'An objective defense of Bayesianism'

by lukeprog1 min read13th Sep 20115 comments

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Recently, Hans Leitgeb and Richard Pettigrew have published a novel defense of Bayesianism:

An Objective Defense of Bayesianism I: Measuring Inaccuracy

One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its sequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm:

Accuracy: An epistemic agent ought to minimize the inaccuracy of her partial beliefs.

In this paper, we make this norm mathematically precise in various ways. We describe three epistemic dilemmas that an agent might face if she attempts to follow Accuracy, and we show that the only inaccuracy measures that do not give rise to such dilemmas are the quadratic inaccuracy measures. In the sequel, we derive the main tenets of Bayesianism from the relevant mathematical versions of Accuracy to which this characterization of the legitimate inaccuracy measures gives rise, but we also show that unless the requirement of Rigidity is imposed from the start, Jeffrey conditionalization has to be replaced by a different method of update in order for Accuracy to be satisfied.

An Objective Defense of Bayesianism II: The Consequences of Minimizing Inaccuracy

In this article and its prequel, we derive Bayesianism from the following norm: Accuracy—an agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we make the norm mathematically precise; in this article, we derive its consequences. We show that the two core tenets of Bayesianism follow from Accuracy, while the characteristic claim of Objective Bayesianism follows from Accuracy together with an extra assumption. Finally, we show that Jeffrey Conditionalization violates Accuracy unless Rigidity is assumed, and we describe the alternative updating rule that Accuracy mandates in the absence of Rigidity.

Richard Pettigrew has also written an excellent introduction to probability.

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Commenting on the fly:

The authors employ the "Ought-Can" principle to defend their assumption that the space of possible worlds should be treated as finite:

Ought-Can: A norm should not demand anything of an agent that is beyond her epistemic reach.

Their argument is essentially this: we (humans) can only divide the space of logical possibilities into finitely many options, so by Ought-Can we do not demand an infinite set of possible worlds.

This is a bit misguided. They should first ask, what is the right answer, cognitive resources be damned? E.g., what is the true probability that an apple will fall on my head tomorrow? Even if this answer is impossible to exactly compute, we need to know that it exists in principle so that we can approximate it in some way. As it stands, they have approximated the true answer, but we don't know what the true answer even looks like, so it's impossible to evaluate how close their approximation is/can be.

(This seems like the sort of mistake you make if you aren't thinking in the back of your head "how would I program an AI to use this epistemology?")

EDIT: At the end of the paper the authors admit that they do need to look into the infinite case so the problem isn't as bad as I initially thought - this paper looks more like tackling a simple case before going after the fully general proof.

I haven't read this either, but the second sounds like Cox's theorem in disguise.

From a quick glance it looks to be the same flavor but a different result. The assumptions they are making seem to be different from Cox's theorem.

We describe three epistemic dilemmas that an agent might face if she attempts to follow Accuracy, and we show that the only inaccuracy measures that do not give rise to such dilemmas are the quadratic inaccuracy measures.

Huh? I don't have the time to look into this, but are they saying that a quadratic inaccuracy measure is superior to entropy?

Yes, basically they're saying given some reasonable (at least to them) assumptions about what an accuracy measure should look like, the only acceptable measure is quadratic.

They make some arbitrary assumptions about how to represent the space of possible worlds and degrees of belief, and it isn't clear if their result depends on these assumptions (they acknowledge this).