## Tuesday, January 20, 2015

### A natural hierarchy of levels of credence

Let a0=1/2. Let an=1−(1−an−1)2. Thus, approximately, the first couple of values are: a0=0.5, a1=0.75, a2=0.938, a3=0.996, a4=0.99998 and a5=0.9999999998.

Think of these as thresholds for credences. There is an interesting property that the items in this sequence have: If a perfect Bayesian rational agent assigns credence an to a proposition p, then she has credence at least an−1 that her credence in p will never fall below an−1. Thus, the agent who assigns credence a1=0.75 to p thinks it's at least as likely as not (a0=0.5) that her credence will always stay at the level of being at least as likely as not. Moreover, it is easy to show that no lower values for an have this property.

So each of these thresholds for belief has the property that it gives the previous degree of confidence in the belief not dipping below the previous degree of confidence. This is a pretty natural property, and it seems like it would provide a good way of dividing up our confidence in our beliefs. Credence from a1 on indicates, maybe, that "likely" the proposition is true. From a2 on, maybe we can be "fairly confident". From a3 on, maybe we can be "quite confident". The level a4 gives us "pretty sure", while a5 maybe makes us "sure", in the ordinary non-philosophical sense of "sure".

How we label the thresholds is a matter of words. But taking these thresholds as natural division points may be a good way to organize beliefs.

Note that each of these thresholds requires approximately twice as much evidence as the preceding one, if we measure the amount of evidence according to the logarithm of the Bayes factor.