**A**n interesting question on X validated reminded me of the epiphany I had some twenty years ago when reading a Annals of Statistics paper by Anirban Das Gupta and Bill Strawderman on shrinkage estimators, namely that some estimators shared the same risk function, meaning their integrated loss was the same for all values of the parameter. As indicated in this question, Stefan‘s instructor seems to believe that two estimators having the same risk function must be a.s. identical. Which is not true as exemplified by the James-Stein (1960) estimator with scale 2(p-2), which has constant risk p, just like the maximum likelihood estimator. I presume the confusion stemmed from the concept of *completeness*, where having a function with constant expectation under all values of the parameter implies that this function is constant. But, for loss functions, the concept does not apply since the loss depends both on the observation (that is complete in a Normal model) and on the parameter.

## Archive for the Statistics Category

## same risk, different estimators

Posted in Statistics with tags Annals of Statistics, complete statistics, hierarchical Bayesian modelling, Jim Berger, shrinkage estimation, William Strawderman on November 10, 2017 by xi'an## probabilities larger than one…

Posted in Statistics with tags conditional density, cross validated, Gibbs sampling, Ising model, normalisation, normalising constant, pmf on November 9, 2017 by xi'an## bridgesampling [R package]

Posted in pictures, R, Statistics, University life with tags Amsterdam, bridge, bridge sampling, bridgesampling, JAGS, R, R package, STAN, University of Amsterdam, warped bridge sampling on November 9, 2017 by xi'an**Q**uentin F. Gronau, Henrik Singmann and Eric-Jan Wagenmakers have arXived a detailed documentation about their * bridgesampling* R package. (No wonder that researchers from Amsterdam favour bridge sampling!)

*[The package relates to a [52 pages] tutorial on bridge sampling by Gronau et al. that I will hopefully comment soon.]*The bridge sampling methodology for marginal likelihood approximation requires

*two*Monte Carlo samples for a ratio of

*two*integrals. A nice twist in this approach is to use a dummy integral that is already available, with respect to a probability density that is an approximation to the exact posterior. This means avoiding the difficulties with bridge sampling of bridging two different parameter spaces, in possibly different dimensions, with potentially very little overlap between the posterior distributions. The substitute probability density is chosen as Normal or warped Normal, rather than a t which would provide more stability in my opinion. The

*package also provides an error evaluation for the approximation, although based on spectral estimates derived from the*

**bridgesampling****package. The remainder of the document exhibits how the package can be used in conjunction with either JAGS or Stan. And concludes with the following words of caution:**

*coda*

“It should also be kept in mind that there may be cases in which the bridge sampling procedure may not be the ideal choice for conducting Bayesian model comparisons. For instance, when the models are nested it might be faster and easier to use the Savage-Dickey density ratio (Dickey and Lientz 1970; Wagenmakers et al. 2010). Another example is when the comparison of interest concerns a very large model space, and a separate bridge sampling based computation of marginal likelihoods may take too much time. In this scenario, Reversible Jump MCMC (Green 1995) may be more appropriate.”

## lazy ABC…what?!

Posted in Kids, pictures, Statistics with tags ABC, Australia, Gold Coast, lazy ABC, pictures, The Australian on November 8, 2017 by xi'an## R package truncnorm

Posted in Statistics with tags accept-reject algorithm, CRAN, John Geweke, R, truncated normal, truncnorm on November 8, 2017 by xi'an**T**his week in Warwick, thanks to a (rather incomprehensible) X validated question, I came across the CRAN R package truncnorm, which provides the “density, distribution function, quantile function, random generation and expected value function for the truncated normal distribution”. The short description of the sampler states that the method follows the accept-reject solution of John Geweke (1991), which I reproduced [independently!] a few years later. I may have missed the right code, but checking on the Github depository associated with this package, I did not find in the C code a trace of our optimal solution via a translated exponential proposal, since the exponential proosal, when used, relies on a scale equal to the left truncation point, *a* in the above picture. Obviously, this does not make a major difference in the execution time (and the algorithm is still correct!).

## O’Bayes in action

Posted in Books, Kids, Statistics, University life with tags bois de Boulogne, Charles de Gaulle, invariance, Jeffreys priors, La Défense, mathematical puzzle, noninformative priors, O-Bayes 2017, objective Bayes, randomisation, RER B, Roissy, Université Paris Dauphine on November 7, 2017 by xi'an**M**y next-door colleague [at Dauphine] François Simenhaus shared a paradox [to be developed in an incoming test!] with Julien Stoehr and I last week, namely that, when selecting the largest number between a [observed] and b [unobserved], drawing a random boundary on a [meaning that a is chosen iff a is larger than this boundary] increases the probability to pick the largest number above ½2…

When thinking about it in the wretched RER train [train that got immobilised for at least two hours just a few minutes after I went through!, good luck to the passengers travelling to the airport…] to De Gaulle airport, I lost the argument: if a<b, the probability [for this random bound] to be larger than a and hence for selecting b is 1-Φ(a), while, if a>b, the probability [of winning] is Φ(a). Hence the only case when the probability is ½ is when a is the median of this random variable. But, when discussing the issue further with Julien, I exposed an interesting non-informative prior characterisation. Namely, if I assume a,b to be iid U(0,M) and set an improper prior 1/M on M, the conditional probability that b>a given a is ½. Furthermore, the posterior probability to pick the right [largest] number with François’s randomised rule is also ½, no matter what the distribution of the random boundary is. Now, the most surprising feature of this coffee room derivation is that these properties only hold for the prior 1/M. Any other power of M will induce an asymmetry between a and b. (The same properties hold when a,b are iid Exp(M).) Of course, this is not absolutely unexpected since 1/M is the invariant prior and since the “intuitive” symmetry only holds under this prior. Power to O’Bayes!

When discussing again the matter with François yesterday, I realised I had changed his wording of the puzzle. The original setting is one with two cards hiding the unknown numbers a and b and of a player picking one of the cards. If the player picks a card at random, there is indeed a probability of ½ of picking the largest number. If the decision to switch or not depends on an independent random draw being larger or smaller than the number on the observed card, the probability to get max(a,b) in the end hits 1 when this random draw falls into (a,b) and remains ½ outside (a,b). Randomisation pays.

## a new paradigm for improper priors

Posted in Books, pictures, Statistics, Travel with tags Alfréd Rényi, Andrei Kolmogorov, axioms of probability, convergence of Gibbs samplers, improper priors, σ-algebra, marginalisation paradoxes, Norway, Trondheim on November 6, 2017 by xi'an**G**unnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.). Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a *random variable* associated with an *infinite mass probability measure*. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion that either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.