## Wednesday, January 21, 2015

### Fear of vacillation

Suppose I assign a credence r>1/2 to some proposition p, and I am a perfectly rational agent (and know for sure that I am). I will diametrically change my mind about p provided that at some future time my credence in p will be as far below 1/2 as it is now above it. I.e., I will diametrically change my mind provided that my credence will get at least as low as 1−r.

Of course, I might diametrically change my mind, either because I am wrong and will get evidence to show it, or because I will get misleading evidence. But it it would be worrisome if rational agents frequently diametrically changed their minds when they were already fairly confident. Fortunately, it doesn't actually happen all that often. Suppose my credence is some number r close to 1. Then my credence that I am wrong is 1−r. It turns out that this is quite close to the upper bound on my probability that I will diametrically change my mind. That upper bound is (1−r)/r (easy to prove using that fact that a perfect Bayesian's credence are a martingale), which for r close to 1 is not much more than 1−r.

So if my credence is close to 1, my concern that I will diametrically change my mind is about the same as my concern that I am wrong. And that's how it should be, presumably.

But perhaps I am more worried that one day I will come to suspend judgment about something I am now pretty confident of, i.e., maybe one day my probability will drop below 1/2. The same martingale considerations show that my probability of this danger is no more than 2(1−r).

Thus, if my current credence is 0.95, my probability that I will one day drop to 1/2 is at most 0.10 while my probability that I will one day drop to the diametrical opposite of 0.05 is at most 0.05/0.95=0.053. If I am a perfect Bayesian agent, I can be pretty confident that I won't have a major change of mind with respect to p if I am pretty confident with respect to p.

So if we are confident we have the truth, we should not be afraid that we will lose it as long as we remain rational.