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.

Qualia and value-laden why-questions

[Note added later: A version of this argument was first discovered by Kahane.]

Consider the famous story of Mary, the neuroscientist raised in a monochrome environment who finally sees an instance of red. It has famously been argued that no matter how much science she knew before she saw red, she learned something new when she saw red, and hence there is something more to the mental life than what science says. I've always been rather sceptical of this line of argument: it just didn't seem to me that a fact was learned.

But I am now thinking--as a result of a social experience--that there is an interesting way to argue that at least in some cases like Mary's one is learning a fact when one experiences a new quale. To know the answer to a why-question is to know a fact. After all, the answer to a why-question encodes an explanation, and explanations are given by means of facts.

Now suppose that Mary instead of leading a monochrome life led a charmed life and never felt any pain. One day she stubs her toe. She learns something by stubbing her toe: what pain feels like. But again we ask: is there a fact that Mary has learned? Here then is an argument:

1. By learning what pain feels like, Mary learned why pain is bad.
2. One learns why something is the case only by learning a fact.
3. So learning what pain feels like is learning a fact.

I give the pain version of the argument not because I find it very plausible, but because I think some readers will find it plausible. I myself am not inclined to think that pain is intrinsically bad, and the reasons why pain is extrinsically bad were available to Mary prior to her stubbing the toe (she knew that pain distracts people from worthwhile pursuits, that it tends to go against people's desires, etc.) But even if I am not convinced by the pain case, I find it pretty plausible that there will be some value-based case where by learning what a quale is like one learns the answer to a why-question. I find particularly plausible aesthetic versions of this. Here's a case where I've had the relevant aesthetic experience: "Why is dark chocolate gustatorily valuable? Because it tastes like that!" Here's one where I haven't. Being largely insensitive to music (more a matter of the brain than the ears, I think), I don't experience music like other people do, and so I don't know why Beethoven is a great composer, though I know on the testimony of others that he is a great composer. But there are possible experiences--namely, those that normal people receive upon listening to Beethoven--such that the what-it-is-like of these experiences answers the question of why Beethoven is a great composer.

Of course, these examples won't help a value-nihilist. But why would anyone be a value-nihilist? (A question with a hook.)

How little we know

At times I am struck by just how little we know (and I don't even put much emphasis on the "know"). I work, inter alia, in philosophy of time and I can't answer my six-year-old's questions about the nature of time. We humans really aren't very smart at all, except at asking questions.

It is not surprising that our ability to ask questions would outpace our ability to find answers. But it is, I think, surprising just how far it outpaces it.

And yet we can know the important thing: that we are made to know and love God.

Liar and truthteller questions

Here are some fun questions:

1. Is the answer to this question negative?
2. Is the answer to this question positive?
3. What is the answer to this question?

The last one is due to my six-year-old.