This essay is, primarily, a discussion of four results about the principle of maximizing entropy (MAXENT) and its connections with Bayesian theory. Result 1 provides a restricted equivalence between the two: where the Bayesian model for MAXENT inference uses an "a priori" probability that is uniform, and where all MAXENT constraints are limited to 0-1 expectations for simple indicator-variables. The other three results report on an inability to extend the equivalence beyond these specialized constraints. Result 2 established a sensitivity of (...) MAXENT inference to the choice of the algebra of possibilities even though all empirical constraints imposed on the MAXENT solution are satisfied in each measure space considered. The resulting MAXENT distribution is not invariant over the choice of measure space. Thus, old and familiar problems with the Laplacian principle of Insufficient Reason also plague MAXENT theory. Result 3 builds upon the findings of Friedman and Shimony (1971; 1973) and demonstrates the absence of an exchangeable, Bayesian model for predictive MAXENT distributions when the MAXENT constraints are interpreted according to Jaynes's (1978) prescription for his (1963) Brandeis Dice problem. Lastly, Result 4 generalizes the Friedman and Shimony objection to cross-entropy (Kullback-information) shifts subject to a constraint of a new odds-ratio for two disjoint events. (shrink)

Can there be good reasons for judging one set of probabilistic assertions more reliable than a second? There are many candidates for measuring "goodness" of probabilistic forecasts. Here, I focus on one such aspirant: calibration. Calibration requires an alignment of announced probabilities and observed relative frequency, e.g., 50 percent of forecasts made with the announced probability of.5 occur, 70 percent of forecasts made with probability.7 occur, etc. To summarize the conclusions: (i) Surveys designed to display calibration curves, from which a (...) recalibration is to be calculated, are useless without due consideration for the interconnections between questions (forecasts) in the survey. (ii) Subject to feedback, calibration in the long run is otiose. It gives no ground for validating one coherent opinion over another as each coherent forecaster is (almost) sure of his own long-run calibration. (iii) Calibration in the short run is an inducement to hedge forecasts. A calibration score, in the short run, is improper. It gives the forecaster reason to feign violation of total evidence by enticing him to use the more predictable frequencies in a larger finite reference class than that directly relevant. (shrink)

According to Williamson , the difference between empirical subjective Bayesians and objective Bayesians is that, while both hold reasonable credence to be calibrated to evidence, the objectivist also takes such credence to be as equivocal as such calibration allows. However, Williamson’s prescription for equivocation generates constraints on reasonable credence that are objectionable. Herein Williamson’s calibration norm is explicated in a novel way that permits an alternative equivocation norm. On this alternative account, evidence calibrated probability functions are recognised as implications of (...) evidence calibrated density functions defined over chance hypotheses. The objective Bayesian equivocates between these calibrated density functions rather than between the calibrated probability functions themselves. The result is an objective Bayesianism that avoids the main problem afflicting Williamson’s original proposal. (shrink)

Examples growing out of the Newcomb problem have convinced many people that decision theory should proceed in terms of some kind of causal probability. I endorse this view and define and investigate a variety of causal probability. My definition is related to Skyrms' definition, but proceeds in terms of objective probabilities rather than subjective probabilities and avoids taking causal dependence as a primitive concept.

In, Levi has laid out the issues involving chances, frequencies, and direct inference with admirable precision. Nevertheless, puzzles remain. The chief puzzle to which I wish to draw attention is this: Under certain circumstances, we can combine knowledge of chances and knowledge of frequencies to yield new knowledge of chances. If Petersen is “drawn at random” from among Swedes, and we also know that the proportion of Protestants among Swedes is 0.9, then we can say that the chance that Petersen (...) is a Protestant is 0.9. But if we apply this principle generally, we are led to generally trivial results: direct inference yields probabilities constrained only to lie in a broad interval like [0, 1]. On the other hand, if we can't always combine knowledge of chances and knowledge of frequencies to get new knowledge of chances, or if this knowledge can be overridden by other considerations, how do we know when we can usefully apply direct inference? (shrink)

The system presented by the author in The Logical Foundations of Statistical Inference suffered from certain technical difficulties, and from a major practical difficulty; it was hard to be sure, in discussing examples and applications, when you had got hold of the right reference class. The present paper, concerned mainly with the characterization of randomness, resolves the technical difficulties and provides a well structured framework for the choice of a reference class. The definition of randomness that leads to this framework (...) is simplified and clarified in a number of respects. It resolves certain puzzles raised by S. Spielman and W. Harper in their contributions to Profiles: Henry E. Kyburg, Jr. and Isaac Levi 1982). (shrink)