Exchangeable probabilities

We shall work with exchangeable probabilities. As a consequence both quotients (3) and (5) have the same value, as we shall see. But to show this we must define the next condition we consider:

Keynes did not consider exchageability, but Johnson did. This author called C2 the combination-postulate (see Johnson, 1932). Exchangeability was studied by Frank Plumpton Ramsey in unpublished work from the late 1920s (see von Plato, 1994, p. 246n). This condition became famous when Bruno de Finetti draw from it his celebrated representation theorem (de Finetti, 1930). Carnap called C2 symmetry and based on it his theory of symmetrical c-functions. Exchangeability has many consequences.

The most important is de Finetti's theorem, but we shall not take it into account because we work with a finite number of individuals, while the representation theorem accounts for denumerable domains.

The first consequence of C2 we consider is the equality of (3) and (5). As we shall see in the next subsection, from C2 follows the equidistribu- tion of random variables, which means that the probability of belonging to a given cell is unaffected by the considered random variable or individual provided it does not belong to the evidence. Thus

and from this follows the equality of (3) and (5).

Another consequence of C2 is that it becomes superfluous to specify in (2) the individuals involved in the hypothesis and in the data. It is enough to consider attributes and statistical distributions. More exactly, we have

where 2d=1 n;=n. Hence n:= (n1 ,...,n;,...,nd) is the vector of the frequencies that the evidence indicates in the various cells; it is the frequency distribution of D. We call the frequency n the occupation number of the cell j, and n the occupation vector of D. As we have said, Keynes did not consider exchangeability, and so did not consider occupation vectors. In contrast, Carnap considered such vectors, calling them "structure descriptions" or "statistical distributions." (7) asserts that probability (2) is a function of a cell, j, and an occupation vector, n; that is, what matters is the cell considered in the hypothesis and the numbers of individuals in the various cells, not which ones they are. Then we can write for short:

(8) reminds us that we are considering an exchangeable probability. Referring to it, we shall speak of the probability j given n.

In the case in which C2 holds, the relevance quotient (3), beside j and g, depends upon n and no longer upon D. We shall recall this by writing Kg (n) and Qg (n) instead of (3) and (5). Further, if C2 and then (6) holds, Kg (n) = Qg (n).

In order to write this quotient using the terminology suggested by (8), it is worth introducing a new symbol for the occupation vector after the new observation, which we suppose relates to an individual belonging to the cell g, has been added to the data. If this is the case the evidence size becomes n + 1 and the occupation number of the cell gbecome ng + 1. We denote by ng = (n1 , ...,ng + 1, ...,nd) the occupation vector of this new evidence. Thus the relevance quotients, in the case of exchangeability, can now be written as

 
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