Abstract
The orthodox view in statistics has it that frequentism and Bayesianism are diametrically opposed—two totally incompatible takes on the problem of statistical inference. This paper argues to the contrary that the two approaches are complementary and need to mesh if probabilistic reasoning is to be carried out correctly.
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Notes
Often—especially when the models in \({\mathbb{M}}\) are indexed by a set of parameters—statisticians use the singular word ‘model’ to refer to the set \({\mathbb{M}}\) itself. In line with the logicians’ use of the term, in this paper the word ‘model’ will be reserved for a specific member of \({\mathbb{M}.}\)
It should be emphasised that such evidence may only be granted defeasibly. If a body of evidence leads to anomalous consequences, its more questionable elements will be withdrawn from the evidence base as they become open to criticism and are no longer taken for granted. See Williamson (2010b, §1.4.1) for further discussion of this notion of evidence.
Advocates of imprecise probability reject even the Probability norm, representing a belief function by a set of probability functions rather than a single probability function. While this sort of view is not normally classified as Bayesian, some versions of this view admit analogies with Bayesianism (see, e.g., Walley 1991).
Miller’s Principle is a similar principle of direct inference. Lewis (1980) put forward his Principal Principle in order to help elucidate the notion of physical probability for subjectivists, though he advocated an independent ‘best-system’ analysis of physical probability understood as single-case chance—see Sect. 7.2 on this point.
Here one should not necessarily think of loss in financial terms. One might suspect that there are times at which one doesn’t care about being financially prudent. For example, betting in a casino might be considered exciting but not financially prudent. In which case one might wonder whether the norms only hold in those cases in which one wishes to be prudent. But prudence is not to be identified with financial prudence: given that one wants excitement it can be prudent to go to a casino—financial losses are outweighed by a lack of excitement. Arguably it is a matter of fact that an ideal action is a prudent action, in the sense of an action that minimises worst-case expected loss, regardless of whether one cares about financial loss.
Depending one’s conception of physical probability, one might hesitate as to whether physical probabilities attach to the macroscopic events of this example. The reader should feel free to reinterpret the terms of this example so as to be comfortable that the relevant physical probabilities are all well defined.
Recall that, in the approach to Bayesian epistemology presented in Sect. 3.1, the agent’s evidence base includes everything she takes for granted in her current context of inquiry. This includes standard modelling assumptions such as the iid assumption. Such assumptions are retracted from her evidence base if they are no longer granted—e.g., if they are called into question by subsequent evidence.
This assumption will be qualified somewhat in Sect. 6.
Note that this inference is only appropriate in cases where \(I(\bar{X}_s,\tau _0)\subseteq [0,1].\) Other cases may warrant higher credence in the claim that \(P^{\ast}(L)\in I(\bar{X}_s,\tau _0);\) see Seidenfeld (1979, Chapter 2) and Mayo (1981, §2) on this point. Expressed in the framework of Sect. 3.2, if \(I(\bar{X}_s,\tau _0))\not\subseteq [0,1]\) then the single-case consequences \({\mathbb{P}^{\ast}_\mathcal{L}}\) of the physical probability information \({\mathbb{P}^{\ast}}\) do not just depend on the explicit information that \(P^{\ast}(P^{\ast}(L)\in I(\bar{X},\tau_{0})) \approx \tau_{0},\) but also on the further information that \(P^{\ast}(P^{\ast}(L)\in I(\bar{X},\tau_{0}))\in [0,1].\) In general, any application of the Calibration norm must respect the single-case consequences of the total evidence, not just of the information that \(P^{\ast}(P^{\ast}(L)\in I(\bar{X},\tau_{0})) \approx \tau_{0}.\) To put it another way, the after-trial evidence differs from the pre-trial evidence, and the fact that \(\bar{X}_s=.41\) may not only be pertinent with regard to the construction of the interval I(.41, τ0), but also in other regards (Hacking 1965, pp. 95–96).
Howson and Urbach are quite right, however, to emphasise that one must guard against substitution failure, as their rebuttal of Miller’s paradox does hinge on substitution failure (Howson and Urbach 1989, §15.e).
Jaynes (1976, §IIIa) maintains that Bayesian interval estimates with respect to a uniform prior are close to, but slightly narrower than, frequentist confidence intervals.
In this paper, the term ‘physical probability’, rather than the more common term ‘objective probability’, is used to refer to non-epistemic probability, in order to avoid confusion in the case of objective Bayesianism, which is objective in the sense that it admits little room for subjective choice, but which cannot be classified as objective in the non-epistemic sense.
Efforts have been directed at resolving this sort of problem in the area of machine learning—e.g., stemming from the ideas of Solomonoff (1964). However, these efforts have been primarily directed at the more restricted problem of balancing simplicity and fit, and even there, nothing approaching consensus has been reached.
The only other justification of single-case applications of confidence-interval methods seems to be Fisher’s fiducial argument; however, this seems to require a calibration principle (Hacking 1965, p. 137), so it is apparently a Bayesian justification. Since the fiducial argument is highly controversial, only applicable in specific situations and hard to apply even there, the more straightforward justification of Sect. 4 is preferred here; the exact relationship between the two justifications remains a question for further research. See Seidenfeld (1979, Chapters 4 and 5) and Haenni et al. (2011, Chapter 5) for further discussion of the fiducial argument.
Recall that it is assumed that \(I(\bar{X}_s,\tau _0)\subseteq [0,1]\) in order for the original inference to be legitimate. See footnote 9.
In statistics, research into predictive probability matching priors is also beginning to show interesting connections between Bayesian priors and frequentist confidence intervals in the non-parametric setting (Sweeting 2008).
References
Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101–117.
de Finetti, B. (1937). Foresight. its logical laws, its subjective sources. In H. E. Kyburg & H. E. Smokler (Eds.). Studies in subjective probability (pp. 53–118). Huntington, New York: Robert E. Krieger Publishing Company. Second (1980) edition.
de Finetti, B. (1970). Theory of probability. New York: Wiley.
Gillies, D. (2000). Philosophical theories of probability. London and New York: Routledge.
Grünwald, P., & Dawid, A. P. (2004). Game theory, maximum entropy, minimum discrepancy, and robust Bayesian decision theory. Annals of Statistics, 32(4), 1367–1433.
Hacking, I. (1965). Logic of statistical inference. Cambridge: Cambridge University Press.
Haenni, R., Romeijn, J.-W., Wheeler, G., & Williamson, J. (2011). Probabilistic logics and probabilistic networks. Synthese library. New York: Springer.
Hoefer, C. (2007). The third way on objective probability: a sceptic’s guide to objective chance. Mind, 116, 549–696.
Howson, C. (2001). The logic of Bayesian probability. In D. Corfield, & J. Williamson (Eds.). Foundations of Bayesianism (pp. 137–159). Dordrecht: Kluwer.
Howson, C., & Urbach, P. (1989). Scientific reasoning: The Bayesian approach. Chicago, IL: Open Court. Second (1993) edition.
Jaynes, E. T. (1976). Confidence intervals vs Bayesian intervals. In W. L. Harper & C. A. Hooker (Eds.). Foundations of probability theory, statistical inference, and statistical theories of science (Vol. 2, pp. 175–257). Dordrecht: D. Reidel.
Kolmogorov, A. N. (1933). The foundations of the theory of probability. New York: Chelsea Publishing Company (1950).
Kyburg, H. E. Jr., & Teng, C. M. (2001). Uncertain inference. Cambridge: Cambridge University Press.
Lewis, D. K. (1980). A subjectivist’s guide to objective chance. In Philosophical papers (Vol. 2, pp. 83–132). Oxford: Oxford University Press (1986).
Lewis, D. K. (1994). Humean supervenience debugged. Mind, 412, 471–490.
Lindgren, B. W., McElrath, G. W., & Berry, D. A. (1957). Introduction to probability and statistics. New York: Macmillan. 1978 Edition.
Mayo, D. G. (1981). In defense of the Neyman-Pearson theory of confidence intervals. Philosophy of Science, 48, 269–280.
Mayo, D. G. (1996). Error and the growth of experimental knowledge. Chicago: University of Chicago Press.
Miller, D. (1966). A paradox of information. British Journal for the Philosophy of Science, 17, 59–61.
Neyman, J. (1955). The problem of inductive inference. Communications on Pure and Applied Mathematics, 8, 13–46.
Ramsey, F. P. (1926). Truth and probability. In H. E. Kyburg & H. E. Smokler (Eds.). Studies in subjective probability (pp. 23–52). Huntington, New York: Robert E. Krieger Publishing Company. Second (1980) edition.
Ramsey, F. P. (1928). Reasonable degree of belief. In D. H. Mellor (Ed.). Philosophical papers. Cambridge: Cambridge University Press. 1990 Edition.
Seidenfeld, T. (1979). Philosophical problems of statistical inference: learning from R. A. Fisher. Dordrecht: Reidel.
Solomonoff, R. (1964). A formal theory of inductive inference. Information and Control, 7(1, 2), 1–22, 224–254.
Sweeting, T. J. (2008). On predictive probability matching priors. In B. Clarke & S. Ghosal (Eds.). Pushing the limits of contemporary statistics: Contributions in honor of Jayanta K. Ghosh (pp. 46–59). Beachwood: Institute of Mathematical Statistics.
von Mises, R. (1928). Probability, statistics and truth. London: Allen and Unwin. Second (1957) edition.
Walley, P. (1991). Statistical reasoning with imprecise probabilities. London: Chapman and Hall.
Wheeler, G., & Williamson, J. (2011). Evidential probability and objective Bayesian epistemology. In P. S. Bandyopadhyay & M. Forster (Eds.). Philosophy of statistics, handbook of the philosophy of science (pp 307–331). Amsterdam: Elsevier.
Williamson, J. (2010a). Bruno de Finetti: Philosophical lectures on probability. Philosophia Mathematica, 18(1):130–135.
Williamson, J. (2010b). In defence of objective Bayesianism. Oxford: Oxford University Press.
Williamson, J. (2011a). An objective Bayesian account of confirmation. In D. Dieks, W. J. Gonzalez, S. Hartmann, T. Uebel, & M. Weber (Eds.). Explanation, prediction, and confirmation. New trends and old ones reconsidered (pp 53–81). Dordrecht: Springer.
Williamson, J. (2011b). Objective Bayesianism, Bayesian conditionalisation and voluntarism. Synthese, 178, 67–85.
Acknowledgments
I am very grateful to the British Academy for supporting this research and to David Corfield, Jan-Willem Romeijn, Jan Sprenger and Gregory Wheeler for helpful comments.
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Williamson, J. Why Frequentists and Bayesians Need Each Other. Erkenn 78, 293–318 (2013). https://doi.org/10.1007/s10670-011-9317-8
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DOI: https://doi.org/10.1007/s10670-011-9317-8