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تاريخ التسجيل : 26/10/2009
وســــــــــام النشــــــــــــــاط : 6
 | | A summary of terminology | |
Throughout the article I adopt the convention of discussing the beliefs of an arbitrary intentional agent whom I shall call “you”. Prominent advocates of IP (including Good and Walley) adopt this convention.This article is about formal models of belief and as such, there needs to be a certain amount of formal machinery introduced. There is a set of states [ltr]Ω[/ltr] which represents the ways the world could be. Sometimes [ltr]Ω[/ltr] is described as the set of “possible worlds”. The objects of belief—the things you have beliefs about—can be represented by subsets of the set of ways the world could be [ltr]Ω[/ltr]. We can identify a proposition [ltr]X[/ltr] with the set of states which make it true, or, with the set of possible worlds where it is true. If you have beliefs about [ltr]X[/ltr] and [ltr]Y[/ltr] then you also have beliefs about “[ltr]X∩Y[/ltr]”, “[ltr]X∪Y[/ltr]” and “[ltr]¬X[/ltr]”; “[ltr]X[/ltr] and [ltr]Y[/ltr]”, “[ltr]X[/ltr] or [ltr]Y[/ltr]” and “it is not the case that [ltr]X[/ltr]” respectively. The set of objects of belief is the power set of [ltr]Ω[/ltr], or if [ltr]Ω[/ltr] is infinite, some measurable algebra of the subsets of [ltr]Ω[/ltr].The standard view of degree of belief is that degrees of belief are represented by real numbers and belief states by probability functions; this is a normative requirement. Probability functions are functions, [ltr]p[/ltr], from the algebra of beliefs to real numbers satisfying:
- [ltr]0=p(∅)≤p(X)≤p(Ω)=1[/ltr]
- If [ltr]X∩Y=∅[/ltr] then [ltr]p(X∪Y)=p(X)+p(Y)[/ltr]
So if your belief state or doxastic state is represented by [ltr]p[/ltr], then your degree of belief in [ltr]X[/ltr] is the value assigned to [ltr]X[/ltr] by [ltr]p[/ltr]; that is, [ltr]p(X)[/ltr].Further, learning in the Bayesian model of belief is effected by conditionalisation. If you learn a proposition [ltr]E[/ltr] (and nothing further) then your post-learning belief in [ltr]X[/ltr] is given by [ltr]p(X∣E)=p(X∩E)/p(E)[/ltr].The alternative approach that will be the main focus of this article is the approach that represents belief by a set of probability functions instead of a single probability. So instead of having some [ltr]p[/ltr]represent your belief state, you have [ltr]P[/ltr], a set of such functions. van Fraassen (1990) calls this your representor, Levi calls it a credal set. I will discuss various ways you might interpret the representor later but for now we can think of it as follows. Your representor is a credal committee: each probability function in it represents the opinions of one member of a committee that, collectively, represents your beliefs.From these concepts we can define some “summary functions” that are often used in discussions of imprecise probabilities. Often, it is assumed that your degree of belief in a proposition, [ltr]X[/ltr], is represented by [ltr]P(X)={p(X):p∈P}[/ltr]. I will adopt this notational convention, with the proviso that I don’t take [ltr]P(X)[/ltr] to be an adequate representation of your degree of belief in [ltr]X[/ltr]. Your lower envelope of [ltr]X[/ltr] is: [ltr]P−(X)=infP(X)[/ltr]. Likewise, your upper envelope is [ltr]P[size=13]¯¯¯¯(X)=supP(X)[/ltr]. They are conjugates of each other in the following sense: [ltr] P¯¯¯¯(X)=1−P−(¬X)[/ltr].[/size] The standard assumption about updating for sets of probabilities is that your degree of belief in [ltr]X[/ltr]after learning [ltr]E[/ltr] is given by [ltr]P(X∣E)={p(X∣E),p∈P,p(E)>0}[/ltr]. Your belief state after having learned [ltr]E[/ltr] is [ltr]P(⋅∣E)={p(⋅∣E),p∈P,p(E)>0}[/ltr]. That is, by the set of conditional probabilities.I would like to emphasise already that these summary functions—[ltr]P(⋅)[/ltr], [ltr]P−(⋅)[/ltr] and [ltr]P[size=13]¯¯¯¯(⋅)[/ltr]—are not properly representative of your belief. Information is missing from the picture. This issue will be important later, in our discussion of dilation.[/size] We shall need to talk about decision making so we shall introduce a simple model of decisions in terms of gambles. We can view bounded real valued functions [ltr]f[/ltr] as “gambles” that are functions from some set [ltr]Ω[/ltr] to real numbers. A gamble [ltr]f[/ltr] pays out [ltr]f(ω)[/ltr] if [ltr]ω[/ltr] is the true state. We assume that you value each further unit of this good the same (the gambles’ pay outs are linear in utility) and you are indifferent to concerns of risk. Your attitude to these gambles reflects your attitudes about how likely the various contingencies in [ltr]Ω[/ltr] are. That is, gambles that win big if [ltr]ω[/ltr] look more attractive the more likely you consider [ltr]ω[/ltr] to be. In particular, consider the indicator function [ltr]I[size=13]X[/ltr]on a proposition [ltr] X[/ltr] which outputs [ltr] 1[/ltr] if [ltr] X[/ltr] is true at the actual world and [ltr] 0[/ltr] otherwise. These are a particular kind of gamble, and your attitude towards them straightforwardly reflects your degree of belief in the proposition. The more valuable you consider [ltr] IX[/ltr], the more likely you consider [ltr] X[/ltr]to be. Call these indicator gambles.[/size] Gambles are evaluated with respect to their expected value. Call [ltr]E[size=13]p(f)[/ltr] the expected value of gamble [ltr] f[/ltr] with respect to probability [ltr] p[/ltr], and define it as:[/size] [ltr]E[size=13]p(f)=∑Ωp(ω)f(ω)[/ltr][/size] | |
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