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A discipline for the avoidance of unnecessary assumptions 1

Published online by Cambridge University Press:  29 August 2014

Lewis H. Roberts
Lewis H. Roberts*
Affiliation:
New York
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Although unnecessary assumptions are something we all try to avoid, advice on how to do so is much harder to come by than admonition. The most widely quoted dictum on the subject, often referred to by writers on philosophy as “Ockham's razor” and attributed generally to William of Ockham, states “Entia non sunt multiplicanda praeter necessitatem”. (Entities are not to be multiplied without necessity.) As pointed out in reference [I], however, the authenticity of this attribution is questionable.

The same reference mentions Newton's essentially similar statement in his Principia Mathematica of 1726. Hume [3] is credited by Tribus [2c] with pointing out in 1740 that the problem of statistical inference is to find an assignment of probabilities that “uses the available information and leaves the mind unbiased with respect to what is not known.” The difficulty is that often our data are incomplete and we do not know how to create an intelligible interpretation without filling in some gaps. Assumptions, like sin, are much more easily condemned than avoided.

In the author's opinion, important results have been achieved in recent years toward solving the problem of how best to utilize data that might heretofore have been regarded as inadequate. The approach taken and the relevance of this work to certain actuarial problems will now be discussed.

Bias and Prejudice

One type of unnecessary assumption lies in the supposition that a given estimator is unbiased when in fact it has a bias. We need not discuss this aspect of our subject at length here since what we might consider the scalar case of the general problem is well covered in textbooks and papers on sampling theory. Suffice it to say that an estimator is said to be biased if its expected value differs by an incalculable degree from the quantity being estimated. Such differences can arise either through faulty procedures of data collection or through use of biased mathematical formulas. It should be realized that biased formulas and procedures are not necessarily improper when their variance, when added to the bias, is sufficiently small as to yield a mean square error lower than the variance of an alternative, unbiased estimator.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 5 , Issue 3 , February 1971 , pp. 374 - 387
Copyright
Copyright © International Actuarial Association 1971

Footnotes

1

Originally presented at the seminar on Mathematical Theory of Risk and allied topics, auspices of the Committee on Mathematical Theory of Risk, Casualty Actuarial Society, November 16, 1966.

References

[1] The authenticity of such attribution is questionable, as observed by Brampton, C. Kenneth, editor of the volume, The De Imperatum et Pontificum Potestate of William of Ockham, (University Press, Oxford, 1927) who states in a note on page 80: “By a curious fate Ockham is in many quarters known solely for his ‘razor’, which Mr. W. M. Thorburn ably proves (Mind, no. 107, July 1918) to be an invention of a later age, occurring first in the works of Condillac less than two centuries ago, and introduced into England by Sir William Hamilton in 1852. But Ockham's meaning is clear enough, that if there is no ‘humanity’ existing apart from the individuals which collectively form it, it is gratuitous to postulate its objective existence (Log. i, cap. lxvi): ‘frustra fit per plura quod potest fieri per pauciora’ (Sent, ii, Dist. 15, o). These words as Mr. Thorburn points out, are actually quoted by Sir Isaac Newton in his third edition of his Principia Mathematica of 1726 (De Mundi Systemate, lib. hi, p. 387). This is Regula i, and continues, ‘Natura enim simplex est et rerum causis superfluis non luxuriat’: but the garbled version in the form, ‘entia non sunt multiplicanda praeter necessitatem’ was invented by John Ponce of Cork in 1639 and took its present shape for the first time in the Logica Vetus et Nova of John Clauberg of Groningen in 1654. Even in his philosophy there is much tha t is untrue in the name, weapon, and formula bestowed upon Ockham by posterity.” The Encyclopaedia Brittanica, however, says that “The famous dictum, ‘pluralites non est ponenda sine necessitae’ (multiplicity ought not to be posited without necessity) has become known as ‘Ockham‘s razor’ though it had already been stressed by other Scholastics,” without commenting upon the variation in wording nor challenging the attribution to Ockham. In the following paragraph it says”… Ockham did not make much of the philosophical arguments of earlier theologians, and applied to theology his famous ‘razor’…” This author relinquishes the task of any further research into the authenticity of ‘Ockham‘s razor’ to qualified medievalists.Google Scholar
[2]Myron, Tribus (a) “The Probability Foundations of Thermodynamics”, Tribus, Myron and Evans, Robert B., Applied Mechanics Review, Vol. 16, No. 10, October 1963.Google Scholar
Myron, Tribus (b) “Why Thermodynamics Is a Logical Consequence of Information Theory”, Tribus, Myron, Shannon, Paul T. and Evans, Robert B., A.I.Ch.E. Journal, March 1966.Google Scholar
Myron, Tribus (c) “Information Theory as the Basis for Thermostatics and Thermodynamics”, Tribus, Myron , Journal of Applied Mechanics, March 1961.Google Scholar
Myron, Tribus (d) “The Maximum Entropy Estimate in Reliability” in Recent Developments in Information and Decision Processes. Macmillan Co. 1962.Google Scholar
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[3]Hume, David, “A Treatise of Human Nature”, 1740. A more pertinent reference, in this author's opinion, is provided in Volume 4 of Hume's Philosophical Works, Edition of 1777. This edition was “corrected by the author for the press, a short time before his death, and which he desired might be regarded as containing his philosophical principles”, according to the “Advertisement” prefacing Volume 1 of the 1854 reprint, published by Little, Brown and Co. of Boston and by Adam and Charles Black of Edinburgh, of the 1777 edition. Most to the point, perhaps, is Hume's rhetorical question (page 35) “All these suppositions are consistent and conceivable. Why then should we give the preference to one, which is no more consistent or conceivable than the rest?” In what follows he argues that past experience is our only guide where no a priori connection can be demonstrated between cause and effect. This author agrees that Hume's discussion of inductive principles is consistent with Tribus's formulation but thinks it may be reading too much into Hume's rather prolix text to find there so clear a statement of the problem as given by Tribus.Google Scholar
[4]Shannon, C. E., “A Mathematical Theory of Communication”, Bell System Technical Journal, Vol. 27, 379, 623. 1948.CrossRefGoogle Scholar
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[6]Bailey, R. A., “Any Room Left for Skimming the Cream”, P.C.A.S. XLVII, 1960.Google Scholar
[7]Kendall, , Maurice, The Advanced Theory of Statistics, Vol. 1, p . 302, Charles Griffin & Sons, Ltd. 1948 *).Google Scholar
[8]Feller, William, An Introduction to Probability Theory and Its Applications, Vol. 1, p. 250, John Wiley & Sons, 1950.Google Scholar
[9]Besides linear programming, possible directions such calculations might take are suggested in Nonlinear Mathematics by Saaty, Thomas L. and Brom, Joseph. McGraw Hill Book Co. 1964.Google Scholar