**W**hile reading Confidence, Likelihood, Probability), by Tore Schweder and Nils Hjort, in the train from Oxford to Warwick, I came upon this unexpected property shown by Lindqvist and Taraldsen (Biometrika, 2005) that to simulate a sample **y** conditional on the realisation of a sufficient statistic, T(**y**)=t⁰, it is sufficient (!!!) to simulate the components of **y** as y=G(u,θ), with u a random variable with fixed distribution, e.g., a U(0,1), and to solve in θ the fixed point equation T(**y**)=t⁰. Assuming there exists a single solution. Brilliant (like an aurora borealis)! To borrow a simple example from the authors, take an exponential sample to be simulated given the sum statistics. As it is well-known, the conditional distribution is then a (rescaled) Beta and the proposed algorithm ends up being a standard Beta generator. For the method to work in general, T(**y**) must factorise through a function of the u’s, a so-called pivotal condition which brings us back to my post title. If this condition does not hold, the authors once again brilliantly introduce a pseudo-prior distribution on the parameter θ to make it independent from the u’s conditional on T(**y**)=t⁰. And discuss the choice of the Jeffreys prior as optimal in this setting even when this prior is improper. While the setting is necessarily one of exponential families and of sufficient conditioning statistics, I find it amazing that this property is not more well-known [at least by me!]. And wonder if there is an equivalent outside exponential families, for instance for simulating a ** t** sample conditional on the average of this sample.

## Archive for sufficient statistics

## fiducial simulation

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags book review, conditional density, English train, fiducial statistics, Jeffreys prior, Monte Carlo Statistical Methods, Oxford, pseudo-random generator, simulation, Student's t distribution, sufficient statistics, University of Warwick on April 19, 2018 by xi'an## Darmois, Koopman, and Pitman

Posted in Books, Statistics with tags cross validated, Don Fraser, exponential families, George Darmois, mathematical statistics, Pitman-Koopman theorem, proof, Stanford University, sufficient statistics on November 15, 2017 by xi'an**W**hen [X’ed] seeking a simple proof of the Pitman-Koopman-Darmois lemma [that exponential families are the only types of distributions with constant support allowing for a fixed dimension sufficient statistic], I came across a 1962 Stanford technical report by Don Fraser containing a short proof of the result. Proof that I do not fully understand as it relies on the notion that the likelihood function itself is a minimal sufficient statistic.

## about the strong likelihood principle

Posted in Books, Statistics, University life with tags ABC, ABC model choice, Alan Birnbaum, Bayesian Analysis, conditioning, sufficient statistics, The Likelihood Principle, weak conditionality principle on November 13, 2014 by xi'an**D**eborah Mayo arXived a Statistical Science paper a few days ago, along with discussions by Jan Bjørnstad, Phil Dawid, Don Fraser, Michael Evans, Jan Hanning, R. Martin and C. Liu. I am very glad that this discussion paper came out and that it came out in Statistical Science, although I am rather surprised to find no discussion by Jim Berger or Robert Wolpert, and even though I still cannot entirely follow the deductive argument in the rejection of Birnbaum’s proof, just as in the earlier version in Error & Inference. But I somehow do not feel like going again into a new debate about this critique of Birnbaum’s derivation. (Even though statements like the fact that the SLP “would preclude the use of sampling distributions” (p.227) would call for contradiction.)

“It is the imprecision in Birnbaum’s formulation that leads to a faulty impression of exactly what is proved.” M. Evans

Indeed, at this stage, I fear that [for me] a more relevant issue is whether or not the debate does matter… At a logical cum foundational [and maybe cum historical] level, it makes perfect sense to uncover if and which if any of the myriad of Birnbaum’s likelihood Principles holds. [Although trying to uncover Birnbaum’s motives and positions over time may not be so relevant.] I think the paper and the discussions acknowledge that *some* version of the weak conditionality Principle does not imply *some* version of the strong likelihood Principle. With other logical implications remaining true. At a methodological level, I am less much less sure it matters. Each time I taught this notion, I got blank stares and incomprehension from my students, to the point I have now stopped altogether teaching the likelihood Principle in class. And most of my co-authors do not seem to care very much about it. At a purely mathematical level, I wonder if there even is ground for a debate since the notions involved can be defined in various imprecise ways, as pointed out by Michael Evans above and in his discussion. At a statistical level, sufficiency eventually is a strange notion in that it seems to make plenty of sense until one realises there is no interesting sufficiency outside exponential families. Just as there are very few parameter transforms for which unbiased estimators can be found. So I also spend very little time teaching and even less worrying about sufficiency. (As it happens, I taught the notion this morning!) At another and presumably more significant statistical level, what matters is information, e.g., conditioning means adding information (i.e., about which experiment has been used). While complex settings may prohibit the use of the entire information provided by the data, at a formal level there is no argument for not using the entire information, i.e. conditioning upon the entire data. (At a computational level, this is no longer true, witness ABC and similar limited information techniques. By the way, ABC demonstrates if needed why sampling distributions matter so much to Bayesian analysis.)

“Non-subjective Bayesians who (…) have to live with some violations of the likelihood principle (…) since their prior probability distributions are influenced by the sampling distribution.” D. Mayo (p.229)

In the end, the fact that the prior may depend on the form of the sampling distribution and hence does violate the likelihood Principle does not worry me so much. In most models I consider, the parameters are endogenous to those sampling distributions and do not live an ethereal existence independently from the model: they are substantiated and calibrated by the model itself, which makes the discussion about the LP rather vacuous. See, e.g., the coefficients of a linear model. In complex models, or in large datasets, it is even impossible to handle the whole data or the whole model and proxies have to be used instead, making worries about the structure of the (original) likelihood vacuous. I think we have now reached a stage of statistical inference where models are no longer accepted as ideal truth and where approximation is the hard reality, imposed by the massive amounts of data relentlessly calling for immediate processing. Hence, where the self-validation or invalidation of such approximations in terms of predictive performances is the relevant issue. Provided we can at all face the challenge…

## Pre-processing for approximate Bayesian computation in image analysis

Posted in R, Statistics, University life with tags ABC, Chamonix, image processing, MCMC, MCMSki IV, Monte Carlo Statistical Methods, path sampling, Potts model, QUT, simulation, SMC-ABC, Statistics and Computing, sufficient statistics, summary statistics on March 21, 2014 by xi'an**W**ith Matt Moores and Kerrie Mengersen, from QUT, we wrote this short paper just in time for the MCMSki IV Special Issue of *Statistics & Computing*. And arXived it, as well. The global idea is to cut down on the cost of running an ABC experiment by removing the simulation of a humongous state-space vector, as in Potts and hidden Potts model, and replacing it by an approximate simulation of the 1-d sufficient (summary) statistics. In that case, we used a division of the 1-d parameter interval to simulate the distribution of the sufficient statistic for each of those parameter values and to compute the expectation and variance of the sufficient statistic. Then the conditional distribution of the sufficient statistic is approximated by a Gaussian with these two parameters. And those Gaussian approximations substitute for the true distributions within an ABC-SMC algorithm à la Del Moral, Doucet and Jasra (2012).

**A**cross 20 125 × 125 pixels simulated images, Matt’s algorithm took an average of 21 minutes per image for between 39 and 70 SMC iterations, while resorting to pseudo-data and deriving the genuine sufficient statistic took an average of 46.5 hours for 44 to 85 SMC iterations. On a realistic Landsat image, with a total of 978,380 pixels, the precomputation of the mapping function took 50 minutes, while the total CPU time on 16 parallel threads was 10 hours 38 minutes. By comparison, it took 97 hours for 10,000 MCMC iterations on this image, with a poor effective sample size of 390 values. Regular SMC-ABC algorithms cannot handle this scale: It takes 89 hours to perform *a single* SMC iteration! (Note that path sampling also operates in this framework, thanks to the same precomputation: in that case it took 2.5 hours for 10⁵ iterations, with an effective sample size of 10⁴…)

**S**ince my student’s paper on Seaman et al (2012) got promptly rejected by *TAS* for quoting too extensively from my post, we decided to include me as an extra author and submitted the paper to this special issue as well.

## ABC with indirect summary statistics

Posted in Statistics, University life with tags ABC, continuous-time stochastic process, econometrics, indirect inference, Ornstein-Uhlenbeck process, semi-parametrics, sufficiency, sufficient statistics on February 3, 2014 by xi'an**A**fter reading Drovandi’s and Pettitt’s Bayesian Indirect Inference, I checked (in the plane to Birmingham) the earlier Gleim’s and Pigorsch’s Approximate Bayesian Computation with indirect summary statistics. The setting is indeed quite similar to the above, with a description of three ways of connecting indirect inference with ABC, albeit with a different range of illustrations. This preprint states most clearly its assumption that the generating model is a particular case of the auxiliary model, which sounds anticlimactic since the auxiliary model is precisely used because the original one is mostly out of reach! This certainly was the original motivation for using indirect inference.

**T**he part of the paper that I find the most intriguing is the argument that the indirect approach leads to sufficient summary statistics, in the sense that they “are sufficient for the parameters of the auxiliary model and (…) sufficiency carries over to the model of interest” (p.31). Looking at the details in the Appendix, I found that the argument is lacking, because the likelihood *as a functional* is shown to be a (sufficient) statistic, which seems both a tautology and irrelevant because this is different from the likelihood considered at the (auxiliary) MLE, which is the summary statistic used *in fine*.

“…we expand the square root of an innovation density h in a Hermite expansion and truncate the infinite polynomial at some integer K which, together with other tuning parameters of the SNP density, has to be determined through a model selection criterion (such as BIC). Now we take the leading term of the Hermite expansion to follow a Gaussian GARCH model.”

**A**s in Drovandi and Pettitt, the performances of the ABC-I schemes are tested on a toy example, which is a very basic exponential iid sample with a conjugate prior. With a gamma model as auxiliary. The authors use a standard ABC based on the first two moments as their benchmark, however they do not calibrate those moments in the distance and end up with poor performances of ABC (in a setting where there is a sufficient statistic!). The best choice in this experiment appears as the solution based on the score, but the variances of the distances are not included in the comparison tables. The second implementation considered in the paper is a rather daunting continuous-time non-Gaussian Ornstein-Uhlenbeck stochastic volatility model à la Barndorf-Nielsen and Shephard (2001). The construction of the semi-nonparametric (why not semi-parametric?) auxiliary model is quite involved as well, as illustrated by the quote above. The approach provides an answer, with posterior ABC-IS distributions on all parameters of the original model, which kindles the question of the validation of this answer in terms of the original posterior. Handling simultaneously several approximation processes would help in this regard.

## ABC model choice [slides]

Posted in pictures, Statistics, Travel, University life with tags ABC, asymptotics, Bayesian model choice, insufficiency, Madrid, Novermber 11, quantile distribution, sufficient statistics, summary statistics, Workshop Métodos Bayesianos 11 on November 7, 2011 by xi'an**H**ere are the slides for my talks both at CREST this afternoon (in ½ an hour!) and in Madrid [on Friday 11/11/11=1^{6}, magical day of the year, especially since I will be speaking at 11:11 CET…] for the Workshop Métodos Bayesianos 11 (no major difference with the slides from Zürich, hey!, except for the quantile distribution example]

## workshop in Columbia [talk]

Posted in Statistics, Travel, University life with tags ABC, ancillary statistics, asymptotics, Bayesian model choice, central limit theorem, Columbia University, New York city, PNAS, sufficient statistics, summary statistics on September 25, 2011 by xi'an**H**ere are the slides of my talk yesterday at the Computational Methods in Applied Sciences workshop in Columbia:

**T**he last section of the talk covers our new results with Jean-Michel Marin, Natesh Pillai and Judith Rousseau on the necessary and sufficient conditions for a summary statistic to be used in ABC model choice. (The paper is about to be completed.) This obviously comes as the continuation of our reflexions on ABC model choice started last January. The major message of the paper is that the statistics used for running model choice cannot have a mean value common to both models, which strongly implies using ancillary statistics with different means under each model. *(I am afraid that, thanks to the mixture of no-jetlag fatigue and of slide inflation [95 vs. 40mn] and of asymptotics technicalities in the last part, the talk was far from comprehensible. I started on the wrong foot with not getting an XL [Xiao-Li’s] comment on the measure-theory problem with the limit in ε going to zero. A peak given that great debate we had in Banff with Jean-Michel, David Balding, and Mark Beaumont, years ago. And our more recent paper about the arbitrariness of the density value in the Savage-Dickey paradox. I then compounded the confusion by stating the empirical **mean was sufficient in the Laplace case…which is not even an exponential family. I hope I will be more articulate next week in Zürich where at least I will not speak past my bedtime!*)