Two problems confronting the eclectic approach to statistics result from its lack of a unifying theoretical foundation. First, there is typically no continuity between a p-value reported as a level of evidence for a hypothesis in the absence of the information needed to estimate a relevant prior on one hand and an estimated posterior probability of a hypothesis reported in the presence of such information on the other hand. Second, the empirical Bayes methods recommended do not propagate the uncertainty due to estimating the prior.

The latter problem is addressed by applying a coherent form of fiducial inference to hierarchical models, yielding empirical Bayes set estimates that reflect uncertainty in estimating the prior. Plugging in the maximum likelihood estimator, while not propagating that uncertainty, provides continuity from single comparisons to large numbers of comparisons.

Filed under: empirical Bayes, fiducial inference, publications ]]>

10th Workshop on Information Theoretic Methods in Science and Engineering

Paris, France

September 11, 2017

David R. Bickel

University of Ottawa

Filed under: imprecise probability, maximum entropy, MDL, model checking, slides ]]>

the goal [of statistical inference in science] is not to infer highly probable claims (in the formal sense)* but claims which have been highly probed and have passed severe probes

Source: Deborah G. Mayo’s Performance or Probativeness? E.S. Pearson’s Statistical Philosophy | Error Statistics Philosophy

Filed under: Fragments ]]>

This is a list of possibly predatory publishers. The kernel for this list was extracted from the archive of Beall’s List at web.archive.org. It will be updated as new information or suggested edits are submitted or found by the maintainers of this site.

Source: List of Predatory Publishers | Stop Predatory Journals (accessed 24 August 2017)

Filed under: Fragments ]]>

Each student will work toward an MSc degree in the Mathematics and Statistics Program at the University of Ottawa. MSc students have the additional option of choosing a Bioinformatics or Biostatistics Specialization. Financial support is available.

Intellectual curiosity and high mathematical aptitude are essential, as is the ability to quickly code and debug computer programs. Strong self motivation and good communication skills are also absolutely necessary. The following qualities are desirable but not required: coursework in bioinformatics, computer science, numerical methods, numerical analysis, software engineering, statistics, and/or biology; familiarly with R, S-PLUS, Stan, JAGS, Mathematica, C, Fortran, and/or LaTeX; experience with UNIX or Linux.

Canadians (by citizenship or permanent residency) are especially encouraged to apply, as are all exceptional students. To be considered, send a PDF CV that has your GPA and contact information of two references to dbickel@uOttawa.ca with a cover letter in the body of the message. Please indicate in the subject line of the message your immigration status (“Canadian citizen,” “Canadian PR,” or “visa”) and, optionally, a specialization (“Bioinformatics” or “Biostatistics”). Only those selected for further consideration will receive a response.

Filed under: applications welcome ]]>

DRB:

Sometimes. A subjective Bayesian encountering completely unexpected data changes the prior:In the philosophy literature, that has been compared to changing the premises of a deductive argument. It has been argued that just as one may revise a premise without abandoning deductive logic as a tool, one may revise a prior without abandoning Bayesian updating as a tool.

(Exerpts from Deborah G. Mayo’s Can You Change Your Bayesian Prior? The one post whose comments (some of them) will appear in my new book | Error Statistics Philosophy)

Filed under: Fragments, model checking ]]>

“Correcting false discovery rates for their bias toward false positives”

Filed under: Fragments ]]>

Hannig, Jan; Iyer, Hari; Lai, Randy C. S.; Lee, Thomas C. M.

Generalized fiducial inference: a review and new results. (English summary)

*J. Amer. Statist. Assoc.* 111 (2016), no. 515, 1346–1361.

62A01 (62F99 62G05 62J05)

Generalized fiducial inference: a review and new results. (English summary)

62A01 (62F99 62G05 62J05)

This review article introduces generalized fiducial inference, the flavor of fiducial statistics developed by the authors and their collaborators since the beginning of the millennium. This research program has been driven by a vision of fiducial distributions as posterior distributions untainted by the subjectivity seen in prior distributions.

Other approaches to fiducial inference bring subjectivity more to the forefront. For example, G. N. Wilkinson had highlighted the incoherence of fiducial distributions formulated in a more Fisherian flavor [J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326]. More recently, R. J. Bowater [AStA Adv. Stat. Anal. 101 (2017), no. 2, 177–197] endorsed an explicitly subjective interpretation of fiducial probability. For the place of generalized fiducial inference in the context of other fiducial approaches, see [D. L. Sonderegger and J. Hannig, in Contemporary developments in statistical theory, 155–189, Springer Proc. Math. Stat., 68, Springer, Cham, 2014; MR3149921] and the papers it {MR3149921} cites.

Other approaches to fiducial inference bring subjectivity more to the forefront. For example, G. N. Wilkinson had highlighted the incoherence of fiducial distributions formulated in a more Fisherian flavor [J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326]. More recently, R. J. Bowater [AStA Adv. Stat. Anal. 101 (2017), no. 2, 177–197] endorsed an explicitly subjective interpretation of fiducial probability. For the place of generalized fiducial inference in the context of other fiducial approaches, see [D. L. Sonderegger and J. Hannig, in Contemporary developments in statistical theory, 155–189, Springer Proc. Math. Stat., 68, Springer, Cham, 2014; MR3149921] and the papers it {MR3149921} cites.

In addition to providing an inspiring exposition of generalized fiducial inference, the authors report these new contributions:

- A weak-limit definition of a generalized fiducial distribution.
- Sufficient conditions for a generalized fiducial distribution to have asymptotic frequentist coverage.
- Novel formulas for computing a generalized fiducial distribution and a fiducial probability of a model.

The fiducial probability of a model is applicable to both model selection and model averaging. A seemingly different fiducial method of averaging statistical models was independently proposed by D. R. Bickel [“A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models”, preprint, Fac. Sci. Math. Stat., Univ. Ottawa, 2015].

Reviewed by David R. Bickel

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This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

This review first appeared at “Generalized fiducial inference: a review and new results. (English summary)” (Mathematical Reviews) with the exception of the clarifying “{MR3149921}” and is used with permission from the American Mathematical Society.

Filed under: fiducial inference, reviews ]]>

Resonance—from probability to epistemology and back.

60A05 (00A30 03A10 62A01)

Look out for an offensive against all major philosophical theories of probability. The assailant draws from an arsenal of postmodern philosophy, a command of probability theory, and a deep reverence for natural science. Disarming readers with a conversational style peppered with Wikipedia references, he confronts them with the thesis that probability depends on resonance, the indescribable intuition and common sense behind human judgments in complex situations.

Burdzy defines probability in terms of six “laws of probability”, intended as an accurate description of how probability is used in science (pp. 8–9, 217). Unlike the axiomatic systems from Kolmogorov onward that are distinct from their potential applications [see A. Rényi, Rev. Inst. Internat. Statist 33 (1965), 1–14; MR0181483], the laws require that mathematical probability by definition agree with features of objective events. Potentially subject to scientific or philosophical refutation (pp. 258–259), the laws are analogous to Maxwell’s equations (p. 222). The testable claim is that they accurately describe science’s use of epistemic probabilities as well as physical probabilities (pp. 259–261).

Laws 3, 4, and 6 are especially physical. Burdzy argues that probability theory could not be applied if symmetries such as physical independence (Law 3) could not be recognized and tentatively accepted by resonance (Section 11.4). Such symmetries do not include the law of the iterated logarithm or many other properties of Martin-Löf sequences, which he finds “totally useless from the practical point of view” (Section 4.14). Law 4, the requirement that assigning equal probabilities should be based on known physical symmetries rather than on ignorance (Section 11.25), echoes R. Chuaqui Kettlun’s Truth, possibility and probability [North-Holland Math. Stud., 166, North-Holland, Amsterdam, 1991 (Sections III.2 and XX.3); MR1159708]. Law 6 needs some qualification or further explanation since it does not apply directly to continuous random variables: “An event has probability 0 if and only if it cannot occur. An event has probability 1 if and only if it must occur” (p. 217).

There is some dissonance in applications to statistics. On the frequentist side, a confidence interval with a high level of confidence should be used to predict that the parameter value lies within the observed confidence interval (Section 11.11, as explained by pp. 292, 294). Even though that generalizes predicting that the parameter values corresponding to rejected null hypotheses are not equal to the true parameter value, Burdzy expresses doubt about how to formalize hypothesis testing in terms of prediction (Section 13.4). His predictive-testing idea may be seen as an application of Cournot’s principle (pp. 22, 278; see [M. R. Fréchet, Les mathématiques et le concret, Presses Univ. France, Paris, 1955 (pp. 201–202, 209–213, 216–217, 221); MR0075110]). On the Bayesian side, Burdzy concedes that priors based on resonance often work well and yet judges them too susceptible to prejudice for scientific use (Section 14.4.3). By ridiculing subjective Bayesian theory as if it legitimized assigning probabilities at will (Section 7.1), Burdzy calls attention to its failure to specify all criteria for rational probability assignment.

Burdzy adds color to the text with random references to religion from the perspective of an atheistic probabilist who left Catholicism (p. 178). Here are some representative examples. First, in contrast to attempts to demonstrate that an objective probability of God’s existence is low [R. Dawkins, The God delusion, Bantam Press, 2006] or high [R. Swinburne, The resurrection of God incarnate, Clarendon Press, Oxford, 2003], he denies the feasibility of computing such a probability (Section 16.7). Second, Burdzy is convinced that religions, like communism, philosophical theories of probability, and other secular ideologies, have inconsistencies to the point of hypocrisy, insisting that his “resonance’ theory” (p. 13) is not an ideology (Chapter 15), much as D. V. Lindley denied that his Bayesianism is a religion [Understanding uncertainty, revised edition, Wiley Ser. Probab. Stat., Wiley, Hoboken, NJ, 2014 (pp. 380–381); MR3236718]. Lastly, Burdzy attributes the infinite consequences of underlying Pascal’s Wager to efforts to deceive and manipulate (Section 16.2.2). However, documenting the historical origins of teachings of eternal bliss and eternal retribution on the basis of primitive Christian and pre-Christian sources lies far beyond the scope of the book.

Under the resonance banner, this probabilist rushes in with a unique barrage of controversial and well-articulated philosophical claims with implications for science and beyond. Those resisting will find themselves challenged to counter with alternative solutions to the problems raised.

Burdzy defines probability in terms of six “laws of probability”, intended as an accurate description of how probability is used in science (pp. 8–9, 217). Unlike the axiomatic systems from Kolmogorov onward that are distinct from their potential applications [see A. Rényi, Rev. Inst. Internat. Statist 33 (1965), 1–14; MR0181483], the laws require that mathematical probability by definition agree with features of objective events. Potentially subject to scientific or philosophical refutation (pp. 258–259), the laws are analogous to Maxwell’s equations (p. 222). The testable claim is that they accurately describe science’s use of epistemic probabilities as well as physical probabilities (pp. 259–261).

Laws 3, 4, and 6 are especially physical. Burdzy argues that probability theory could not be applied if symmetries such as physical independence (Law 3) could not be recognized and tentatively accepted by resonance (Section 11.4). Such symmetries do not include the law of the iterated logarithm or many other properties of Martin-Löf sequences, which he finds “totally useless from the practical point of view” (Section 4.14). Law 4, the requirement that assigning equal probabilities should be based on known physical symmetries rather than on ignorance (Section 11.25), echoes R. Chuaqui Kettlun’s Truth, possibility and probability [North-Holland Math. Stud., 166, North-Holland, Amsterdam, 1991 (Sections III.2 and XX.3); MR1159708]. Law 6 needs some qualification or further explanation since it does not apply directly to continuous random variables: “An event has probability 0 if and only if it cannot occur. An event has probability 1 if and only if it must occur” (p. 217).

There is some dissonance in applications to statistics. On the frequentist side, a confidence interval with a high level of confidence should be used to predict that the parameter value lies within the observed confidence interval (Section 11.11, as explained by pp. 292, 294). Even though that generalizes predicting that the parameter values corresponding to rejected null hypotheses are not equal to the true parameter value, Burdzy expresses doubt about how to formalize hypothesis testing in terms of prediction (Section 13.4). His predictive-testing idea may be seen as an application of Cournot’s principle (pp. 22, 278; see [M. R. Fréchet, Les mathématiques et le concret, Presses Univ. France, Paris, 1955 (pp. 201–202, 209–213, 216–217, 221); MR0075110]). On the Bayesian side, Burdzy concedes that priors based on resonance often work well and yet judges them too susceptible to prejudice for scientific use (Section 14.4.3). By ridiculing subjective Bayesian theory as if it legitimized assigning probabilities at will (Section 7.1), Burdzy calls attention to its failure to specify all criteria for rational probability assignment.

Burdzy adds color to the text with random references to religion from the perspective of an atheistic probabilist who left Catholicism (p. 178). Here are some representative examples. First, in contrast to attempts to demonstrate that an objective probability of God’s existence is low [R. Dawkins, The God delusion, Bantam Press, 2006] or high [R. Swinburne, The resurrection of God incarnate, Clarendon Press, Oxford, 2003], he denies the feasibility of computing such a probability (Section 16.7). Second, Burdzy is convinced that religions, like communism, philosophical theories of probability, and other secular ideologies, have inconsistencies to the point of hypocrisy, insisting that his “resonance’ theory” (p. 13) is not an ideology (Chapter 15), much as D. V. Lindley denied that his Bayesianism is a religion [Understanding uncertainty, revised edition, Wiley Ser. Probab. Stat., Wiley, Hoboken, NJ, 2014 (pp. 380–381); MR3236718]. Lastly, Burdzy attributes the infinite consequences of underlying Pascal’s Wager to efforts to deceive and manipulate (Section 16.2.2). However, documenting the historical origins of teachings of eternal bliss and eternal retribution on the basis of primitive Christian and pre-Christian sources lies far beyond the scope of the book.

Under the resonance banner, this probabilist rushes in with a unique barrage of controversial and well-articulated philosophical claims with implications for science and beyond. Those resisting will find themselves challenged to counter with alternative solutions to the problems raised.

Reviewed by David R. Bickel

This review first appeared at “Resonance—from probability to epistemology and back” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

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