Abstract
We examine the possibility of justifying the principle of maximum relative entropy (MRE) considered as an updating rule by looking at the value of learning theorem established in classical decision theory. This theorem captures an intuitive requirement for learning: learning should lead to new degrees of belief that are expected to be helpful and never harmful in making decisions. We call this requirement the value of learning. We consider the extent to which learning rules by MRE could satisfy this requirement and so could be a rational means for pursuing practical goals. First, by representing MRE updating as a conditioning model, we show that MRE satisfies the value of learning in cases where learning prompts a complete redistribution of one’s degrees of belief over a partition of propositions. Second, we show that the value of learning may not be generally satisfied by MRE updates in cases of updating on a change in one’s conditional degrees of belief. We explain that this is so because, contrary to what the value of learning requires, one’s prior degrees of belief might not be equal to the expectation of one’s posterior degrees of belief. This, in turn, points towards a more general moral: that the justification of MRE updating in terms of the value of learning may be sensitive to the context of a given learning experience. Moreover, this lends support to the idea that MRE is not a universal nor mechanical updating rule, but rather a rule whose application and justification may be context-sensitive.
Highlights
Let the probability functions P and P 0 represent, respectively, an agent’s prior and posterior degrees-of-belief functions over an algebra of propositions F generated by a set of possible worldsW
P and some constraint χ imposed on P 0, which P 0 should the agent choose from the set of her posterior degrees-of-belief functions that satisfy χ? A given constraint χ imposed on P 0 is supposed to represent a learning experience, and we associate with every learning experience a set Pχ of posterior degrees-of-belief functions singled out by χ, i.e., Pχ = {P 0 : P 0 satisfies χ}
We show that updating by maximum relative entropy (MRE) satisfies the value of learning in cases where the constraint reporting one’s learning experience concerns a complete redistribution of one’s degrees of belief over a partition of propositions
Summary
Let the probability functions P and P 0 represent, respectively, an agent’s prior and posterior degrees-of-belief functions over an algebra of propositions F generated by a set of possible worlds. We show that updating by MRE satisfies the value of learning in cases where the constraint reporting one’s learning experience concerns a complete redistribution of one’s degrees of belief over a partition of propositions. Minimizing RE under some constraints imposed on posterior degrees of belief is a way, but by no means the only way, to make the idea of modesty more precise: the agent adopts the posterior degree-of-belief function that meets the constraints reporting her learning experience and is RE-closest to her prior degree-of-belief function Under this procedure, the existence of a uniquely maximally modest P 0 satisfying a given constraint is guaranteed, since Pχ is a closed convex set. What this paper shows is that it is not always true that revising degrees of belief by dint of MRE leads to modest new degrees of belief that are expected to be helpful and never harmful for one’s decisions
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