Interpreting IP

Imprecise probabilities aren’t a radically new theory. They are merely a slight modification of existing models of belief for situations of ambiguity. Often your credences will be precise enough, and your available actions will be such that you act more or less as if you were a strict Bayesian. One might analogize imprecise probabilities as the “Theory of Relativity” to the strict Bayesian “Newtonian Mechanics”: all but indistinguishable in all but the most extreme situations. This analogy goes deeper: in both cases, the theories are “empirically indistinguishable” in normal circumstances, but they both differ radically in some conceptual respects. Namely, the role of absolute space in Newtonian mechanics/GR; how to model ignorance in the strict/imprecise probabilist case. Howson (2012) makes a similar analogy between modelling belief and models in science. Both involve some requirement to be somewhat faithful to the target system, but in each case faithfulness must be weighed up against various theoretical virtues like simplicity, computational tractability and so on. Likewise Hosni (2014) argues that what model of belief is appropriate is somewhat dependent on context. There is of course an important disanalogy in that models of belief are supposed to be normative as well as descriptive, whereas models in science typically only have to play a descriptive role. Walley (1991) discusses a similar view but is generally sceptical of such an interpretation.

3.4.1 What is a belief?

One standard interpretation of the probability calculus is that probabilities represent “degrees of belief” or “credences”. This is more or less the concept that under consideration so far. But what is a degree of belief? There are a number of ways of cashing out what it is that a representation of degree of belief is actually representing.
One of the most straightforward understandings of degree of belief is that credences are interpreted in terms of an agent’s limiting willingness to bet. This is an idea which goes back toRamsey (1926) and de Finetti (1964, 1990 [1974]). The idea is that your credence in [ltr]X[/ltr] is [ltr]α[/ltr] just in case [ltr]α[/ltr] is the value at which you are indifferent between the gambles:

Win [ltr]1−α[/ltr] if [ltr]X[/ltr], lose [ltr]α[/ltr] otherwise
Lose [ltr]α−1[/ltr] if [ltr]X[/ltr], win [ltr]α[/ltr] otherwise

This is the “betting interpretation”. This is the interpretation behind Dutch book arguments: this interpretation of belief makes the link between betting quotients and belief strong enough to sanction the Dutch book theorem’s claim that beliefs must be probabilistic. Williamson in fact takes issue with IP because IP cannot be given this betting interpretation (2010: 68–72). He argues that Smith’s and Walley’s contributions notwithstanding (see formal appendix), the single-value betting interpretation makes sense as a standard for credence in a way that the one-sided betting interpretation doesn’t. The idea is that you may refuse all bets unless they are at extremely favourable odds by your lights. Such behaviour doesn’t speak to your credences. However, if you were to offer a single value then this tells us something about your epistemic state. There is something to this idea, but it must be traded off against the worry that forcing agents to have such single numbers systematically misrepresents their epistemic states. As Kaplan puts it

اقتباس :: The mere fact that you nominate [ltr][size=18]0.8[/ltr] under the compulsion to choose some determinate value for [[ltr]p(X)[/ltr]] hardly means that you have a reason to choose [ltr]0.8[/ltr]. The orthodox Bayesian is, in short, guilty of advocating false precision. (Kaplan 1983: 569, Kaplan’s emphasis)[/size]

A related interpretation of credence is to understand credence as being just a representation of an agent’s dispositions to act. This interpretation sees credence as that function such that your elicited preferences and observed actions can be represented as those of an expected utility maximiser with respect to that probability function (Briggs 2014: section 2.2). Your credencesjust are that function that represents you as a rational agent. For precise probabilism, “rational agent” means “expected utility maximiser”. For imprecise probabilism, rational agent must mean something slightly different. A slightly more sophisticated version of this sort of idea is to understand credence to be exactly that component of the preference structure that the probability function represents in the representation theorem. Recall the discussion of incompleteness (section 2.2). IP represents you as the agent conflicted between all the [ltr]p∈P[/ltr] such that unless the [ltr]p[/ltr] agree that [ltr]X[/ltr] is better than [ltr]Y[/ltr] or vice versa, you find them incomparable. What a representation theorem actually proves is a matter of some dispute (see Zynda 2000; Hájek 2008; Meacham and Weisberg 2011).
One might take the view that credence is modelling some kind of mental or psychological quantity in the head. Strength of belief is a real psychological quantity and it is this that credence should measure. Unlike the above views, this interpretation of credence isn’t easy to operationalise. It also seems like this understanding of strength of belief distances credence from its role in understanding decision making. The above behaviourist views take belief’s role in decision making to be central to or even definitional of what belief is. This psychological interpretation seems to divorce belief from decision. Whether there are such stable neurological structures is also a matter of some controversy (Fumagalli 2013; Smithson and Pushkarskaya forthcoming).
A compromise between the behaviourist views and the psychological views is to say that belief is characterised in part by its role in decision making. This leaves room for belief to play an important role in other things, like assertion or reasoning and inference. So the answer to the question “What is degree of belief?” is: “Degree of belief is whatever psychological factors play the role imputed to belief in decision making contexts, assertion behaviour, reasoning and inference”. There is room in this characterisation to understand credence as measuring some sort of psychological quantity that causally relates to action, assertion and so on. This is a sort of functionalist reading of what belief is. Eriksson and Hájek (2007) argue that “degree of belief” should just be taken as a primitive concept in epistemology. The above attempts to characterise degree of belief then fill in the picture of the role degree of belief plays.

3.4.2 What is a belief in [ltr][size=19]X[/ltr]?[/size]

So now we have a better idea of what it is that a model of belief should do. But which part of our model of belief is representing which part of the belief state? The first thing to say is that [ltr]P(X)[/ltr]is not an adequate representation of the belief in [ltr]X[/ltr]. That is, one of the values of the credal set approach is that it can capture certain kinds of non-logical relationships between propositions that are lost when focusing on, say, the associated set of probability values. For example, consider tossing a coin of unknown bias. [ltr]P(H)=P(T)=[0,1][/ltr], but this fails to represent the important fact that [ltr]p(H)=1−p(T)[/ltr] for all [ltr]p∈P[/ltr]. Or that getting a heads on the first toss is at least as likely as heads on two consecutive tosses. These facts that aren’t captured by the sets-of-values view can play an important role in reasoning and decision.
[ltr]P(X)[/ltr] might be a good enough representation of belief for some purposes. For example in the Ellsberg game these sets of probability values (and their associated sets of expectations) are enough to rationalise the non-probabilistic preferences. How good the representation needs to be depends on what it will be used for. Representing the sun as a point mass is a good enough representation for basic orbital calculations, but obviously inadequate if you are studying coronal mass ejections, solar flares or other phenomena that depend on details of the internal dynamics of the sun.

Interpreting IP

3.4.1 What is a belief?

3.4.2 What is a belief in [ltr][size=19]X[/ltr]?[/size]

Interpreting IP :: تعاليق

Interpreting IP