Prior and Posterior Probabilities and Semantic Information

Abstract

According to logicians, high ‘semantic information’ must be desired by a scientist. It is associated with ‘surprise’. Indeed, if scientist is defined as one for whom all errors are equally undesirable and all correct statements are equally desirable, then it does follow that in the case of mutually exclusive propositions:

• (A) the scientist (though not a decision-maker in general) receiving perfect evidence gains least when it is a priori most probable (and thus ‘least surprising’): and

• (B) he gains on the average most from that imperfect evidence which assigns to some proposition a higher probability a posteriori than that assigned to any proposition by any other such evidence.

Of these results, (A) is probably, and (B) possibly, an adequate rendering, in terms of decision theory, of the logicians’ discussion related to ‘semantic information’ of non-compatible propositions.

Notes

I owe much to discussions conducted by Professor G. Menges in his Heidelberg Seminar in the summer 1972, and to his paper (1972), and to discussions which I had with Y. Bar-Hillel and D. W. Peterson in the same summer.