calculations enable extensions and predictions to be made, not with certainty, but, at least, with the doubt confined within measurable limits. The ability thus to project and extend with a measurable margin of error, thus to judge future probabilities from past experience, is fundamental in practical statistical work, and, it may be argued, is of basic importance in the ordinary conduct of life.1

The controversy over this process of deriving numerical probabilities from observed frequencies centers about the question as to whether empirical evidence alone is sufficient. The possibility of determining probabilities in this way has been denied by more than one logician. Among recent writers, J. M. Keynes has argued strongly against the process of determining probabilities from purely empirical evidence, and has questioned the validity of methods frequently employed in statistical investigations. Keynes' criticisms cannot but have a salutary effect on practical statistical work. The emphasis he places upon the logical foundations of statistical method may serve to awaken statisticians to the need of a thorough understanding of the assumptions on which they base their work. But he has gone farther, it would seem, in throwing empiricism overboard and in denying the validity of inverse probabilities than the practical statistician can go.

M. C. Rorty has said that all business and most engineering operations are based fundamentally on probabilities, and it may be added. that the probabilities which enter into these operations are derived primarily from experience. The wide-spread activities of life and fire insurance rest upon probabilities which are largely empirical. "The real importance of the theory of probability in regard to mass phenomena consists," says Tschuprow, "in determining the mathematical relations of the various probabilities not in a deductive but in an empirical manner-without an a priori exhaustive knowledge of the mutual relations and actions between cause and effect-by means of statistical enumeration of the frequency of the observed event. The conception of a probability finds its justification in the close relation between the mathematical probabilities and the relative frequencies as determined in a purely empirical way."2 Certainly

1 "The practical man is always working by comparative probabilities, even if he has not reduced his appreciations to numbers. He would certainly scoff at the idea that a first sample should not influence his judgment of what a second sample or what the bulk would be like. For such an idea would render most actions in life impossible." Karl Pearson, "The Fundamental Problem of Practical Statistics," Biometrika, xiii, 3.

2 Quoted by Arne Fisher, The Mathematical Theory of Probabilities, 82.

this is true as regards social and economic problems, in which fields the strict a priori method has practically no applications. Karl Pearson, while admitting that an ultimate logical demonstration of the approximate stability of statistical ratios may be impossible, contends that this principle of inverse probabilities "rests on the foundation of common sense and the experience of what follows its disregard." 1

Yet strict empiricism in the field of probabilities rests upon a rather dubious foundation. Without going as far as Keynes, the existence of an a priori element may be recognized as a base upon which empirical methods build. It may be called "common sense, but something apart from the mere objective evidence enters as a rational basis for empirical probabilities. In his Mathematical Theory of Probabilities Arne Fisher has clarified the issues in this old problem and, while recognizing the dominant part that empirical probabilities must play in the actual application of statistical methods, has admitted the necessity of a subjective element. Neither a strict dependence upon a priori probabilities nor a strict empiricism will serve as a basis for practical investigations. Probabilities may be derived from experience, though the subjective elements of reason and common sense must be ever present. "We do not need to limit our investigations to problems where we are able to determine a priori the probability for the happening of an event in a single trial, but limit ourselves to postulate the existence of such an a priori probability." By empirical methods an approximation to this postulated a priori probability may be secured, and for practical purposes such an approximation is all that is required.

We must, in all this, recognize the validity of Keynes' chief criticism, and grant that some degree of doubt must attach to probabilities based upon empirical evidence, that the statistician's "probable error," when used in an inductive sense, represents a spurious accuracy. Since, such a doubtful measure must be used, in default of better, it is incumbent upon the statistician to recognize the fictitious accuracy of this measure when applied inductively, and to make every possible effort to reduce the degree of doubt by strengthening the evidence, by increasing the amount of relevant knowledge. If he does not do this, if he trusts merely to his mathematical computations, failing to take the precautions which are essential to the validity of any induction, failing to recognize that the weight of

1 "On the Influence of Past Experience on Future Expectations." Phil. Mag., xiii (6th series), 1907.

2 Arne Fisher, loc. cit., 114.

statistical evidence alone is inadequate, the foundation of his argument will be of the veriest quicksand.

Granting the empirical justification and the practical necessity of statistical induction, the question arises as to how the doubt which attaches to the conclusions of this type of reasoning may be minimized. The probabilities derived from the application of Bayes' theorem represent the probable limits of error in generalizing from statistical results, on the assumption that certain conditions have been satisfied. The extent to which these conditions have been fulfilled can never be determined precisely, but in order that confidence may attach to statistical inductions every effort must be made to satisfy these conditions and to make as accurate as possible the measure of probability derived from the induction. The methods of thus validating statistical induction are in part familiar in every-day practice, but their importance cannot be over-emphasized, in view of the doubt which must adhere to the conclusions of any induction.

A first and commonly recognized requirement is that methods of random selection should have been employed in choosing the sample studied. With respect to the generalization in question, the members included in the sample should be random members of the larger group. How this may be attained need not concern us here.

A second method of increasing the reliability of the induction involves the testing by every possible means of the stability of the average, frequency ratio, coefficient of correlation, or equation of relationship. It is a virtue of Keynes' book that he has stressed the importance of such statistical tests of stability, as opposed to the mathematical method of computing probabilities from a single sample. Logically, this means verifying the assumption about natureabout the universe of facts from which the sample was drawnwhich is variously labelled the uniformity of nature, the stability of large numbers, or the regularity of averages. This stability is always assumed but is seldom verified by comprehensive statistical methods. By breaking up a sample into significant sub-groups, by including new cases classified according to some significant principle, and by testing the stability of the given measure as computed for these minor groups, the soundness of the original assumption may be determined and the whole inductive process validated. Keynes has pointed out that this testing of stability in the results, when the data are thus re-grouped on the basis of definite class-characteristics, corresponds to the process of improving the analogy which is fundamental in any induction. Technical methods of testing this stability

have been developed by Lexis. The necessity of such analysis has been generally recognized, but in the drawing of inductive conclusions from actual statistical results these tests are generally neglected. If the unmeasurable element of doubt which attaches to any induction is to be minimized, there must be present such evidence of stability under varying conditions. Such a test, if successful, proves the essential homogeneity of the sample with respect to the characteristic or association in question, and creates a presumption in favor of the homogeneity of the larger group to which the results are to be applied. A mere increase in the number of cases will not serve the same purpose, and will not strengthen the argument to the same extent.

The inductive process, as based upon statistical results, is thus subject to the same sort of limitations as the general process of induction. Statistical measures are better adapted to generalization, in a world marked by variation, than are the conclusions of universal inductions; they constitute a far more useful scientific tool, but they cannot be used without regard to these limitations. A statistical induction resembles a universal induction in that it can never carry complete confidence unless there be present some a priori element, something in addition to the mere statistical evidence. Actually, this is recognized in the formal terms of statistical procedure. The group must be "homogeneous," the members included in the sample must be chosen "at random." The statistical evidence alone will never furnish complete proof that these conditions have been realized; an a priori factor must be present to substantiate the statistical evidence. A recognition of the need of such confirmation and of the other limitations to statistical induction will not make the tools of statistics less useful; on the contrary, greater weight will attach to the well-tested and properly qualified conclusions of legitimate induction.2

1 Cf. Keynes, loc. cit., Chap. XXXII.

2 The above discussion has a bearing upon the relation between rational equations or laws on the one hand and empirical equations or laws on the other, a matter of some importance to the present argument. The former class consists of those mathematical expressions of relationship which may be deduced from known laws, while empirical equations are those having no theoretical justification, no known relation to accepted principles or laws. A sharp distinction is usually drawn between these two types. Little weight is given to empirical laws; rational equations are supposed to differ in some fundamental respect from the empirical type, and to apply in a rigorous and invariant manner. A true conception may perhaps be gained from the preceding discussion. The purely empirical law and the purely a priori law, if there be any

5. SUMMARY AND CONCLUSIONS

The foregoing arguments may be briefly summarized. Nature is so organized and human faculties are so circumscribed that complete knowledge of the historical or descriptive type is impossible. We cannot discriminate or handle scientifically the multitude of unique objects and events which make up the universe of facts in any given field. We must deal with phenomena in the aggregate, sacrificing detailed knowledge of individual things and events for the wider, more useful, though less precise knowledge of averages. Knowledge of single events, moreover, is of little practical use because of variation in nature, because individual phenomena, even of such, would stand as far apart as the poles. Actually, they shade off into each other by imperceptible degrees. As Edgeworth writes, formule which have not and those which have an a priori basis . . . are as opposite as night and day; yet as night insensibly passes into day, so it is impossible to fix the point at which the presumptions of deductive reasoning become effective." ("On the Representation of Statistics by Mathematical Formulae," Jour. of the Royal Statistical Society, lxii, 546.) To carry weight there must be some a priori element in any empirical law, and this element will be present in varied proportions in scientific generalizations.

As the deductive element increases, greater weight will ordinarily attach to generalizations, provided the empirical test of correspondence with the facts be met. The greater weight ordinarily attaching to the law with a rational or deductive basis does not necessarily arise from a closer agreement with the facts. "As regards the accuracy of representing the observations," says Steinmetz, "no material difference exists between a rational and an empirical equation. An empirical equation frequently represents the observations with great accuracy, while inversely a rational equation usually does not rigidly represent the observations, for the reason that in nature the conditions on which the rational law is based are rarely perfectly fulfilled." (Engineering Mathematics, 210.) That a law is rational, therefore, does not mean that it describes an invariant relationship. The stronger logical base, not a closer correspondence with the facts or a closer "fit" in the mathematical sense, accounts for the greater weight attaching to generalizations of this type and for the greater confidence with which curves based upon rational equations are projected.

In actually improving scientific generalizations we do not jump from the empirical to the rational, but pass from first approximations, which may be largely empirical and limited in scope, to more inclusive and perhaps simpler statements with a stronger deductive base, supported by wider knowledge and weightier a priori arguments. Our confidence in such generalizations must rest upon both empirical evidence and a priori reasoning. The conditions which determine our judgment of a scientific theory are summarized by Cournot in these words: "Le jugement que la raison porte sur la valeur intrinsèque de cette théorie est un jugement probable, dont la probabilité tient d'une part à la simplicité de la formule théorique, d'autre part au nombre des faits ou des groupes de faits qu'elle relié." (Quoted by Henry L. Moore, Laws of Wages, 10.)