A rude assembly

Seamus Bradley's blog

Huber and Schmidt-Petri: Degrees of Belief

This is the first in what I hope to be a series of posts on the book Degrees of Belief edited by Franz Huber and Cristoph Schmidt-Petri. There are some really interesting papers in there so this should be fun. In fact, this is the first in what I hope to be a series of series of posts on books I am reading. Indeed, I hope to make a habit of writing about books I’m reading. This seems like a good way of actually making this blog work for me, rather than it just being an exercise in procrastination.

Anyway, the book is about degrees of belief. The clue is in the title. The book is split into three parts. The first part is on the relationship between degrees of belief and categorical, or full belief. The second part is on formal models of degrees of belief: probability theory and friends. And the third part is on “logical approaches”. I haven’t read that section, so I don’t really know what that means yet.

This post is really just to introduce the series of posts, which will start next week when I get back from my holiday. So in this post I’ll restrict myself to a couple of comments on the introductory chapter by Franz Huber.

Huber basically sets out to survey the material that will be covered by the other contributors to the book. Along the way, he makes one or two interesting comments I’d like to pick up on.

On page 10, Huber introduces Dempster-Shafer belief theory by saying that

The theory of Dempster-Shafer belief functions …
rejects the claim that degrees of belief
can be measured by the epistemic agent’s
betting behaviour.

This is a slightly odd claim to make. I don’t know whether Dempster or Shafer make this claim. As I understand it, DS belief theory was designed as a theory to amalgamate different sources of evidence. Hence all the “Dempster’s rule of combination” stuff. Decision making, and thus betting, doesn’t come into it. So might that be what Huber means?

I guess what Huber is driving at is that adopting any kind of non-probabilistic model of belief requires getting around the Dutch book argument. Now, one way of doing that is to simply deny that belief can be measured by betting behaviour. But this isn’t the only way. Indeed, for a suitably modified betting scenario, DS beliefs are measured by betting behaviour as I showed in Dutch book arguments and imprecise probabilities [PDF].

Well, that’s a minor point. And one that I only really brought up so I could mention my own paper.

Huber makes an interesting point a little later on, when discussing how to interpret the DS belief and plausibility functions $\operatorname{Bel}$ and $\operatorname{Plaus}$. He suggests that $\operatorname{Bel}(A)$ measures the degree to which the evidence you have supports $A$. And $\operatorname{Plaus}(A)$ measures the degree to which the evidence doesn’t support $\neg A$. So far, so conventional. Now, the interesting point that I’d not seen mentioned before is this: on this understanding, probabilism is just the claim that all evidence supports either $A$ or $\neg A$.

If we understand probabilism as the claim that degrees of belief ought to be probabilities, and we notice that probability functions are a special case of DS functions where $\operatorname{Bel}(A) = \operatorname{Plaus}(A)$ for all $A$, then probabilism just is the claim that all evidence that does not speak for $A$ speaks against $A$. I think this is a really nice way of understanding something of the relationship between the frameworks. DS theory (and similar frameworks) allow an agent to suspend judgement in a way that a probabilist simply cannot. I especially like this, because it fits perfectly with something I’ve been writing in my thesis about the Objective Bayesian norm of Equivocation, and how wrong it is. But I’ll save that for another time.

Huber then goes on to discuss some other similar formal models of belief: possibility theory, ranking theory. I can’t say I fully understand what advantages these have over, say DS theory, but I’ll save my gripes until I’ve read the contributions that properly cover those models.

Huber’s paper ends with a discussion of belief revision and non-monotonic logic. Again, I will have more to say about those topics when we get to the full sections on them.

So, there you have it: a not entirely satisfactory summary of a summary of a book. Tune in next time for some discussion of the Lockean thesis and full belief!