Predictability of future credence changes

Suppose you update your credences via Bayesian update at discrete moments of time (i.e., at any future time, your credence is the result of a finite number of Bayesian updates from your present credence). Then it can be proved that you cannot be sure (i.e., assign probability one) that your credence will ever be higher than it is now, and similarly you cannot be sure that your credence will ever be lower than it is now.

The same is not true for continuous Bayesian update, as is shown by Alan Hajek’s Cable Guy story. Cable Guy will come tomorrow between 8:00 am and 4:00 pm, with 4:00 pm included but 8:00 am excluded. Your current credence that they will come in the morning is 1/2 and your current credence that they will come in the afternoon is also 1/2.

Then it is guaranteed that there will be a time after 8:00 am when Cable Guy hasn’t come yet. At that time, because you have ruled out some of the morning possibilities but none of the afternoon possibilities, your credence that the Cable Guy will come in the afternoon will have increased and your credence that the Cable Guy will come in the morning will have decreased.

Proof of fact in first paragraph: A Bayesian agent’s credences are a martingale. To obtain a contradiction, suppose there is probability 1 that the credences will go above their current value. Let C_n be the agent’s credence after the nth update, and consider everything from the point of view of the agent right now, before the updates, with current credence r. Let τ be the first time such that C_τ > r (this is defined with probability one). By Doob’s Optional Sampling Theorem, E[C_τ] = r. But this contradicts the inequality C_τ > r.

A tale of two forensic scientists

Curley and Arrow are forensic scientist expert witnesses involved in very similar court cases. Each receives a sum of money from a lawyer for one side in the case. Arrow will spend the money doing repeated tests until the money (which of course also pays for his time) runs out, and then he will present to the court the full test data that she found. Curley, on the other hand, will do tests until such time as either the money runs out or he has reached an evidential level sufficient for the court to come to the decision that the lawyer who hired him wants. Of course, Curley isn't planning to commit perjury: he will truthfully report all the tests he actually did, and hopes that the court won't ask him why he stopped when he did. Curley reasons that his method of proceeding has two advantages over Arrow's:

1. if he stops the experiments early, his profits are higher and he has more time for waterskiing; and
2. he is more likely to confirm the hypothesis that his lawyer wants confirmed, and hence he is more likely to get repeat business from this lawyer.

Now here is a surprising fact which is a consequence of the martingale property of Bayesian investigations (or of a generalization of van Fraassen's Reflection Principle). When Curley and Arrow reflect on what their final credences will be with respect to what they are each testing, if they are perfect Bayesian agents, their expectation for their future credence equals their current credence. This thought may lead Curley to think himself quite innocent in his procedures. After all, on average, he expects to end up with the same credence as he would if he followed Arrow's more onerous procedures.

So why do we think Curley crooked? It's because we do not just care about the expected values of credences. We care about whether credences reach particular thresholds. In the case at hand, we care about whether the credence reaches the threshold that correlates with a particular court decision. And Curley's method does increase the probability that that credence level will be reached.

What happens is that Curley, while favoring a particular conclusion, sacrifices the possibility of reaching evidence that confirms that conclusion to a degree significantly higher than his desired threshold, for the sake of increasing the probability of reaching the threshold. For when he stops his experiments once the level of confirmation has reached the desired threshold, he is giving up on the possibility—useless to him or to the side that hired him—that the level of confirmation will go up even higher.

I think it helps that in real life we don't know what the thresholds are. Real-life experts don't know just how much evidence is needed, and so there is some an incentive to try to get a higher level of confirmation, rather than to stop once one has reached a threshold. But of course in the above I stipulated there was a clear and known threshold.

Van Fraassen's reflection principle

Van Fraassen's reflection principle (RP) says that if a rational agent is certain she will assign credence r to p, then she should now assign r to p.

As I was writing on being-sure yesterday, I was struck by the fact (and I wasn't the first person to be struck by it) that for Bayesian agents, the RP is a special case of the fact that the sequence of continually updated credences forms a martingale whose filtration is defined by the information one is updating on the basis of.

Indeed, martingale considerations give us the following generalization of RP:

• (ERP) For any future time t, assuming I am certain that I will remain rational, my current credence in p should equal the expected value of my credence at t.

(Van Fraassen himself formulates ERP in addition to RP.) In RP, it is assumed that I am certain that my credence at t will be r, and of course then the expected value of that future credence is r. But ERP generalizes this to cases where I don't know exactly what my future credence will be.

But we can get an even further generalization of RP. I understand that ERP and RP apply when there is a specific future time at which one knows what one's credence will be. But suppose instead we have some method of determining a variable future time T. The one restriction on that determination is that it can only depend on the data available to us up to and including that time. For instance, we might not know exactly when we will perform some set of experiments in the next couple of years, and we might let T be a time at which those experiments have been performed. The generalization of ERP then is:

• (GERP) For any variable future time T in a future human life bounded by the normal bounds on human life and such that whether T has been reached is guaranteed to be dependent only on data gathered up to time T, my current credence in p should equal the expected value of my credence at T.

This follows from Doob's optional sampling theorem (given that human life has a normal upper bound of about 200 years) and the martingale property of Bayesian epistemic lives.

Now GERP seems like a quite innocent generalization of ERP when we are merely thinking about the fact that we don't know when we will do an experiment. But now imagine a slightly crooked scientist out to prove a pet theory. She gets a research grant that suffices for a sequence of a dozen experiments. She is not so crooked that she will fake experimental data or believe contrary to the evidence, but she resolves that she will stop experimenting as soon as she has enough experiments to confirm her theory—or at the end of the dozen experiments, if worst comes to worst. This is intuitively a failure of scientific integrity—she seems to be biasing one's research plan to favor the pet theory. One might think that the slightly crooked scientist would be irrational to set her current credence according to her expected value of her credence at her chosen stopping point. But according to GERP, that's exactly what she should do. Indeed, according to GERP, the expected value of her credence at the end of a series of experiments does not depend on how she chooses when to stop the experiments. Nonetheless, she is being crooked, as I hope to explain in a future post.