Abstract
One of the most popular theories of probability and statistics is the “frequency theory” originating in the eighteenth century. This data-driven view is developed here through a chief proponent, Richard von Mises. Von Mises is concerned in a major way that laws of large numbers and the central limit theorem follow from any account of probability and statistics. He defines a “collective” containing unrelated or fairly inconsequential data. He thus does not capture trends, correlations, perturbations, and a host of phenomena from the realm of statistics. Frequency theory also ignores extreme values or outliers. The chapter ultimately concludes that the theory assumes a convergence at infinity that will never be experienced. However, frequency theory vies with the Bayesian theory in current-day popularity because of claimed successes.
This book is about a new, fourth paradigm for science based on data-intensive computing. In such scientific research, we are at a stage of development that is analogous to when the printing press was invented (p. xiii in Bell, Gordon, 2009 , “Foreword,” pp. xiii–xvii in Hey, Tony, Stewart Tansley, and Kristin Tolle, ed, The Fourth Paradigm: Data-Intensive Scientific Discovery, Redmond, Washington: Microsoft Research ).
Learning to use a “computer” of this scale may be challenging. But the opportunity is great: The new availability of huge amounts of data, along with the statistical tools to crunch these numbers, offers a whole new way of understanding the world. Correlation supersedes causation, and science can advance even without coherent models, unified theories, or really any mechanistic explanation at all. There’s no reason to cling to our old ways. It’s time to ask: What can science learn from Google? (These deliberately provocative words are from Chris Anderson, 2008 , “The End of Theory: The Data Deluge Makes the Scientific Method Obsolete,” Wired, 6/23/08.)
As Bellerophon’s fame grew, so did his hubris. Bellerophon felt that because of his victory over the Chimera he deserved to fly to Mount Olympus, the realm of the gods. However, this presumption angered Zeus and he sent a gad-fly to sting the horse causing Bellerophon to fall all the way back to Earth (From “Bellerophon,” Wikipedia, the free encyclopedia, accessed 10/17/2013).
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- 1.
A very helpful approach to efficiencies is found in Chap. 23 of Efron, Bradley, and Robert J. Tibshirani, 1993, An Introduction to the Bootstrap, New York: Chapman & Hall. In general, the topic is one of variance reduction techniques, and control functions provide potentially very great simulation efficiency gains for light-tailed distributions. One need not combine these with the very useful bootstrap modeling to produce, for instance, confidence intervals that can account for mean [fractile] estimates as well as say 5th and 95th centile estimates.
- 2.
On pages vii, 156, 158, 159, and 163 in Probability, Statistics and Truth, New York: Dover Publications, Inc., 1957, Richard von Mises opposes small sample theory including Bayesian theory and the use of case studies.
- 3.
- 4.
From von Mises, 1957, ibid., pp. 8, 30, 53, 54, 100, and 166.
- 5.
From von. Mises, 1957, ibid., pp. 80, 113, and 125.
- 6.
From von Mises, 1957, Ibid., pp. 11, 18, 102, and135.
- 7.
From von Mises, 1957, Ibid., pp. 14, 33, and 127.
- 8.
From pp. 131–132 in Gannon, Dennis and Dan Reed, 2009, “Parallelism and the Cloud,” pp. 131–135 in Hey, Tony et al., op. cit.
- 9.
From Van De Sompel, Herbert and Carl Lagoze, 2009, “All Aboard: Toward a Machine-Friendly Scholarly Communication System,” pp. 193–199 in Hey, Tony et al., op cit.
- 10.
From Von Mises, 1957, op cit., pp. 53, 54, 67, and 80.
- 11.
See Von Mises, 1957, op. cit., pp. 141–145.
- 12.
- 13.
From Von Mises, 1957, Ibid., pp. 24, 25.
- 14.
- 15.
- 16.
Elaborating on this idea, on p. 6 Applied Chaos Theory: A Paradigm for Complexity, San Diego CA: Academic Press. 1993, A. B Cambel states
Because a set of numbers appears to be random does not mean that it is. Statistical analyses must be undertaken to establish the nature of the data. For example, …in card games a deck must be shuffled seven times before the odds that a card may be in any position are the same. In turn, two decks must be shuffled nine times…
Further elaborating on this idea on p. 18, in Randomness, Cambridge, Massachusetts: Harvard University Press. 1998, Deborah J. Bennett maintains that
Von Mises defined randomness in a sequence of observations in terms of the inability to devise a system to predict where in a sequence a particular observation will occur without prior knowledge of the sequence….Yet certainly every sequence conforms to some rule—we may simply not know what the rule is ahead of time….In a 1963 paper Andrei Kolmogorov was able to show that if only simple formulas, rules, or laws of production are allowed, then von Mises-type sequences would exist….[For Kolmogorov] a random sequence is one with maximal complexity…if the shortest formula which computes it is extremely long. …The problem is, How do we ever know if we have found the shortest formula? … Common to all of these views is the unpredictability of future events based on past events.
The reader who wishes to learn about random coin tossing should read [quoted in J. Ford, 1983, “How Random Is a Coin Toss?” Physics Today, April, pp. 40–47; note that the first known six-sided dice are dated as being from the East in 2750 B.C. See also von Mises, 1957, op. cit., pp. 69, 74, and 85 on how the manufacturer, toss and eventual wear and tear of the die will impact the results.
- 17.
- 18.
From Von Mises, 1957, Ibid., pp. 109–113, 116, 134. See pp. 163 and 199 in Carnap, Rudolf, 1962, Logical Foundations of Probability, Chicago: University of Chicago Press. It turns out that Carnap would agree: one can safely speak about the estimate and then and only then the ultimate outcome relative to the estimate.
- 19.
From Von Mises, 1957, ibid., pp. 80, 104, 105, and 108.
- 20.
Von Mises recognizes how small changes may accumulate to produce large effects, as when, in chaos theory, small changes in initial conditions have dramatic effects. 1957, Ibid., p. 180 and 182.
- 21.
From Von Mises, 1957, Ibid., p. 127.
- 22.
- 23.
From p. 83 in Bernstein, Peter L., 1996, Against the Gods: The Remarkable Story of Risk, New York: John Wiley & Sons, Inc.
- 24.
From Keynes, 1921, op. cit., p. 103.
- 25.
From Keynes, 1921, op. cit., p. 335.
- 26.
In 1962, on p.188 of The Silent Spring, Boston: Houghton Mifflin Company, Rachel Carson maintained that “it is simply impossible to predict the effects of lifetime exposure to chemical and physical agents that are part of the biological experience of man.”
- 27.
As a result of all the practical activities that those engaged in data mining participate in, they do not need to worry about having the fate of Bellerophon, who, after having been a hero who killed the Chimera, tried to fly to Mt. Olympus on Pegasus, was thrown off Pegasus by Zeus, and “wandered alone about the plain of Aleios, eating his heart out, skulking aside from the trodden track of humanity.” (From The Iliad, book VI, translated by Richard Lattimore, Chicago: Phoenix Books, University of Chicago Press, 1951).
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Taylor, C.E. (2015). The Frequency Theory of Probability. In: Robust Simulation for Mega-Risks. Springer, Cham. https://doi.org/10.1007/978-3-319-19413-4_3
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