Archive for fiducial inference

revisiting marginalisation paradoxes [Bayesian reads #1]

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on February 8, 2019 by xi'an

As a reading suggestion for my (last) OxWaSP Bayesian course at Oxford, I included the classic 1973 Marginalisation paradoxes by Phil Dawid, Mervyn Stone [whom I met when visiting UCL in 1992 since he was sharing an office with my friend Costas Goutis], and Jim Zidek. Paper that also appears in my (recent) slides as an exercise. And has been discussed many times on this  ‘Og.

Reading the paper in the train to Oxford was quite pleasant, with a few discoveries like an interesting pike at Fraser’s structural (crypto-fiducial?!) distributions that “do not need Bayesian improper priors to fall into the same paradoxes”. And a most fascinating if surprising inclusion of the Box-Müller random generator in an argument, something of a precursor to perfect sampling (?). And a clear declaration that (right-Haar) invariant priors are at the source of the resolution of the paradox. With a much less clear notion of “un-Bayesian priors” as those leading to a paradox. Especially when the authors exhibit a red herring where the paradox cannot disappear, no matter what the prior is. Rich discussion (with none of the current 400 word length constraint), including the suggestion of neutral points, namely those that do identify a posterior, whatever that means. Funny conclusion, as well:

“In Stone and Dawid’s Biometrika paper, B1 promised never to use improper priors again. That resolution was short-lived and let us hope that these two blinkered Bayesians will find a way out of their present confusion and make another comeback.” D.J. Bartholomew (LSE)

and another

“An eminent Oxford statistician with decidedly mathematical inclinations once remarked to me that he was in favour of Bayesian theory because it made statisticians learn about Haar measure.” A.D. McLaren (Glasgow)

and yet another

“The fundamentals of statistical inference lie beneath a sea of mathematics and scientific opinion that is polluted with red herrings, not all spawned by Bayesians of course.” G.N. Wilkinson (Rothamsted Station)

Lindley’s discussion is more serious if not unkind. Dennis Lindley essentially follows the lead of the authors to conclude that “improper priors must go”. To the point of retracting what was written in his book! Although concluding about the consequences for standard statistics, since they allow for admissible procedures that are associated with improper priors. If the later must go, the former must go as well!!! (A bit of sophistry involved in this argument…) Efron’s point is more constructive in this regard since he recalls the dangers of using proper priors with huge variance. And the little hope one can hold about having a prior that is uninformative in every dimension. (A point much more blatantly expressed by Dickey mocking “magic unique prior distributions”.) And Dempster points out even more clearly that the fundamental difficulty with these paradoxes is that the prior marginal does not exist. Don Fraser may be the most brutal discussant of all, stating that the paradoxes are not new and that “the conclusions are erroneous or unfounded”. Also complaining about Lindley’s review of his book [suggesting prior integration could save the day] in Biometrika, where he was not allowed a rejoinder. It reflects on the then intense opposition between Bayesians and fiducialist Fisherians. (Funny enough, given the place of these marginalisation paradoxes in his book, I was mistakenly convinced that Jaynes was one of the discussants of this historical paper. He is mentioned in the reply by the authors.)

information maximising neural networks summaries

Posted in pictures, Statistics with tags , , , , , , , , on February 6, 2019 by xi'an

After missing the blood moon eclipse last night, I had a meeting today at the Paris observatory (IAP), where we discussed an ABC proposal made by Tom Charnock, Guilhem Lavaux, and Benjamin Wandelt from this institute.

“We introduce a simulation-based machine learning technique that trains artificial neural networks to find non-linear functionals of data that maximise Fisher information : information maximising neural networks.” T. Charnock et al., 2018
The paper is centred on the determination of “optimal” summary statistics. With the goal of finding “transformation which maps the data to compressed summaries whilst conserving Fisher information [of the original data]”. Which sounds like looking for an efficient summary and hence impossible in non-exponential cases. As seen from the description in (2.1), the assumed distribution of the summary is Normal, with mean μ(θ) and covariance matrix C(θ) that are implicit transforms of the parameter θ. In that respect, the approach looks similar to the synthetic likelihood proposal of Wood (2010). From which an unusual form of Fisher information can be derived, as μ(θ)’C(θ)⁻¹μ(θ)… A neural net is trained to optimise this information criterion at a given (so-called fiducial) value of θ, in terms of a set of summaries of the same dimension as the data. Which means the information contained in the whole data (likelihood) is not necessarily recovered, linking with this comment from Edward Ionides (in a set of lectures at Wharton).
“Even summary statistics derived by careful scientific or statistical reasoning have been found surprisingly uninformative compared to the whole data likelihood in both scientific investigations (Shrestha et al., 2011) and simulation experiments (Fasiolo et al., 2016)” E. Ionides, slides, 2017
The maximal Fisher information obtained in this manner is then used in a subsequent ABC step as the natural metric for the distance between the observed and simulated data. (Begging the question as to why being maximal is necessarily optimal.) Another question is about the choice of the fiducial parameter, which choice should be tested by for instance iterating the algorithm a few steps. But having to run simulations for a single value of the parameter is certainly a great selling point!

distributions for parameters [seminar]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on January 22, 2018 by xi'an
Next Thursday, January 25, Nancy Reid will give a seminar in Paris-Dauphine on distributions for parameters that covers different statistical paradigms and bring a new light on the foundations of statistics. (Coffee is at 10am in the Maths department common room and the talk is at 10:15 in room A, second floor.)

Nancy Reid is University Professor of Statistical Sciences and the Canada Research Chair in Statistical Theory and Applications at the University of Toronto and internationally acclaimed statistician, as well as a 2014 Fellow of the Royal Society of Canada. In 2015, she received the Order of Canada, was elected a foreign associate of the National Academy of Sciences in 2016 and has been awarded many other prestigious statistical and science honours, including the Committee of Presidents of Statistical Societies (COPSS) Award in 1992.

Nancy Reid’s research focuses on finding more accurate and efficient methods to deduce and conclude facts from complex data sets to ultimately help scientists find specific solutions to specific problems.

There is currently some renewed interest in developing distributions for parameters, often without relying on prior probability measures. Several approaches have been proposed and discussed in the literature and in a series of “Bayes, fiducial, and frequentist” workshops and meeting sessions. Confidence distributions, generalized fiducial inference, inferential models, belief functions, are some of the terms associated with these approaches.  I will survey some of this work, with particular emphasis on common elements and calibration properties. I will try to situate the discussion in the context of the current explosion of interest in big data and data science. 

on confidence distributions

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , on January 10, 2018 by xi'an

As Regina Liu gave her talk at ISI this morning on fusion learning and confidence distributions, this led me to think anew about this strange notion of confidence distributions, building a distribution on the parameter space without a prior to go with it, implicitly or explicitly, and vaguely differing from fiducial inference. (As an aside, the Wikipedia page on confidence distributions is rather heavily supporting the concept and was primarily written by someone from Rutgers, where the modern version was developed. [And as an aside inside the aside, Schweder and Hjort’s book is sitting in my office, waiting for me!])

Recall that a confidence distribution is a sample dependent distribution on the parameter space, which is uniform U(0,1) [in the sample] at the “true” value of the parameter. Used thereafter as a posterior distribution. (Again, almost always without a prior to go with it. Which is an incoherence from a probabilistic perspective. not mentioning the issue of operating without a pre-defined dominating measure. This measure issue is truly bothering me!) This seems to include fiducial distributions based on a pivot, unless I am confused. As noted in the review by Nadarajah et al. Moreover, the concept of creating a pseudo-posterior out of an existing (frequentist) confidence interval procedure to create a new (frequentist) procedure does not carry an additional validation per se, as it clearly depends on the choice of the initialising procedure. (Not even mentioning the lack of invariance and the intricacy of multidimensional extensions.)

fiducial inference

Posted in Books, Mountains, pictures, Running, Statistics, Travel with tags , , , , , , , , , , on October 30, 2017 by xi'an

In connection with my recent tale of the many ε’s, I received from Gunnar Taraldsen [from Tronheim, Norge] a paper [jointly written with Bo Lindqvist and just appeared on-line in JSPI] on conditional fiducial models.

“The role of the prior and the statistical model in Bayesian analysis is replaced by the use of the fiducial model x=R(θ,ε) in fiducial inference. The fiducial is obtained in this case without a prior distribution for the parameter.”

Reading this paper after addressing the X validated question made me understood better the fundamental wrongness of fiducial analysis! If I may herein object to Fisher himself… Indeed, when writing x=R(θ,ε), as the representation of the [observed] random variable x as a deterministic transform of a parameter θ and of an [unobserved] random factor ε, the two random variables x and ε are based on the same random preimage ω, i.e., x=x(ω) and ε=ε(ω). Observing x hence sets a massive constraint on the preimage ω and on the conditional distribution of ε=ε(ω). When the fiducial inference incorporates another level of randomness via an independent random variable ε’ and inverts x=R(θ,ε’) into θ=θ(x,ε’), assuming there is only one solution to the inversion, it modifies the nature of the underlying σ-algebra into something that is incompatible with the original model. Because of this sudden duplication of the random variates. While the inversion of this equation x=R(θ,ε’) gives an idea of the possible values of θ when ε varies according to its [prior] distribution, it does not account for the connection between x and ε. And does not turn the original parameter into a random variable with an implicit prior distribution.

As to conditional fiducial distributions, they are defined by inversion of x=R(θ,ε), under a certain constraint on θ, like C(θ)=0, which immediately raises a Pavlovian reaction in me, namely that since the curve C(θ)=0 has measure zero under the original fiducial distribution, how can this conditional solution be uniquely or at all defined. Or to avoid the Borel paradox mentioned in the paper. If I get the meaning of the authors in this section, the resulting fiducial distribution will actually depend on the choice of σ-algebra governing the projection.

“A further advantage of the fiducial approach in the case of a simple fiducial model is that independent samples are produced directly from independent sampling from [the fiducial distribution]. Bayesian simulations most often come as dependent samples from a Markov chain.”

This side argument in “favour” of the fiducial approach is most curious as it brings into the picture computational aspects that do not have any reason to be there. (The core of the paper is concerned with the unicity of the fiducial distribution in some univariate settings. Not with computational issues.)

all those ε’s…

Posted in Kids, pictures, Statistics, University life with tags , , , , , , on October 25, 2017 by xi'an

A revealing [and interesting] question on X validated about ε’s… The question was about the apparent contradiction in writing Normal random variates as the sum of their mean and of a random noise ε in the context of the bivariate Normal variate (x,y), since using the marginal x conditional decomposition led to two different sets of ε’s. Which did not seem to agree. I replied about these ε’s having to live in different σ-algebras, but this reminded me of some paradoxes found in fiducial analysis through this incautious manipulation of ε’s…

fiducial on a string

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , on June 26, 2017 by xi'an

A very short note in arXiv today by Gunnar Taraldsen and Bo Henry Lindqvist (NTU, Norway). With the above title. I find the note close to unreadable, I must say, as the notations are not all or well- defined. The problem starts from Teddy Seidenfeld [whom I met in Harvard around Dutch book arguments] arguing about the lack of unicity of fiducial distributions in a relatively simple setting. Actually the note is also inspired from Bayes, Fiducial and Frequentist, and comments from Teddy, a talk I apparently missed by taking a flight back home too early!

What I find surprising in this note is that the “fiducial on a string” is a conditional distribution on the parameter space restricted to a  curve, derived from the original fiducial distribution by a conditioning argument. Except that since the conditioning is on a set of measure zero, this conditional is not only not-unique, but it is completely undefined and arbitrary, since changing it does not modify the properties of the joint distribution.