¹ The recent paper, “Economic Theory, Statistical Inference, and Economic History,” by John R. Meyer and Alfred H. Conrad (Journal of Economic History, Vol. XVII, No. 4 (Dec., 1957), pp. 524-544) discusses these same problems in a less formal manner and treats many preliminary questions on the role of measurement and of causation in history that are only touched on here. I am in entire agreement with this excellent paper, and its authors and I agree that the model here presented (which was conceived before I had read their paper) can be considered as a rigorous formalization of their major views.

² We use this term always to refer to information gained from outside the problem at hand, not in the technical sense of knowledge gained before experience.

³ See, for example, William Feller, Probability Theory and Its Applications, New York, John Wiley and Sons, 1950, Vol. I, p. 307 ff.

⁴ For our purposes, a tensor of rank k is an ordered set of tensors of rank k-1, all with the same dimensions. A scalar (here a real number) is defined as a tensor of rank 0. Thus a vector is an ordered set of scalars: it is a tensor of rank 1; a matrix is an ordered set of vectors, all having the same dimensionality: it is a tensor rank 2—a rectangular array of scalars; an ordered set of matrices with the same dimensions (a rectangular solid of scalars) is a tensor of rank 3, and so forth. (It happens that in the present instance, all dimensions are equal so that the Markov matrix—the multiple Markov tensor of rank 2—is a square array of scalars, the multiple Markov tensor of rank 3 is a cubical array, etc.) Multiplication and addition of these tensors are easy to define, but we shall not need to do so here.

We give the name “multiple Markov” to the tensor under discussion because it is a generalization of the simple Markov matrix in exactly the same way as the multiple Markov process is a generalization of the simple Markov process. See J. L. Doob, Stochastic Processes, New York, John Wiley and Sons, 1953, pp. 89-90.

⁵ The criterion for successful description here is essentially arbitrary and should be decided upon in advance. For example, two reasonably plausible rules are as follows: 1) Adopt some arbitrary standard of statistical significance or size of standard errors of forecast of variables; 2) Successful description has been achieved when adding another rank to the tensor does not improve prediction of states of nature beyond the initial time periods. The reason that all such rules must be arbitrary is that what one considers “success” depends on what things one considers it important to explain and on what level of inaccuracy of prediction one is willing to tolerate.

⁶ See Doob, loc. cit., for the technical statement of this.

⁷ Suppose that the values of some variables cannot be so ordered. Let that variable have k+1 (not necessarily a finite or denumerable number) distinct values. We then substitute for that variable k new variables, one for each of the values of the original variable save one, the ith new variable taking on the value 1 when the original variable takes on its ith value and being 0 otherwise. The values of each of these new variables are (trivially) ordinally ordered, and together they contain all information given by the old variable. (Of course, the reason for choosing one less new variable than the old variable has values is that we wish to keep an independent set. If all our new variables are 0, then the old variable must have the remaining value.)

For example, suppose that it were impossible to order the variable color along an ordinal scale (it isn't). We should then define a variable, red, to be one when color was red and zero otherwise. Similarly for other values of the color variable. All these new variables would be ordinally ordered, and a glance at their values would tell us the value of the original color variable which we can then discard.

Note that we do not assume a finite or even a denumerable number of variables.

⁸ This is not much of a restriction, since we can consider very small time intervals. The whole business could undoubtedly be developed for continuous time, but the results seem hardly to justify the extra complexity. Of course, the time interval chosen affects the view of the world that results.

⁹ For the definition of these terms for a simple Markov matrix (we should define them quite analogously for the multiple Markov tensor), see Feller, op. cit., pp. 320-321.

¹⁰ See, for example, St. George Stock, Stoicism, London, Archibald Constable, 1908, pp. 91-92.

¹¹ A perfect analogy to this property is given by Robert R. Bush, Frederick Mosteller, and Gerald Thompson, “A Formal Structure for Multiple-Choice Situations,” Chapter VIII in Decision Processes (R. M. Thrall, C. H. Coombs, and R. L. Davis, eds.), New York, John Wiley and Sons, 1954.

¹² One way of defining a convex set is to say that motion in any direction (not just parallel to one of the axes), if it takes a point out of the set, does not return it when carried further. Note that, whereas convexity is not invariant with respect to monotonie transformations of the axes, quasi-convexity is.

Professor Dorfman has pointed out, however, that quasi-convexity is not independent of the choice of axes (see the first paragraph of this section). A subset which is quasi-convex with one set of variables may not be so with an equivalent set. This is precisely the point made above, that the way in which the problem is conceived influences the approach and the results. The Bayesian subsets corresponding to a given problem for one set of categories may not be the same as those for another set.

¹³ Professor Meyer has pointed out that if we, as investigators, were really interested solely in a given Bayesian problem, there would be no reason to require quasi-convexity. The reason we do require it is that we are also interested in the relation of the results of one problem to the investigation of another—a question to which we shall return below. In the example given in the text, we should not care about distinguishing perfect competition from perfect monopoly were it not for the fact that we are interested in other problems besides that of transportation arrangements.

¹⁴ Remember that such conditional probability distributions will involve only ones and zeros, if one takes a determinist view of the original historical process. We have not yet omitted any variable which would form part of the error term in a regression equation.

¹⁵ Note that a set of negligible variables need not be a negligible set.

¹⁶ I am indebted to Professor Conrad for a slightly different form of this illustration.

¹⁷ Relevant, that is, in terms of the Bayesian reduction with the same definition of states of Nature at time t.

¹⁸ Of course, anyone is free to disagree, both here and later. One man's Weltanschauung is another man's poison.

¹⁹ If ten volumes can be called “manageable.” Again, I come to classify Toynbee, not to praise him.

²⁰ See Doob, Loc. cit., for the proof of this statement. Professor Dorfman has suggested that an example of violation of the assumption of this particular theorem would be the possibility of setting off two different world destroying bombs.

²¹ One way of looking at the recent researches of Professor Guy H. Orcutt (described in his “A New Type of Socio-Economic System,” Review of Economics and Statistics, vol. XXXIX, no. 2 (May, 1957), p. 116.) is as an unusually direct attack on this specification problem by the use of the stochastic properties of a highly simplified tensor (which can be multiple Markov) in “Monte Carlo” experiments with high speed computing equipment.

²² That interdependence has been described in a strikingly similar if not quite so rigorous way by Wassily Leontief, in his “Note on the Pluralistic Interpretation of History and the Problem of Interdisciplinary Cooperation,” Journal of Philosophy, vol. XLV, no. 23 (Nov. 4, 1948), pp. 617-623. I am indebted to W. Eric Gustafson for this reference.

²³ Define negligible in this context analogous to the definition of irrelevance in footnote 17.

²⁴ See Meyer and Conrad, op. cit., for an equivalent statement in terms of the error terms in regression equations.