The answer to the question in the title of this post may seem obvious (after all, isn't Bayesianism all about probabilities?), but I think that the long discussion that followed Lauren's post on van Fraassen's objection to Bayesianism from quantum mechanics shows that it isn't clear at all - or at least, that it wasn't clear to either of us as we were discussing the issue. I think that I now understand why. In this post, I'm going to give three answers to this question, which I will call The Primitivist Account (P), The Kripkean Possible Worlds Account (KPW), and the Lewisian Possible Worlds Account (LPW). This post will discuss what each view means, and where vagueness enters each account. I will also be identifying three crucial problems with (P) and showing how each of the other views answers these difficulties.
Here are brief definitions of each view, and how each one relates subjective degrees of rational confidence to probabilities (I will explain in more depth later).
Part of the reason for the previous confusion is that I was more or less assuming (P), and I think that Lauren had noticed some serious problems with it. First, a word on the reason for my assumption, and then I will try to state Lauren's objections.
(P) may be the dominant interpretation of Bayesianism. I don't really know. But there is good reason why someone reading the literature might think it's the dominant interpretation: it maps especially well to how Bayesian philosophers actually apply Bayesianism. Most philosophers who apply Bayesian reasoning (myself included) do it by simply making up numbers that are supposed to represent their degrees of confidence. Where do these numbers come from? We simply observe that we have varying degrees of confidence about different beliefs, and map these degrees of confidence to the real numbers between 0 and 1. Vagueness comes in from the fact that we don't have mathematically precise degrees of confidence, and our numbers are simply made up from our imprecise degrees of confidence, rather than computed somehow.
Now, as I have said, I believe that three important questions came out in our previous discussion which this theory leaves unanswered:
(P) does not answer these questions. This is, of course, to be expected from a theory called "primitivism," but I think the third question is particularly problematic. In the previous discussion, it was van Fraassen's assumption that we should do this very thing that brought up the issue. However, Bayesianism really needs these principles. Is it possible to provide an analysis of degrees of rational confidence that adequately answers these questions? (KPW) and (LPW) will attempt this very thing.
(KPW) is inspired by Kripke's treatment of possible worlds in terms of state spaces in the 1980 preface to Naming and Necessity, pp. 15-20. Kripke here argues that possible worlds are the same sorts of things as the "states" used in school probability theory, the difference being that possible worlds are maximally specific. Now, consider a view according to which the state space of Bayesian reasoning is the space of all epistemically possible worlds - that is, all the world-states (which are abstractions just like dice-states) which might for all we know be actual. Note that not all of these may be really possible. For instance, the Anselmian God either exists or does not exist, with logical necessity, but his existence and non-existence may both be epistemically possible for a particular person. So, when we say that we have subjective degree of confidence .5 for a given proposition, we are saying that that proposition holds in half of all epistemically possible worlds.
This view will be helped by Lewis's observation about the relationship between propositions and possible worlds: namely, that every proposition picks out a set of possible worlds, the worlds in which it obtains. (Lewis wants this to be a reductive analysis of propositions, but we need not do that.) So, consider any given proposition you believe. There is a set of possible worlds in which that proposition obtains. The set of epistemically possible worlds (for you) is the intersection of the sets for all the propositions you believe.
The (KPW) answer to question (1) has already been given - Bayesian degrees of confidence are probabilities. Let's proceed to give an interpretation of the math on (KPW).
Bayes' theorem is a relation between an initial probability - a probability over some state space S - and a conditional probability - a probability over some subset S' of S. Usually, we consider some proposition p and some evidence e. We already have assigned a particular degree of confidence to p and we want to adjust our confidence in light of learning the new evidence e. We use Bayes's theorem to calculate P(p|e). What has happened here? The new evidence e has eliminated certain formerly epistemically possible worlds - namely, all the worlds according to which ~e. In order to computer P(p|e) we have to know something about the relationship between p and e. In particular, we have to know P(e|p), P(p) and P(e) (all of them over the initial state space S). This involves knowing how many of our epistemically possible worlds certain conditions obtain in.
The (KPW) answer to question (3) should now be quite clear. Probabilities for events like dice rolls are, on this view, actually just special cases of degrees of subjective confidence. Why is there a 1/6 probability of a single die rolling 1? Because in 1/6 of all epistemically possible worlds it will land 1. (We should think of these world-states as covering the whole history of the world, so future events can be handled the same as past or present events.) In our school probability exercises, we simplify the case by supposing there are only 6 worlds. In fact, there are 6 sets of worlds. We know that the worlds will divide more or less evenly (we assume we know with certainty that the die will be rolled), because most of the propositions we are uncertain about vary independently of the result of the die roll. The ones that don't vary independently (e.g. propositions stating that the die is unfair in some particular way) are, for all we know, as likely to favor one side of the die as another.
Vagueness enters (KPW) by virtue of the fact that the worlds are created by us. They don't exist objectively. As such, there is vagueness as to how many worlds there are, and there is vagueness as to whether certain propositions are true in certain worlds. These things are simply not fully defined. We should nevertheless be able to fix upper and lower bounds by considering all of the possible resolutions of the vagueness (actually, we can probably do better by figuring out in advance which resolutions will lead to high values, and which to low values). In practice, however, we do something more like the die role case: we eliminate all the propositions that vary (more or less) independently (as far as we know) of the propositions under consideration, to divide the epistemically possible worlds into sets, and then consider each set as a single underspecified world.
(LPW) is very similar to (KPW) in its answers to the three questions. (LPW) holds that we are talking about real possible worlds, which are epistemically accessible - that is, which might, for all we know, be the world we're in. Vagueness on (LPW) is different. Because the worlds are fully defined and there is an objective truth about how many there are, there is only one source of vagueness properly so-called: vagueness about whether a given world is epistemically accessible. However, there is also second-order uncertainty - uncertainty about whether a certain world is genuinely possible, or whether a given proposition obtains in a certain world.
These two theories improve on (P) by providing explanations for why we use Bayesian reasoning the way we do, and why it works like probability theory at all. They also allow us to define our degrees of confidence much more clearly.Posted by Kenny at November 30, 2007 9:10 AM
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