The Dempster-Shafer belief function does not satisfy that desideratum. The pair (Bel(A), Plaus(A)) where Plaus(A)=1-Bel(~A) does, trivially; is this what you have in mind?

I agree that it’s good to use real numbers, rather than pairs etc. I’m asking what’s so bad about your real number being the Dempster-Shafer belief function, which doesn’t satisfy (4).

]]>In my second post on the topic I describe why I prefer to use real numbers to represent uncertainty rather than, say, pairs of real numbers or lattices or what have you. In that post I recommend reading Kevin Van Horn’s guide to Cox’s theorem (which you can find at http://ksvanhorn.com/bayes/papers.html along with comments by Shafer and a rejoinder) for a more in-depth discussion of these issues. IIRC I got the observation that Dempster-Shafer theory does in effect satisfy (4) from Van Horn, but I don’t remember exactly where.

]]>Consider Bel(A) for a proposition A, where Bel is the Dempster-Shafer belief function. (Dempster-Shafer theory is often described as giving two values to each proposition, the “belief” and the “plausibility”, but we’re going to look at just the belief function).

(4) is where Bel fails, and thus is not isomorphic to probability theory.

I’d like to know your thoughts on, does this really mean that Bel is an inconsistent or irrational belief function?

For example, let’s say my degrees of belief are given by Bel. I say, “Bel(Next coin flip is heads) = 0.5, and Bel(Next coin flip is tails) = 0.5. Furthermore, Bel(Trump wins the next election) =0.5, and Bel(Trump loses the next election)=0.3”. Cox happens to be present, and exclaims, “let S be the function mapping your belief in a proposition to your belief in its negation. First you say S(0.5)=0.5, and then you say S(0.5)=0.3. Inconsistent!” To which I reply, “there does not exist such a function. Why should there?”

So, would you back up Cox in this argument, and say I have made some mistake? If so, how would you explain it to me?

People actually seem to use Dempster-Shafer theory. It’s not OBVIOUSLY bad to fail (4).

]]>“in social science it’s notorious that every variable is correlated with every other variable, at least a little bit. I imagine that this makes Pearl-style causal inference a big pain — all of the causal graphs would end up totally connected, or close to”

I’m surprised you would make this point. A consequence of Pearl-ian concepts such as “back-door paths” are how connectivity induces correlations. You only need a sparsely connected graph to obtain correlations everywhere. i.e. correlations being dense does not mean that causality is not sparse.

]]>I don’t know of any good ideas about how this limiting operation is supposed to happen in real problems, so I’m personally leery of the continous case.

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