading it. While I appreciate the effort made by Murray Aitkin to place his theory within a coherent Bayesian framework, I remain unconvinced of the said coherence, for reasons exposed below.

**T**he main chapters of the book are Chapter 2 about the “Integrated Bayes/likelihood approach” and Chapter 4 about the “Unified analysis of finite populations”, Chapter 7 also containing a new proposal about “Goodness of fit and model diagnostics”. Chapter 1 is a nice introduction to frequentist, likelihood and Bayesian approaches to inference and the four remaining chapters are applications of Murray Aitkin‘s principles to various models. The style of the book is quite pleasant although slightly discursive in what I (a Frenchman!) would qualify as an English style in that it is often relying on intuition to develop concepts. I also think that the argument of being close to the frequentist decision (aka the p-value) too often serves as a justification in the book (see, e.g., page 43 “the p-value has a direct interpretation as a posterior probability”). As an aside, Murray Aitkin is a strong believer in plotting cdfs rather than densities to provide information about a distribution and hence cdf plots abound throughout the book. (I counted 82 pictures of them.) While the book contains a helpful array of examples and datasets, the captions of the (many) figures are too terse for my taste: The figures are certainly not self-contained and even with the help of the main text they do not always make complete sense. Continue reading

## Savage-Dickey published

Posted in Statistics, University life with tags Dickey-Savage ratio, EJS, Electronic Journal of Statistics on July 12, 2010 by xi'an**W**e got this email on Saturday about our Savage-Dickey resolution:

Your article “On resolving the Savage–Dickey paradox” was published in the Electronic Journal of Statistics 2010, Vol. 4, 643-654.

You may access electronic version of your paper in Euclid by DOI link http://dx.doi.org/10.1214/10-EJS564

**N**o extreme wonder that it appeared that quickly (when considering it was written in November and submitted to EJS in February) since EJS is an electronic journal but nice nonetheless!

## Savage-Dickey paper accepted

Posted in Statistics, University life with tags Bayes factor, Dickey-Savage ratio, Electronic Journal of Statistics, Monte Carlo methods, simulation on June 3, 2010 by xi'an**A**fter our second (light) round of revision, the [rearXived] paper on the Savage-Dickey paradox was accepted by the ** Electronic Journal of Statistics**. Great! This is actually my first paper in

**. In fact, I managed to include a short comment inspired by Geoff Nicholls, following a conversation we had at CRiSM. Namely, the three expressions we recover for the Monte Carlo approximations to the Bayes factor can all be seen as different avatars of the bridge sampling family of estimators. Therefore, it could be possible to compare those approaches against their asymptotic variance, or even to improve upon them…**

*EJS*## Snapshots from CRiSM workshop

Posted in Statistics, University life with tags Bayesian model choice, classification, CRiSM, Dickey-Savage ratio, MCMC, model uncertainty, particle filters, University of Warwick, Zeeman building on May 31, 2010 by xi'an**T**he workshop is a mix of theory, methodology, computational techniques and applications that makes it quite exciting. Although I already gave several versions of my talk in the past year, I still got interesting feedback, in particular a connection between our Savage-Dickey representation and bridge sampling. Among the interesting talks I attended, a few snapshots; both Jim Berger and Dongchu Sun presented new closed form expressions for effective numbers of parameters in complex linear models, Antonio Lijoi covered some mathematics of species estimation models [with some possible connections with Geoff Nicholls’ earlier talk on language classification], Chris Holmes exposed the advances he had made since Edinburgh (at least for those of us who attended both meetings), highlighting an interesting link with LDA, both Chris and Matthew Stephens considering direction selection for discrimination and clustering, Peter Müller talked about the analysis of a genetic pathway model represented as a graph, Robert Kohn zoomed through an adaptive particle MCMC algorithm where the likelihood was estimated (with the interesting side comment that what looks as an unbiased estimator from one perspective is also an auxiliary joint likelihood from another perspective) and David Spiegelhalter gave a both hilarious and thought-provoking talk on the “deeper uncertainty” that surrounds statistical models.