Abstract
In this chapter a well known class of inductive methods, i.e., GC-systems,and a family of probability distributions widely used in BS, i.e., Dirichlet distributions,will be considered.
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Notes
The terms “Generalized Carnapian systems” and “GC-systems” - together with many other terms, symbols and definitions used in this section - are borrowed from Kuipers (1978, 1980).
As Pietarinen (1972, pp. 60–61) points out, C-systems were anticipated in 1925 by W. E. Johnson in a paper published posthumously in 1932.
The requirement (RR) also implies that the predictive probabilities do not depend upon the variety of evidence a„. More precisely, (RR) implies that the special values p(Q,/e„)do not depend upon the number c of different kinds of individuals exemplified in a„, i.e., the number c of categories Q, such that n, > 0.
At the turn of the century several students - including Hardy (1889), Withworth (1897, pp. 224–225), Gini (1911) and Lidstone (1920) - suggested that Beta distributions might be the appropriate priors for the Bayesian analysis of binomial inferences: see Good, 1965, p. 17;
Zabell, 1982, p. 1092; and Costantini, 1978, pp. 313–314; 1984, p. 154. For a description of the main features of Beta distributions see Wilks (1962, pp. 173–175).
The uniform distribution for q1 is represented by the pdf “f(q 1 ) = 1“ which attributes equal density to all possible values of q1.
Lidstone (1920) was probably the first to suggest to use Dirichlet distribution in multinomial inferences (see Good, 1965, p. 25). For an exposition of the main features of Dirichlet distributions see Wilks (1962, pp. 177–182).
The uniform distribution on q is represented by the pdf “f(gh,qk1) = 1” which attributes equal density to all possible values of q.
For a long time the equivalence between GC-systems on the one hand and Beta and Dirichlet distributions on the other, apparently was not noticed. As far as I know this equivalence was pointed out first by Good (1965, p. 17): “It seems possible that G. F. Hardy [who defined Beta distributions] was the first to suggest a `continuum of inductive methods’, to use Camap’s phrase.” Later, the equivalence between GC-systems and Dirichlet (Beta) distributions was proved by Jamison (1970, pp. 47–49), Rosenkrantz (1977, pp. 71–73), and others.
The only difference is that p o (Q,/e„) = n,/n is not defined in the case where n = 0, whereas it follows from (34) thatp r .o(Qi/eo) =p 7 .o(Q,) = y.
Several arguments against the straight rule and (y,0) have been given (see Carnap, 1952, § 14; 1980, pp. 85–86 and 95; Good 1965, pp. 15–18, 23, 28 and 36; Kuipers, 1978, p. 52). Note, for instance, that the apparently plausible requirement of adequacy (Reg) (see Chapter 2.1) is not satisfied by extreme inductive methods (y,0) since they attribute prior probability zero to any `heterogeneous’ sequence a„, i.e., to any sequence containing more than one kind of trial.
In (38) the term “GC-systems” is interpreted to include the extreme GC-systems (y,0) and (y,00) with X-values)` = 0 and X = co. The class of `proper’ GC-systems, where 0 < < co, is obtained by adding (i) the axiom of regularity (Reg), which leads to the exclusion of X = 0, and (ii) the axiom of positive relevance (PR), which leads to the exclusion of X = co.
Johnson (1932, pp. 421–423) anticipated not only Camap’s C-systems (cf. note 2) but also their axiomatizations: see Good (1965, pp. 25–26) and Zabell (1982).
Indeed (RL) - as opposed to (RR), which becomes tautological when k = 2 - imposes a genuine restriction on the predictive probabilities p(Q,/e„) also when k = 2 (cf. Good, 1965, p. 26).
It follows from (39) that a unified axiomatization of GC-systems would consist in adopting the requirement of linearity (RL) for any k x 2. However, a serious defect of this axiomatization is that the intuitive meaning of (RL) is much less clear than that of the requirement of restricted relevance (RR) used in (38) (cf. Carnap, 1980, pp. 101–102).
Note that (CIRQ), if applied when k > 2, immediately excludes any analogy by similarity. However (CIRQ), as opposed to (RR), enacts a genuine restriction on the predictive probabilities p(Q 1 /e„) also when k = 2.
A possible criticism of Costantini’s axiomatization is that the new axiom (CIRQ) is less intuitively plausible than the `old’ axiom (RR).
Besides the objective probabilities governing a multivariate Bernoulli process, many other types of proportions are considered in the empirical sciences. In fact, proportions are of interest whenever “some unit is broken into parts, such as proportions (by weight) of various chemical constituents of a substance” (Connor and Mosimann, 1969, p. 194). For some examples concerning the biological sciences see Mosimann (1962, pp. 77–81) and Connor and Mosimann (1969, pp. 200–205).
Given the constraint Eq, = 1, the proportions q,,…,qk cannot be independent of each other in the usual sense (as defined in (3.4)); indeed “statisticians and biologists alike have been wary of the correlations existing among proportions or percentages since Pearson’s (1897) paper on spurious correlations” (Mosimann, 1962, p. 65). However, very often these correlations do not denote any `genuine’ correlation among the proportions but simply “serve as measures of the correlation due to the constraint [Eq, = 1]” (Connor and Mosimann, 1968, p. 78).
The concept of neutrality aims to individuate those cases where there is no genuine correlation among the proportions gl,…,gk. More precisely: when all the proportions qh are neutral w.r.t. q=(q,,…,qk), then the correlations existing between the different couples of proportions depend uniquely on the constraint Eq1 = 1.
Other `neutrality-based’ parametric requirements conceptually akin to (CM) have been used to formulate other axiomatizations of Dirichlet distributions (Darroch and Ratcliff, 1971; Doksum, 1974, pp. 187–188; and James and Mosimann, 1980).
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© 1993 Springer Science+Business Media Dordrecht
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Festa, R. (1993). GC-systems and Dirichlet Distributions. In: Optimum Inductive Methods. Synthese Library, vol 232. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8131-8_6
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