Abstract
In this chapter some features of the contextual view of optimum inductive methods (optimum prior distributions) will be elucidated by comparing this view with others.
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Notes
Here Carnap would appear to imply that k-based inductive estimates of the success of inductive methods cannot be used to select the optimum inductive method. Carnap may have supposed, rightly, that any inductive method would immodestly evaluate itself as optimum.
This mistake was identified by Spielman (1972, p. 376) and immediately accepted by Lewis (1974, p. 84).
This is somewhat problematical: if the prior depends on the data, then the posterior is no longer proportional to the product of the prior and the likelihood: therefore Bayes’s theorem no longer holds. This observation was made by Prof. Molenaar.
This particular aspect of my approach to EPO appears to meet a desideratum expressed by Pietarinen (1974). Indeed, concerning Lewis’s hyperempiricist view, Pietarinen (1974, p. 197) remarks: “When the relative frequency of a kind [Q,] in a sample is [n,/n], the estimation formula [suggested by Lewis] recommends a certain method [X]. But whether or not the method is rational depends on the situation, in particular, on the question of [the degree of uniformity of the universe]. This cannot be decided on the basis of the frequencies in the sample alone; it is tempting to say that somehow it must be presupposed.… In Lewis’s procedure… the only relevant evidential information consists of the size and the degree of uniformity of the sample. But… to know these things is not enough to estimate the degree of uniformity of the universe. Ordinary inductive situations… may differ in respect of [the degree of uniformity], and this is relevant for the choice of the value of X.” According to Pietarinen (1974, p. 198) both Carnap’s aprioristic view and Lewis’s hyperempiricist view “are defective” since they are unable to take account of “the degree of uniformity of the things under study”. On the contrary, my contextual approach to EPO shows a possible way of considering the estimated degree of (dis)order of the universe under examination.
This procedure had already been described by Kuipers (1984) who analyzed the limit-process in the special case where X0 = 0.
Remember that the `extreme’ X-values are k = 0 and k = 00 (see Chapter 6.4). Kuipers (ibicl., pp. 40–41) also shows that, for some possible e„ and Xo, the result of the limit-process is an extreme X-value. Kuipers leaves unanswered the question whether or not this is a reason not to use certain values of X0 as possible starting values of the process.
There is a strong resemblance between my immodesty-based interpretation of Kuipers’s limit-process and Lewis’s description (1971, p. 62) of the behaviour of a person using a modest inductive method (cf. Chapter 4.2).
Remember that, for any X # co, pl,(Q;Q;/e„) > [pl,(Qi/e„)]2: cf. Chapter 3, note 16. It would be interesting to ascertain whether some limit-process of the kind described by Kuipers could also be generated by the X„-based Bayesian estimates E7,0(C(q) I e„). However, even if the nonBayesian estimate in (1) were replaced by E7,0(C(q) (e„), the whole procedure (1)-(2) would remain non-Bayesian. In fact, the usual Bayesian procedures allow calculation of the expected error - w.r.t. an inductive method X0 - of all inductive methods (including X0) relative to any evidence e„. Hence, a Bayesian who `starts’ with method X0 and evidence e„ would select that method X1 which minimizes the expected error - relative to X0 and e„ - as the most appropriate inductive method. But theorem (4.11) implies that any inductive method X0 immodestly indicates itself as being maximally appropriate. This means, in Kuipers’s terminology, that the `estimate X1’ suggested by the starting method a0 is… X0!
Kuipers’s second solution to EPO is formally similar to my V1-solution (see Chapter 8.5). However, there is a substantial difference between the two solutions. Whilst est(C(q) I e„) is a `formal’ e„-based estimate, my external estimate Est[C(q)] is an `informal’ estimate based on the background information BK.
A hyperempiricist view also appears to underlie Hintikka’s assumption (1987, p. 305) that a researcher “might initially choose a large Carnapian index of caution X. But if the overwhelming majority of observed individuals belongs to a small number of cells, [he] might very well be led to acknowledge that his caution was excessive and that he or she ought to have opted for a smaller k in the first place.”
Several proposals in line with the hyperempiricist view are advanced by Good (1965, pp. 28 and 35–38). They involve estimating the parameter a* of a symmetrical Dirichlet distribution Dir(a*,…,a*) on the basis of the empirical sample e„. For instance, if I understand Good (ib., p. 35) correctly, the value a* might be selected so as to maximize the prior predictive probability p(e„) derived using formula (3.48) from Dir(a*,…,a*).
Regarding the nature of the new probability system, Kuipers (ibid.) points out that “it is equally easy to see that the resulting system, for k 2 3, cannot have the property that p(Q1/e„) is only a function of n and n1, being a structural property of all X-systems”. It is not clear to me whether such a system is exchangeable although I guess it is not. Note that in Kuipers’s new system the special values p(Q1/e„) depend, besides on n and n;, only on the degree of order E(n1/n)2 of a„.
Rosenkrantz (1982, p. 93), referring to the family Q = (Q,non-Q), points out that even the extreme inductive methods X = 0 and k = oe can be adopted when certain `presuppositions’ about the degree of homogeneity of the considered population are made. Indeed he points out that “[X = 0] does seem appropriate for somebody who knows that the population is completely homogeneous, and… [X = cc] seems apt for one who knows that half the population is Q and half non-Q.”. (Cf. the proposal made in Chapter 8, note 10.)
For this purpose one could use some technical results obtained by Walk (1963, pp. 523528 and 1966, pp. 76–78) who showed that “the optimal value of)` is a monotonic function of the entropy in one’s universe” (Hintikka, 1987, p. 303).
This view appears to be shared by Horwich (1982, p. 81) who claims that “a reasonable inductive practice must be demonstrably reliable: there must be reason to think that it is good at directing us towards the truth.… So it can be reasonable to adopt one practice rather than another only if it can be shown that the one is more likely to be successful”.
Note that according to Friedman (ibid., p. 369), “the kind of probability at issue is objective or physical probability. We are requiring that there be a lawlike statistical correlation between the property of being reached by [a given inductive method] and the property of being true, in the very same sense in which there is a lawlike statistical correlation… between smoking and lung cancer.” A strong argument against Friedman’s formulation of the verisimilitude approach to scientific method is given by Niiniluoto ( 1980, pp. 453–454 ).
Here “representative” is not used in the standard statistical sense regarding the sampling procedure used for drawing the sample from the population. Rather the sample is ”representative of the entire universe“ if its composition is similar to that of the universe.
Rosenkrantz (1982, p. 93) points out that Carnap’s preference for small values of X can be justified only “by confining it to contexts where prior information about the population is lacking. It then becomes a special case of the principle favoring adoption of a diffuse prior in the absence of prior information.”
Rosenkrantz (1982, p. 93), indeed, points out that “the problem of induction… is to justify that preference.”
Regarding the cases where the available background knowledge BK is very poor, see Chapter 8, note 4.
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© 1993 Springer Science+Business Media Dordrecht
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Festa, R. (1993). The Contextual Approach to EPO: Comparisons with Other Views. In: Optimum Inductive Methods. Synthese Library, vol 232. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8131-8_9
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