confidence in confidence

[This is a ghost post that I wrote eons ago and which got lost in the meanwhile.]

Following the false confidence paper, Céline Cunen, Niels Hjort & Tore Schweder wrote a short paper in the same Proceedings A defending confidence distributions. And blame the phenomenon on Bayesian tools, which “might have unfortunate frequentist properties”. Which comes as no surprise since Tore Schweder and Nils Hjort wrote a book promoting confidence distributions for statistical inference.

“…there will never be any false confidence, and we can trust the obtained confidence! “

Their re-analysis of Balch et al (2019) is that using a flat prior on the location (of a satellite) leads to a non-central chi-square distribution as the posterior on the squared distance δ² (between two satellites). Which incidentally happens to be a case pointed out by Jeffreys (1939) against the use of the flat prior as δ² has a constant bias of d (the dimension of the space) plus the non-centrality parameter. And offers a neat contrast between the posterior, with non-central chi-squared cdf with two degrees of freedom

and the confidence “cumulative distribution”

Cunen et al (2020) argue that the frequentist properties of the confidence distribution 1-C(R), where R is the impact distance, are robust to an increasing σ when the true value is also R. Which does not seem to demonstrate much. A second illustration of B and C when the distance δ varies and both σ and |y|² are fixed is even more puzzling when the authors criticize the Bayesian credible interval for missing the “true” value of δ, as I find the statement meaningless for a fixed value of |y|²… Looking forward the third round!, i.e. a rebuttal by Balch et al (2019)

false confidence, not fake news!

“…aerospace researchers have recognized a counterintuitive phenomenon in satellite conjunction analysis, known as probability dilution. That is, as uncertainty in the satellite trajectories increases, the epistemic probability of collision eventually decreases. Since trajectory uncertainty is driven by errors in the tracking data, the seemingly absurd implication of probability dilution is that lower quality data reduce the risk of collision.”

In 2019, Balch, Martin, and Ferson published a false confidence theorem [false confidence, not false theorem!] in the Proceedings of the Royal [astatistical] Society, motivated by satellite conjunction (i.e., fatal encounter) analysis. But discussing in fine the very meaning of a confidence statement. And returning to the century old opposition between randomness and epistemic uncertainty, aleatory versus epistemic probabilities.

“…the counterintuitiveness of probability dilution calls this [use of epistemic probability] into question, especially considering [its] unsettled status in the statistics and uncertainty quantification communities.”

The practical aspect of the paper is unclear in that the opposition of aleatory versus epistemic probabilities does not really apply when the model connecting the observables with the position of the satellites is unknown. And replaced with a stylised parametric model. When ignoring this aspect of uncertainty, the debate is mostly moot.

“…the problem with probability dilution is not the mathematics (…) if (…)  inappropriate, that inappropriateness must be rooted in a mismatch between the mathematics of probability theory and the epistemic uncertainty to which they are applied in conjunction analysis.”

The probability dilution phenomenon as described in the paper is that, when (posterior) uncertainty increases, the posterior probability of collision eventually decreases, which makes sense since poor precision implies the observed distance is less trustworthy and the satellite could be anywhere. To conclude that increasing the prior or epistemic uncertainty makes the satellites safer from collision is thus fairly absurd as it only concerns the confidence in the statement that there will be a collision. But I agree with the conclusion that the statement of a low posterior probability is a misleading risk metric because, just like p-values, it is a.s. taken at face value. Bayes factors do relativise this statement [but are not mentioned in the paper]. But with the spectre of Lindley-Jeffreys paradox looming in the background.

The authors’ notion of false confidence is formally a highly probable [in the sample space] report of a high belief in a subset A of the parameter set when the true parameter does not belong to A. Which holds for all epistemic probabilities in the sense that there always exists such a set A. A theorem that I see as related to the fact that integrating an epistemic probability statement [conditional on the data x] wrt the true sampling distribution [itself conditional on the parameter θ] is not coherent from a probabilistic standpoint. The resolution of the paradox follows a principle set by Ryan Martin and Chuanhai Liu, such that “it is almost a tautology that a statistical approach satisfying this criterion will not suffer from the severe false confidence phenomenon”, although it sounds to me that this is a weak patch on a highly perforated tyre, the erroneous interpretation of probabilistic statements as frequentist ones.

look, look, confidence! [book review]

As it happens, I recently bought [with Amazon Associate earnings] a (used) copy of Confidence, Likelihood, Probability (Statistical Inference with Confidence Distributions), by Tore Schweder and Nils Hjort, to try to understand this confusing notion of confidence distributions. (And hence did not get the book from CUP or anyone else towards purposely writing a review. Or a ½-review like the one below.)

“Fisher squared the circle and obtained a posterior without a prior.” (p.419)

Now that I have gone through a few chapters, I am no less confused about the point of this notion. Which seems to rely on the availability of confidence intervals. Exact or asymptotic ones. The authors plainly recognise (p.61) that a confidence distribution is neither a posterior distribution nor a fiducial distribution, hence cutting off any possible Bayesian usage of the approach. Which seems right in that there is no coherence behind the construct, meaning for instance there is no joint distribution corresponding to the resulting marginals. Or even a specific dominating measure in the parameter space. (Always go looking for the dominating measure!) As usual with frequentist procedures, there is always a feeling of arbitrariness in the resolution, as for instance in the Neyman-Scott problem (p.112) where the profile likelihood and the deviance do not work, but considering directly the distribution of the (inconsistent) MLE of the variance “saves the day”, which sounds a bit like starting from the solution. Another statistical freak, the Fieller-Creasy problem (p.116) remains a freak in this context as it does not seem to allow for a confidence distribution. I also notice an ambivalence in the discourse of the authors of this book, namely that while they claim confidence distributions are both outside a probabilisation of the parameter and inside, “producing distributions for parameters of interest given the data (…) with fewer philosophical and interpretational obstacles” (p.428).

“Bias is particularly difficult to discuss for Bayesian methods, and seems not to be a worry for most Bayesian statisticians.” (p.10)

The discussions as to whether or not confidence distributions form a synthesis of Bayesianism and frequentism always fall short from being convincing, the choice of (or the dependence on) a prior distribution appearing to the authors as a failure of the former approach. Or unnecessarily complicated when there are nuisance parameters. Apparently missing on the (high) degree of subjectivity involved in creating the confidence procedures. Chapter 1 contains a section on “Why not go Bayesian?” that starts from Chris Sims‘ Nobel Lecture on the appeal of Bayesian methods and goes [softly] rampaging through each item. One point (3) is recurrent in many criticisms of B and I always wonder whether or not it is tongue-in-cheek-y… Namely the fact that parameters of a model are rarely if ever stochastic. This is a misrepresentation of the use of prior and posterior distributions [which are in fact] as summaries of information cum uncertainty. About a true fixed parameter. Refusing as does the book to endow posteriors with an epistemic meaning (except for “Bayesian of the Lindley breed” (p.419) is thus most curious. (The debate is repeating in the final(e) chapter as “why the world need not be Bayesian after all”.)

“To obtain frequentist unbiasedness, the Bayesian will have to choose her prior with unbiasedness in mind. Is she then a Bayesian?” (p.430)

A general puzzling feature of the book is that notions are not always immediately defined, but rather discussed and illustrated first. As for instance for the central notion of fiducial probability (Section 1.7, then Chapter 6), maybe because Fisher himself did not have a general principle to advance. The construction of a confidence distribution most often keeps a measure of mystery (and arbitrariness), outside the rather stylised setting of exponential families and sufficient (conditionally so) statistics. (Incidentally, our 2012 ABC survey is [kindly] quoted in relation with approximate sufficiency (p.180), while it does not sound particularly related to this part of the book. Now, is there an ABC version of confidence distributions? Or an ABC derivation?) This is not to imply that the book is uninteresting!, as I found reading it quite entertaining, with many humorous and tongue-in-cheek remarks, like “From Fraser (1961a) and until Fraser (2011), and hopefully even further” (p.92), and great datasets. (Including one entitled Pornoscope, which is about drosophilia mating.) And also datasets with lesser greatness, like the 3000 mink whales that were killed for Example 8.5, where the authors if not the whales “are saved by a large and informative dataset”… (Whaling is a recurrent [national?] theme throughout the book, along with sport statistics usually involving Norway!)

Miscellanea: The interest of the authors in the topic is credited to bowhead whales, more precisely to Adrian Raftery’s geometric merging (or melding) of two priors and to the resulting Borel paradox (xiii). Proposal that I remember Adrian presenting in Luminy, presumably in 1994. Or maybe in Aussois the year after. The book also repeats Don Fraser’s notion that the likelihood is a sufficient statistic, a point that still bothers me. (On the side, I realised while reading Confidence, &tc., that ABC cannot comply with the likelihood principle.) To end up on a French nitpicking note (!), Quenouille is typ(o)ed Quenoille in the main text, the references and the index. (Blame the .bib file!)

complex Cauchys

During a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as

f(x;θ¹,θ²) = |θ²| / |x-θ|²

when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,

C(aθ+b/cθ+d)

[when X is C(θ)] follows rather readily. But then comes the remark that

“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”

which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that  a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).

distributions for parameters [seminar]

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.