Abstract
Generalized probabilistic learning takes place in a black-box where present probabilities lead to future probabilities by way of a hidden learning process. The idea that generalized learning can be partially characterized by saying that it doesn’t foreseeably lead to harmful decisions is explored. It is shown that a martingale principle follows for finite probability spaces.
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Notes
The arguments presented here do not hold generally in infinite probability spaces if the probability measure is not countably additive (Kadane et al. 1996, 2008). Many scholars—de Finetti (1974) first and foremost among them—think that countable additivity should not be thought of as a coherence requirement. See Seidenfeld (2001) for a recent overview of reasons to take finite additivity as basic. See also Zabell (2002) and references therein on finitely additive martingale theory, which would be relevant for an extension of our results to more general situations.
If the utilities don’t depend on the outcomes of the experiment, the terms here reduce to the earlier ones.
Graves (1989) presents a generalization for probability kinematics.
That each posterior p f is a probability function means that you expect to be statically coherent.
See Myrvold (2012) for a similar treatment with epistemic utility functions.
More formally, suppose that there are n · m intervals around zero where m is the number of states. For i = 1,…, n, j = 1,…, m let u ij be an arbitrary number in the ijth interval. We assume that for all u ij there are acts A 1,…, A n such that u(A i &S j ) = u ij .
It follows that this issue does not arise in Skyrms’ value of knowledge theorem. If the martingale principle is assumed to hold, then p f (S) = p(S|p f ) = 0 if p f is not the posterior in state S.
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Huttegger, S.M. Learning experiences and the value of knowledge. Philos Stud 171, 279–288 (2014). https://doi.org/10.1007/s11098-013-0267-7
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DOI: https://doi.org/10.1007/s11098-013-0267-7