The Principle of Indifference (PoI) is plausibly defined as follows:

• (Pol): Each member of a set of propositions should be assigned the same probability (of truth) in the absence of any reason to assign them different probabilities. (Castell 1998: 387)

(PoI) is a principled way to assign probabilities in situations of epistemic ignorance. When you have no reason to assign probabilities to a set of propositions in one way versus another way (PoI) instructs you to assign a uniform distribution of probabilities across the propositions in the partition. Despite being a principled way to assign probabilities, given ignorance, there are problems with (PoI).

A well-known problem with (PoI) is Bertrand’s paradox. The upshot of Bertrand’s paradox is that unless (PoI) is somehow restricted it results in inconsistent assignments of probabilities to the same event. Equally valid ways of carving up the outcome space (i.e., the propositions in the partition) result in (PoI) assigning different uniform distributions to the same event. How the outcome space is described changes the probability value (PoI) recommends. (PoI) is description-dependent and inconsistent as a result. A lesser-known problem with (PoI) involves reliance on the notion of a ‘reason’ in the definition of PoI (i.e., “in the absence of any reason…”). This is the issue that I want to explore.

As Paul Castell (1998: 388) points out, you can generate different iterations of (PoI) based on the strength that you assign to the notion of an epistemic reason. How strong does a consideration in favor of believing that p have to be in order to count as a reason to believe that p?

Epistemic reasons can vary in strength. I might believe that Sheriff Chance is corrupt because I saw her take a bribe from an ex-convict named Stumpy, or I might believe that Sheriff Chance is corrupt because I suspect that she is corrupt. Assuming my suspicion is merely a suspicion, and not based on solid evidence, the former reason to believe that the Sheriff is corrupt is stronger than the latter reason.

In general, adopting a weaker understanding of ‘reason’ generates a stronger (i.e., more stringent) version of (PoI). Such a version of (PoI) is more stringent because it is a more demanding principle. To qualify as being in a state of epistemic ignorance requires not having any reason to assign different probabilities to the propositions. If, for instance, a mere suspicion qualifies as a ‘reason’, then you cannot possess any suspicion that one proposition is more likely than the others. If you have such a suspicion, then you have a reason to assign them different probabilities and (PoI) does not apply. This puts a high bar on what it takes for (PoI) to apply to a situation of uncertainty. The inverse also holds: a stronger understanding of ‘reason’ generates a weaker (i.e., less stringent) version of (PoI). If, for instance, only knowledge qualifies as a ‘reason’, then you cannot possess any knowledge that one proposition is more likely than the others. Because knowledge is a stronger epistemic concept than mere suspicion it will be easier to qualify as being in a state of epistemic ignorance (i.e., possessing no reasons). This places a low bar on what it takes for (PoI) to apply to a situation of uncertainty. These generalizations are as follows:

• High Bar: A weak interpretation of ‘reason’ yields a strong interpretation of (PoI);
• Low Bar: A strong interpretation of ‘reason’ yields a weak interpretation of (PoI).

Before trying to set the bounds of interpretations of (PoI) by finding iterations of (PoI) that satisfy High Bar and Low Bar it would be good to get a grip on how these considerations create problems for (PoI). One aspect of the worry is that (PoI) generates probability distributions (i.e., quantities) contingent upon the interpretation of a qualitative notion (i.e., a reason). If you think traditional epistemology should inform formal epistemology, then this may not be much of a worry. However, if you think things should work in the opposite direction or not at all (i.e., formal epistemology should inform traditional epistemology or they should be regarded as domains that don’t meaningfully interact), then this may be regarded as a worry. However, the deeper worry is akin to the problem Bertrand’s paradox causes (PoI).

What I call the ‘Many Interpretations’ problem for (PoI) results in inconsistent assignments of probabilities to the same event. The problem is not generated based on a redescription of the propositions in the partition. Rather, the problem is generated based on a reinterpretation of (PoI). For instance, this can occur when a Low Bar interpretation of (PoI) is applicable to an epistemic situation, so it recommends a uniform distribution of probabilities, but a High Bar interpretation of (PoI) is not applicable to the same epistemic situation, and a non-uniform distribution of probabilities results. How (PoI) is interpreted determines whether or not it is applicable to one and the same epistemic situation, which results in inconsistent application of the principle, and inconsistent assignment of probability values. Is the Many Interpretations problem something that Bayesians need to worry about?

At first glance it appears that Subjective Bayesians aren’t impacted by the Many Interpretations problem. Subjective Bayesians do not require a ‘reason’ to assign probabilities to propositions. Such Bayesians regard (PoI) as unnecessary, no matter how it is interpreted. As Castell (1998) explains about this Bayesian position:

The thought is that where you do not feel in a position to make a (warranted) probabilistic judgement given the available evidence, the proper thing to do is simply to abstain from judgement. According to this view, it is unreasonable to wheel in a principle that provides probabilities where you judge none to be warranted: there is no role for (Pol) within Bayesianism. (p. 388)

Castell argues that Bayesians must (and in fact do) rely on (PoI) in assigning probabilities. In the interest of space I will not rehearse the details of his argument. However, there is a good case to be made that Bayesians rely, even if only implicitly, on (PoI), and in certain situations Bayesians must rely on (PoI) because no frequency data is available in the facts of the case or in the background information. So, I think that, despite professions otherwise, this objection to (PoI) is pressing for Bayesians as well.

Let’s explore a few iterations of (PoI). Castell (1998) articulates two iterations of (PoI) in an attempt to find a High Bar version of (PoI). The first iteration of (PoI) involves judgment (J):

• (PoI-J): Each member of a set of propositions should be assigned the same probability in the absence of a subjective judgement to the contrary. (p. 388 n.4)

Castell recognizes that (PoI-J) is problematic. We don’t make probabilistic judgments about many things. This is often because we have not thought about the matter. Do we really assign a uniform distribution to things that we have never thought about? Castell thinks that we need to introspect on such things. This generates the second iteration of (PoI), which involves introspection and judgment (IJ):

• (PoI-IJ): Each member of a set of propositions should be assigned the same probability if due consideration (introspection) yields no subjective judgement to the contrary. (p. 388 n.4)

Though I wouldn’t put (PoI-IJ) at the upper limit of High Bar, (PoI-IJ) is a viable option for a High Bar version of (PoI). This is because the notion of a ‘reason’ is relatively weak. As Castell says about (PoI-IJ), “where an agent feels unable to make any judgements (however weakly based on evidence), it directs him to adopt the uniform distribution” (p. 388 n.4). (PoI-IJ) is a more stringent principle because it requires you to not have any subjective judgment “however weakly based on evidence” in order to assign a uniform distribution of probabilities. By contrast, a viable option for a Low Bar version of (PoI) involves knowledge (K):

• (PoI-K): Each member of a set of propositions should be assigned the same probability if due consideration (introspection) yields no knowledge to the contrary.

(PoI-K) has a strong interpretation of ‘reason’ and generates a weak version of (PoI). It requires you to not have any knowledge that one (or more) of the propositions should be assigned a different probability. Such knowledge is harder to come by, so (PoI-K) is easier to satisfy. There is a Low Bar regarding the applicability of (PoI-K) to a situation of uncertainty.

Along the High Bar/Low Bar spectrum there is a range of interpretations of (PoI). Such interpretations include having no intuition, no reasonable doubt, and no belief to the contrary.

The Many Interpretations problem calls for a restriction of (PoI) to prevent inconsistent assignments of probabilities to the same event. Is a Low Bar or a High Bar interpretation of (PoI) more likely to be correct? If it’s possible to argue that one interpretation is the correct interpretation, then other interpretations can be ruled out. This delimits the permissiveness of (PoI) and prevents multiple interpretations from generating different probabilities. However, such a move greatly reduces the scope and power of (PoI). It will not be possible to apply (PoI) to many situations of epistemic uncertainty, situations that are not ruled out under a permissive understanding of (PoI).

Reference

Castell, Paul. 1998. A Consistent Restriction of the Principle of Indifference. The British Journal for the Philosophy of Science 49: 387-95.