Comparing the epistemic relevance of measurements

Suppose P is a regular probability on (the powerset of) a finite space Ω representing my credences. A measurement M is a partition of Ω into disjoint events E₁, ..., E_n, with the result of the experiment being one of these events. In a given context, my primary interest is some subalgebra F of the powerset of Ω.

Note that a measurement can be epistemically relevant to my primary interest without any of the events in in the measurement being something I have a primary interest in. If I am interested in figuring out whether taller people smile more, my primary interest will be some algebra F generated by a number of hypotheses about degree to which height and smiliness are correlated in the population. Then, the measurement of Alice’s height and smiliness will not be a part of my primary interest, but it will be epistemically relevant to my primary interest.

Now, some measurements will be more relevant with respect to my primary interest than others. Measuring Alice’s height and smiliness will intuitively be more relevant to my primary interest about height/smile correlation, while measuring Alice’s mass and eye color will be less so.

The point of this post is to provide a relevance-based partial ordering on possible measurements. In fact, I will offer three, but I believe they are equivalent.

First, we have a pragmatic ordering. A measurement M₁ is at least as pragmatically relevant to F as a measurement M₂, relative to our current (prior) credence assignment P, just in case for every possible F-based wager W, the P-expected utility of wagering on W after a Bayesian update on the result of M₁ is at least as big as that of wagering of W after updating on the result of M₂, and M₁ is more relevant if for some wager W the utility of wagering after updating on the result of M₁ is strictly greater.

Second, we have an accuracy ordering. A measurement M₁ is at least as accuracy relevant to F as a measurement M₂ just in case for every proper scoring rule s on F, the expected score of updating on the result of M₁ is better than or equal to the expected score of updating on the result of M₂, and M₁ is more relevant when for some scoring rule the expected score is better in the case of M₁.

Third, we have a geometric ordering. Let H_P, F(M) be the horizon of a measurement M, namely the set of all possible posterior credence assignments on F obtained by starting with P, conditionalizing on one of the possible events in that M partitions Ω into, and restricting to F. Then we say that M₁ is at least as (more) geometrically relevant to F as M₂ just in case the convex hull of the horizon of M₁ contains (strictly contains) the convex hull of the horizon of M₂.

I have not written out the details, but I am pretty sure that all three orderings are equivalent, which suggests that I am on to something with these concepts.

An interesting special case is when one’s interest is binary, an algebra generated by a single hypothesis H, and the measurements are binary, i.e., partitions into two sets. In that case, I think, a measurement M₁ is at least as (more) relevant as a measurement M₂ if and only if the interval whose endpoints are the Bayes factors of the events in M₁ contains (strictly contains) the interval whose endpoints are the Bayes factors of the events in M₂.