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.

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