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Bayes and the ‘old evidence’ problem

Summary:
Bayes and the ‘old evidence’ problem Among the many achievements of Newton’s theory of gravitation was its prediction of the tides and their relation to the lunar orbit. Presumably the success of this prediction confirmed Newton’s theory, or in Bayesian terms, the observable facts about the tides e raised the probability of Newton’s theory h. But the Bayesian it turns out can make no such claim. Because the facts about the tides were already known when Newton’s theory was formulated, the probability for e was equal to one. It follows immediately that both C (e ) and C (e |h ) are equal to one (the latter for any choice of h ). But then the Bayesian multiplier is also one, so Newton’s theory does not receive any probability boost from its prediction of

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Bayes and the ‘old evidence’ problem

Among the many achievements of Newton’s theory of gravitation was its prediction of the tides and their relation to the lunar orbit. Presumably the success of this prediction confirmed Newton’s theory, or in Bayesian terms, the observable facts about the tides e raised the probability of Newton’s theory h.

Bayes and the ‘old evidence’ problemBut the Bayesian it turns out can make no such claim. Because the facts about the tides were already known when Newton’s theory was formulated, the probability for e was equal to one. It follows immediately that both C (e ) and C (e |h ) are equal to one (the latter for any choice of h ). But then the Bayesian multiplier is also one, so Newton’s theory does not receive any probability boost from its prediction of the tides. As either a description of actual scientific practice, or a prescription for ideal scientific practice, this is surely wrong.

The problem generalizes to any case of “old evidence”: If the evidence e is received before a hypothesis h is formulated then e is incapable of boosting the probability of h by way of conditionalization. As is often remarked, the problem of old evidence might just as well be called the problem of new theories, since there would be no difficulty if there were no new theories, that is, if all theories were on the table before the evidence began to arrive. Whatever you call it, the problem is now considered by most Bayesians to be in urgent need of a solution. A number of approaches have been suggested, none of them entirely satisfactory.

A recap of the problem: If a new theory is discovered midway through an inquiry, a prior must be assigned to that theory. You would think that, having assigned a prior on non-empirical grounds, you would then proceed to conditionalize on all the evidence received up until that point. But because old evidence has probability one, such conditionalization will have no effect. The Bayesian machinery is silent on the significance of the old evidence for the new theory.

Michael Strevens

Lars Pålsson Syll
Professor at Malmö University. Primary research interest - the philosophy, history and methodology of economics.

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