Exchangeability, independence and relevance quotient

A particular case of great importance is the relevance quotient at V, that is to say, the relevance quotient determined with respect to a void evidence. For Keynes's coefficent this is:

which we shall call the relevance quotient at V. As is shown by the alternative name Keynes used for the relevance quotient, he recognizes that this quotient is a way to deal with stochastic dependence. If we recall that exchangeability is a particular case of stochastic dependence, it is natural to imagine that exchangeability can be defined via the relevance quotient. With a proviso, this is actually the case. For the sake of simplicity, we shall consider only two cells, say H (head) and T (tail), and the relevance quotient at V. What we are saying can be immediately extended to the general case. In the case of two individuals, say X_{1} and X_{2} , we consider the probabilities of X_{t} = H = H_{t} , X_{t} = T = T_{i} . If this is the case two relevance quotients can be taken into account

Now let us suppose that the relevance quotient is symmetric, that is
For (9) and (10) we have

When P(H_{1} ) = x and thus P(T_{1} ) = 1 - x, and P(H_{2}) = y and thus P(T_{2} ) = 1 — y, it is easy to check that

Exchangeability implies P(T_{1} л H_{2} ) = P(H_{1} л T_{2} ), which holds true only if in (11), besides the condition of symmetry QH(V)=QH(V)=q, we have x = y, that is, equidistribution of X_{1} and X_{2}. Hence the symmetry of KH(V) is not sufficient for exchangeability.

Futher, we consider

then, by probability calculus,

that is

In only two cases does this equality hold: either q = 1 or q Ф 1 л y = x. The case q = 1, from (11), implies:

that is, X_{1} and X_{2} are stochastically independent. If this is the case, x and y, that is P(X_{1} = H) and P(X_{2} = T) can be chosen at pleasure. This means that P(X_{:} = T) Ф P(X_{2} = T) and P(X_{t} = H л X_{2} = T) Ф P(X_{t} = T л X_{2} = H ). In fact, as is well known, stochastic independence does not implies equidistribution.

In the second case q Ф 1 л y = x (where P(X_{1} = H) = P(X_{2} = H), that is equidistribution), the symmetry of the K-relevance quotient implies exchangeability, that is, P(X_{:} = T) = P(X_{2} = T ) and P(X_{:} = H л X_{2} = T ) = P(X_{1} = T л X_{2} = H ). Thus the symmetry of the K-relevance quotient, that is, (10) implies exchangeability only if equidistribution is added.

In this case, when the K-relevance quotient is equal to 1, then independence and equidistribution hold.

The relationship between exchangeability and symmetry is simpler and more powerful with regard to Q^{g}(•). In fact, from Q^{g}(D) = Qj_{g}(D) it follows immediately that P(X_{n+2} = j a X„_{+1} = g a D) = P(X_{n+1}= jD) P(X_{n+1} = gD) = P(X_{n+2} = g a X_{n+1} = i a D), that is, exchangebility holds true. It means that the symmetry of Carnap's Q^{g}(-) implies exchangeability and the symmetry of Keynes's K^{g} (•), the converse not being true. For this reason Carnap's coefficient has relegated Keynes's in the characterization of predictive inferences. In all the following we shall use Carnap's symmetric coefficient Q^{g} (•), but Keynes's followers can appreciate that its value is equal to Keynes's K^{g} (•).

Recall that Q^{g} (•) is an etero-relevance quotient; it is a measure of the variation of the probability of an attribute after a failure. If it is less (greater) than 1 it denotes, from a subjective or logical standpoint, a decrease (increase) in the degree of belief or confirmation after the failure. It follows that the probability of observing an attribute is an increasing (decreasing) function of the observed successes. If it is equal to 1, the observations are independent and equidistributed. Hence the relevance quotient is a companion of the correlation coefficient.