The problem of priors

Counterfactuals about scientific practice reveal some curious facts about our prior probabilities. Our handling of experimental suggests an approximate flatness in our prior distributions of various constants (cf. this). But the flatness is not perfect. Suppose we are measuring some constant k in a law of nature, a constant that is either dimensionless or expressed in a natural unit system, and we come back with 2.00000. Then we will assign a fairly high credence to the hypothesis that k is exactly 2. But any kind of continuous prior distribution will assign zero prior to k being exactly 2, and the posterior will still be zero, so our prior for 2 must have been non-zero and non-infinitesimal. But for most numbers, the prior for k being that number must be zero or infinitesimal, or else the probabilities won’t add up to 1.

More generally, our priors favor simpler theories. And they favor them in a way that is tuned (finely or not). If our prior for k being exactly 2 were high, then we wpould believe that k = 2 even after a measurement of 3.2 (experimental error!). If our prior were too low, then we wouldn’t ever conclude that k = 2, no matter how many digits after the “2.” we measured to be zero.

There are is now an interesting non-normative question about the priors:

• Why are human priors typically so tuned?

There is, of course, an evolutionary answer—our reasoning about the world wouldn’t work if we didn’t have a pattern of priors that was so tuned. But there is a second question that the evolutionary story does not answer. To get to the second question, observe that our priors ought to be so tuned. Someone whose epistemic practices involve the rejection of the confirmation of scientific theories on the basis of too strong a prejudice for simple theories (“There is only one thing, and it’s round—everything else is illusion”) or too weak a preference for simple theories (“There are just as many temperature trends where there is a rise for a hundred years and then a fall for a hundred years as where there is a rise for two hundred years, so we have no reason at all to think global warming will continue”) is not acting as she ought.

So now we have this normative question:

• Why is it that our priors ought to be so tuned?

These give us the first two desiderata on a theory of priors:

1. The theory should explain why our priors are tuned with respect to simplicity as they are.

2. The theory should explain why our priors should be so tuned.

Here is another desideratum:

3. The theory should exhibit a connection between priors and truth.

Next, observe that our priors are pretty vague. They certainly aren’t numerically precise, and they shouldn’t be, because beings with our capacity couldn’t reason with precise numerical credences in the kinds of situations where we need to.

4. The theory should not imply that our having those priors we should requires us to always have numerically precise priors.

Further, there seems to be something to subjective Bayesianism, even if we should not go all the way with the subjective Bayesians. Which we should not, because then we cannot rationally criticize the person who has too strong or too weak an epistemic preference for simple theories.

5. The theory should not imply a unique set of priors that everyone should have.

Next, different kinds of agents should have different priors. For instance, agents like us typically shouldn’t be numerically precise. But angelic intellects that are capable of instantaneous mathematical computation might do better with numerically precise priors. Moreover, and more controversially, beings that lived in a world with simpler or less simple laws shouldn’t be held hostage to the priors that work so well for us.

6. The theory should allow for the possibility that priors vary between kinds of agents.

And then, of course, we have standard desiderata on all theories, such as that they be unified.

Finally, observe the actual methodology of philosophy of science: We observe how working scientists make inferences, and while we are willing at times to offer corrections, we use the actual inferential practices as evidence for how the inferential practices ought to go. In particular, we extract the kinds of priors that people have from their epistemic behavior when it is at its best:

7. The theory should allow for the methodology of inferring what kinds of priors we ought to have from looking at actual epistemic behavior.

Subjective Bayesianism fails with respect to desiderata 2 and 3, and if it satisfies 1, it is only by being conjoined with some further story, which decreases the unity of the story. Objective Bayesianism fails with respect to desiderata 5 and 6, and some versions of it have trouble with 4. Moreover, to satisfy 1, it needs to be conjoined with a further story. And it’s not clear that objective Bayesianism is entitled to the methodology advocated in 7.

What we need is something in between subjective and objective Bayesianism. Here is such a theory: Aristotelian Bayesianism. On general Aristotelian principles, we have natures which dictate a range of normal features with an objective teleology. For instance, the nature of a sheep specifies that they should have four legs in support of quadrapedal locomotion. Moreover, in Aristotelian metaphysics, the natures also explain the characteristic structure of beings with that nature. Thus, the nature of a sheep is not only that in virtue of which a sheep ought to have four legs, but also has guided the embryonic development of typical sheep towards a four-legged state. Finally, in an Aristotelian picture, when things act normally, they tend to achieve the goals that their nature assigns to that activity.

Now, in my Aristotelian Bayesianism, our human nature leads to characteristic patterns of epistemic behavior for the telos of truth. From the patterns of behavior that are compatible with our nature, one can derive constraints on priors—namely, that they be such as to underwrite such behavior. These priors are implicit in the patterns of behavior.

We can now take the desiderata one by one:

1. Our priors are tuned as they are since our development is guided by a nature that leads to epistemic behavior that determines priors to be so tuned.

2. Our priors ought to be so tuned, because all things ought to act in the way that their nature makes natural.

3. Natural behavior is teleological, and our epistemic behavior is truth-directed.

4. The the priors we ought to have are back-calculated from the epistemic behaviors we ought to have, and our behaviors cannot have precise numbers attached to them in such a way as to yield precise numerical priors.

5. Nothing in the theory requires that unique priors be derivable from what epistemic behavior is characteristic. Typically, in Aristotelian theories, there is a range of normalcy—a ratio of length of legs to length of arms between x and y, etc.

6. Different kinds of beings have different natures. Sheep ought to have four legs and we ought to have two. We are led to expect that different kinds of agents would have different appropriate priors. Moreover, animals tend to be adapted to their environment, so we would expect that in worlds that are sufficiently different, different priors would be appropriate.

7. Since beings have a tendency towards acting naturally, the actual behavior of beings—especially when they appear to be at their best—provides evidence of the kind of behavior that they ought to exhibit. And from the kind of epistemic behavior we ought to exhibit, we can back-calculate the kinds of priors that are implicit in that behavior.

This post is inspired by Barry Loewer saying in discussion that I was Kantian because I think there are objective constraints on priors. I am not Kantian. I am Aristotelian.