Some important distinctions

There are a number of distinctions that it is important to make in what follows.
An important parameter in an IP theory is the normative force the theory is supposed to have. Is imprecision obligatory or is it merely permissible? Is it always permissible/obligatory, or only sometimes? Or we might be interested in a purely descriptive project of characterising the credal states of actual agents, with no interest in normative questions. This last possibility will concern us little in this article.
It is also helpful to distinguish belief itself from the elicitation of that belief and also from your introspective access to those beliefs. The same goes for other attitudes (values, utilities and so on). It may be that you have beliefs that are not amenable to (precise) elicitation, in practice or even in principle. Likewise, your introspective access to your own beliefs might be imperfect. Such imperfections could be a source of imprecision. Bradley (2009) distinguishes many distinct sources of imperfect introspection. The imperfection could arise from your unawareness of the prospect in question, the boundedness of your reasoning, ignorance of relevant contingencies, or because of conflict in your evidence or in your values (pp. 240–241). See Bradley and Drechsler (2014) for further discussion of types of uncertainty.
There are a variety of aspects of a body of evidence that could make a difference to how you ought to respond to it. We can ask how much evidence there is (weight of evidence). We can ask whether the evidence is balanced or whether it tells heavily in favour of one hypothesis over another (balance of evidence). Evidence can be balanced because it is incomplete: there simply isn’t enough of it. Evidence can also be balanced if it is conflicted: different pieces of evidence favour different hypotheses. We can further ask whether evidence tells us something specific—like that the bias of a coin is 2/3 in favour of heads—or unspecific—like that the bias of a coin is between 2/3 and 1 in favour of heads. This specificity should be distinguished from vagueness or indeterminacy of evidence: that a coin has bias about 2/3 is vague but specific, while that a coin has bias definitely somewhere between 2/3 and 1 is determinate but unspecific. Likewise, a credal state could be indeterminate, fuzzy, or it could be unspecific, or it could be both. It seems like determinate but unspecific belief states will be rarer than indeterminate ones.
Isaac Levi (1974, 1985) makes a distinction between “imprecise” credences and “indeterminate” credences (the scare quotes are indicating that these aren’t uses of the terms “imprecise” and “indeterminate” that accord with the usage I adopt in this article). The idea is that there are two distinct kinds of belief state that might require a move to an IP representation of belief. An “imprecise” belief in Levi’s terminology is an imperfectly introspected or elicited belief in mine, while an “indeterminate” belief is a (possibly) perfectly introspected belief that is still indeterminate or unspecific (or both). Levi argues that the interesting phenomenon is “indeterminate” credence. Walley (1991) also emphasises the distinction between cases where there is a “correct” but unknown probability from cases of “indeterminacy”.
There is a further question about the interpretation of IP that cross-cuts the above. This is the question of whether we understand [ltr]P[/ltr] as a “complete” or “exhaustive” representation of your beliefs, or whether we take the representation to be incomplete or non-exhaustive. Let’s talk in terms of the betting interpretation for a moment. The exhaustive/non-exhaustive distinction can be drawn by asking the following question: does [ltr]P[/ltr] capture all and only your dispositions to bet or does [ltr]P[/ltr] only partially capture your dispositions to bet? Walley emphasises this distinction and suggests that most models are non-exhaustive.
Partly because of Levi’s injunction to distinguish “imprecise” from “indeterminate” belief, some have objected to the use of the term “imprecise probability”. Using the above distinction between indeterminate, unspecific and imperfectly introspected belief, we can keep separate the categories Levi wanted to keep separate all without using the term “imprecise”. We can then use “imprecise” as an umbrella term to cover all these cases of lack of precision. Conveniently, this allows us to stay in line with the wealth of formal work on “Imprecise Probabilities” which term is used to cover cases of indeterminacy. This usage goes back at least to Peter Walley’s influential book Statistical Reasoning with Imprecise Probabilities (Walley 1991).
So, “Imprecise” is not quite right, but neither is “Probability” since the formal theory of IP is really about previsions (sort of expectations) rather than just about probability (expectations of indicator functions). Helpfully, if I abbreviate Imprecise Probability to “IP” then I can exploit some useful ambiguities.

Some important distinctions

Some important distinctions :: تعاليق

Some important distinctions