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.