Corroboration, Explanation, Evolving Probability, Simplicity and a Sharpened Razor

My aim is to discuss (degrees of) corroboration, explanatory power, and complexity, and the relationships between them, and to suggest a new form of Ockham's razor which is applicable in particular to practical statistical problems. It will be found necessary to distinguish between two meanings of 'explanatory power' but nearly unique explicata will be suggested in terms of probability. These explicata will be derived from reasonable desiderata or axioms. Corroboration, explanatory power, and complexity can be expressed in terms of one another, and probability can be defined in terms of them, but I believe that the best practical method for estimating any of these things is to make judgments about all of them, and perhaps other things as well, and then to make deductions and modify the judgments in the light of inconsistencies that emerge. Among the conclusions are (a) (degree of) corroboration can be identified with 'weight of evidence' (in a sense not to be confused with the sense in which it was used, for example, by Popper, 1959), where the identification preserves all ordering relations (inequalities) between degrees of corroboration; (b) the explanatory power in the 'weak sense' of a hypothesis H, for explaining evidence E, namely the explanatory power that is unaffected by cluttering up H with irrelevancies, can be identified with the amount of mutual information between H and E; (c) the explanatory power in the strong sense, that is affected by the cluttering mentioned, can be obtained by subtracting from the weak explanatory power a term proportional to the information in H; (d) the complexity of a proposition can be identified with the amount of information in it, that is, minus the logarithm of its probability; (e) a sharpened form of 'Ockham's razor' is proposed which is that if our primary purpose is explanation we should select the hypothesis (among those we know) which has the maximum strong explanatory power; (f) this improvement of Ockham's razor can be applied to the estimation of statistical parameters. It will typically give rise to interval estimates, but it would give point estimates if weak explanatory power Received 27.xii.67

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