## Archive for epistemic probability

## baseless!

Posted in Books, Statistics with tags 1922, Bayes theorem, epistemic probability, frequency properties, George Boole, history of statistics, inverse probability, normalised maximum likelihood, Pierre Simon Laplace, R.A. Fisher, Siméon Poisson on July 13, 2021 by xi'an## false confidence, not fake news!

Posted in Books, Statistics with tags Bayes factors, confidence distribution, epistemic probability, Jeffreys-Lindley paradox, Proceedings of the Royal Society, Royal Society on May 28, 2021 by xi'an

“…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.”

**I**n 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, thatinappropriateness must be rooted in a mismatch between the mathematics of probability theoryand 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]

Posted in Books, Statistics, University life with tags ABC, amazon associates, Bayesian foundations, BibTeX, book review, confidence distribution, confidence intervals, epistemic probability, fiducial distribution, frequentist coverage, Neyman-Scott problem, Nobel Prize, Norway, prior free posterior, Quenouille, survey, whales on April 23, 2018 by xi'an**A**s 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!)

## SAS on Bayes

Posted in Books, Kids, pictures, R, Statistics, University life with tags asymptotics, Bayesian inference, credible intervals, cross validated, epistemic probability, plug-in resolution, SAS, Stack Exchange on November 8, 2016 by xi'an**F**ollowing a question on X Validated, I became aware of the following descriptions of the pros and cons of Bayesian analysis, as perceived by whoever (Tim Arnold?) wrote SAS/STAT(R) 9.2 User’s Guide, Second Edition. I replied more specifically on the point

It [Bayesian inference] provides inferences that are conditional on the data and are exact, without reliance on asymptotic approximation. Small sample inference proceeds in the same manner as if one had a large sample. Bayesian analysis also can estimate any functions of parameters directly, without using the “plug-in” method (a way to estimate functionals by plugging the estimated parameters in the functionals).

which I find utterly confusing and not particularly relevant. The other points in the list are more traditional, except for this one

It provides interpretable answers, such as “the true parameter θ has a probability of 0.95 of falling in a 95% credible interval.”

that I find somewhat unappealing in that the 95% probability has only relevance wrt to the resulting posterior, hence has no absolute (and definitely no frequentist) meaning. The criticisms of the prior selection

It does not tell you how to select a prior. There is no correct way to choose a prior. Bayesian inferences require skills to translate subjective prior beliefs into a mathematically formulated prior. If you do not proceed with caution, you can generate misleading results.

It can produce posterior distributions that are heavily influenced by the priors. From a practical point of view, it might sometimes be difficult to convince subject matter experts who do not agree with the validity of the chosen prior.

are traditional but nonetheless irksome. Once acknowledged there is no correct or true prior, it follows naturally that the resulting inference will depend on the choice of the prior and has to be understood conditional on the prior, which is why the credible interval has for instance an epistemic rather than frequentist interpretation. There is also little reason for trying to convince a fellow Bayesian statistician about one’s prior. Everything is conditional on the chosen prior and I see less and less why this should be an issue.