## Fiducial nonparametrics

Sonderegger, Derek L.; Hannig, Jan

Fiducial theory for free-knot splines. Contemporary developments in statistical theory, 155–189,

Springer Proc. Math. Stat., 68, Springer, Cham, 2014.

62F12 (62F10 62F99 65D07)

The research reported reflects the recent surge in developments of Fisher’s fiducial argument [S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595]. The work of this chapter is carried out within the framework of generalized fiducial inference [J. Hannig, Statist. Sinica 19 (2009), no. 2, 491–544; MR2514173 (2010h:62071)], which is built on the functional-model formulation of fiducial statistics [A. P. Dawid, M. Stone and M. Stone, Ann. Statist. 10 (1982), no. 4, 1054–1074; MR0673643 (83m:62008)] rather than on the broadly equivalent confidence-based tradition beginning with [G. N. Wilkinson, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326 (58 #31491)] and generalized by [E. E. M. van Berkum, H. N. Linssen and D. Overdijk, J. Statist. Plann. Inference 49 (1996), no. 3, 305–317; MR1381161 (97k:62007)].

{For the entire collection see MR3149911.}

Reviewed by David R. Bickel

This review first appeared at “Fiducial theory for free-knot splines” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## The likelihood principle as a relation

Evans, Michael

What does the proof of Birnbaum’s theorem prove? (English summary)

Electron. J. Stat. 7 (2013), 2645–2655.

62A01 (62F99)

The author formalizes the theorem in terms of set theory to say that the likelihood relation is the equivalence relation generated by the union of the sufficiency relation and the conditionality relation. He finds the result trivial because it relies on extending the conditionality relation, itself intuitively appealing, to the equivalence relation it generates, which conflicts with usual frequentist reasoning and which may even be meaningless for statistical practice. This viewpoint is supported with a counterexample.

While some would regard the irrelevance of the theorem as repelling an attack on frequentist inference, emboldening the advancement of novel methods rooted in fiducial probability [R. Martin and C. Liu, Statist. Sci. 29 (2014), no. 2, 247–251; MR3264537; cf. J. Hannig, Statist. Sci. 29 (2014), no. 2, 254–258; MR3264539; S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595], the author criticizes the conditionality principle as formalized by the conditionality relation. The problem he sees is that the equivalence relation generated by the conditionality relation and needed for the applicability of the theorem “is essentially equivalent to saying that it doesn’t matter which maximal ancillary we condition on and it is unlikely that this is acceptable to most frequentist statisticians”.

The author concludes by challenging frequentists to resolve the problems arising from the plurality of maximal ancillary statistics in light of the “intuitive appeal” of the conditionality relation. From the perspective of O. E. Barndorff-Nielsen [Scand. J. Statist. 22(1995), no. 4, 513–522; MR1363227 (96k:62010)], that might be accomplished by developing methods for summarizing and weighing “diverse pieces of evidence”, with some of that diversity stemming from the lack of a unique maximal ancillary statistic for conditional inference.

Reviewed by David R. Bickel

**References**

- Barndorff-Nielsen, O. E. (1995) Diversity of evidence and Birnbaum’s theorem (with discussion). Scand. J. Statist., 22(4), 513–522. MR1363227 MR1363227 (96k:62010)
- Birnbaum, A. (1962) On the foundations of statistical inference (with discussion). J. Amer. Stat. Assoc., 57, 269–332. MR0138176 MR0138176 (25 #1623)
- Cox, D. R. and Hinkley, D. V. (1974) Theoretical Statistics. Chapman and Hall. MR0370837 MR0370837 (51 #7060)
- Durbin, J. (1970) On Birnbaum’s theorem on the relation between sufficiency, conditionality and likelihood. J. Amer. Stat. Assoc., 654, 395–398.
- Evans, M., Fraser, D. A. S. and Monette, G. (1986) On principles and arguments to likelihood (with discussion). Canad. J. of Statistics, 14, 3, 181–199. MR0859631 MR0859631 (87m:62017)
- Gandenberger, G. (2012) A new proof of the likelihood principle. To appear in the British Journal for the Philosophy of Science.
- Halmos, P. (1960) Naive Set Theory. Van Nostrand Reinhold Co. MR0114756 MR0114756 (22 #5575)
- Helland, I. S. (1995) Simple counterexamples against the conditionality principle. Amer. Statist., 49, 4, 351–356. MR1368487 MR1368487 (96h:62003)
- Holm, S. (1985) Implication and equivalence among statistical inference rules. In Contributions to Probability and Statistics in Honour of Gunnar Blom. Univ. Lund, Lund, 143–155. MR0795054 MR0795054 (86k:62002)
- Jang, G. H. (2011) The conditionality principle implies the sufficiency principle. Working paper.
- Kalbfleisch, J. D. (1975) Sufficiency and conditionality. Biometrika, 62, 251–259. MR0386075 MR0386075 (52 #6934)
- Mayo, D. (2010) An error in the argument from conditionality and sufficiency to the likelihood principle. In Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science (D. Mayo and A. Spanos eds.). Cambridge University Press, Cambridge, 305–314. MR2640508 MR2640508
- Robins, J. and Wasserman, L. (2000) Conditioning, likelihood, and coherence: A review of some foundational concepts. J. Amer. Stat. Assoc., 95, 452, 1340–1346. MR1825290 MR1825290

This review first appeared at “What does the proof of Birnbaum’s theorem prove?” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## Causality, Probability, and Time (by Kleinberg)—a review

Kleinberg, Samantha

Causality, probability, and time. Cambridge University Press, Cambridge, 2013. viii+259 pp. ISBN: 978-1-107-02648-3

60A99 (03A05 03B48 62A01 62P99 68T27 91G80 92C20)

Arguably an equally valuable contribution of the book is its integration of relevant work in philosophy, computer science, and statistics. While the first two disciplines have benefited from the productive interactions exemplified in [J. Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference, Morgan Kaufmann Ser. Represent. Reason., Morgan Kaufmann, San Mateo, CA, 1988; MR0965765 (90g:68003)] and [J. Williamson, Bayesian nets and causality, Oxford Univ. Press, Oxford, 2005; MR2120947 (2005k:68198)], the statistics community has developed its own theory of causal inference in relative isolation. Rather than following S. L. Morgan and C. Winship [Counterfactuals and causal inference: methods and principles for social research, Cambridge Univ. Press, New York, 2007] and others in bringing that theory into conversation with that of Pearl [op. cit.], the author creatively employs recent developments in statistical inference to identify causes.

For the specific situation in which many putative causes are tested but only a few are true causes, she explains how to estimate the local rate of discovering false causes. In this context, the local false discovery rate (LFDR) corresponding to a putative cause is a posterior probability that it is not a true cause. This is an example of an empirical Bayes method in that the prior distribution is estimated from the data rather than assigned.

Building on [P. Suppes, A probabilistic theory of causality, North-Holland, Amsterdam, 1970; MR0465774 (57 #5663)], the book emphasizes the importance for prediction not only of whether something is a cause but also of the strength of a cause. A cause is ε–significant if its causal strength, defined in terms of changing the probability of its effect, is at least ε, where ε is some nonnegative number. Otherwise, it is ε-insignificant.

The author poses an important problem and comes close to solving it, i.e., the problem of inferring whether a cause is ε-significant. The solution attempted in Section 4.2 confuses causal significance (ε-significance) with statistical significance (LFDR estimate below some small positive number α). This is by no means a fatal criticism of the approach since it can be remedied in principle by defining a false discovery as a discovery of an ε-insignificant cause. This tests the null hypothesis that the cause is ε-insignificant for a specified value of ε rather than the book’s null hypothesis, which in effect asserts that the cause is limε→0ε-insignificant, i.e., ε-insignificant for all ε>0. In the case of a specified value of ε, a cause should be considered ε-significant if the estimated LFDR is less than α, provided that the LFDR is defined in terms of the null hypothesis of ε-insignificance. The need to fill in the technical details and to answer more general questions arising from this distinction between causal significance and statistical significance opens up exciting opportunities for further research guided by insights from the literature on seeking substantive significance as well as statistical significance [see, e.g., M. A. van de Wiel and K. I. Kim, Biometrics 63 (2007), no. 3, 806–815; MR2395718].

Reviewed by David R. Bickel

This review first appeared at Causality, Probability, and Time (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## Multivariate mode estimation

Hsu, Chih-Yuan; Wu, Tiee-Jian

Efficient estimation of the mode of continuous multivariate data. (English summary)

Comput. Statist. Data Anal. 63 (2013), 148–159.

62F10 (62F12)

The authors cite several papers indicating the need for such multivariate mode estimation in applications. They illustrate the practical use of their estimator by applying it to climatology and handwriting data sets.

Simulations indicate a large variety of distributions and dependence structures under which the proposed estimator performs substantially better than its competitors. An exception is the case of contamination with data from a distribution that has a different mode than the mode that is the target of inference.

Reviewed by David R. Bickel

This review first appeared at “Efficient estimation of the mode of continuous multivariate data” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## Integrated likelihood in light of de Finetti

Coletti, Giulianella; Scozzafava, Romano; Vantaggi, Barbara

Integrated likelihood in a finitely additive setting. (English summary) Symbolic and quantitative approaches to reasoning with uncertainty, 554–565, Lecture Notes in Comput. Sci., 5590, Springer, Berlin, 2009.

62A01 (62A99)

Interpreting the likelihood function under the coherence framework of de Finetti, this paper mathematically formulates the problem by defining the likelihood of a simple or composite hypothesis as a subjective probability of the observed data conditional on the truth of the hypothesis. In the probability theory of this framework, conditional probabilities given a hypothesis or event of probability zero are well defined, even for finite parameter sets. That differs from the familiar probability measures that Kolmogorov introduced for frequency-type probabilities, each of which, in the finite case, can only have zero probability mass if its event cannot occur. (The latter but not the former agrees in spirit with Cournot’s principle that an event of infinitesimally small probability is physically impossible.) Thus, in the de Finetti framework, the likelihood function assigns a conditional probability to each simple hypothesis, whether or not its probability is zero.

When the parameter set is finite, every coherent conditional probability of a sample of discrete data given a composite hypothesis is a weighted arithmetic mean of the conditional probabilities of the simple hypotheses that together constitute the composite hypothesis. In other words, the coherence constraint requires that the likelihood of a composite hypothesis be a linear combination of the likelihoods of its constituent simple hypotheses. Important special cases include the maximum and the minimum of the likelihood over the parameter set. They are made possible in the non-Kolmogorov framework by assigning zero probability to all of the simple hypotheses except those of maximum or minimum likelihood.

The main result of the paper extends this result to infinite parameter sets. In general, the likelihood of a composite hypothesis is a mixture of the likelihoods of its component simple hypotheses.

{For the entire collection see MR2907743 (2012j:68012).}

Reviewed by David R. Bickel

This review first appeared at “Integrated likelihood in a finitely additive setting” (Mathematical Reviews) and is used with permission from the American Mathematical Society.