Last week I claimed that the saying "extraordinary claims require extraordinary evidence" is appropriate anytime people too easily make more extreme claims than their evidence can justify. Eliezer, however, whom I respect, thought the saying appropriate anytime people make claims with a very low prior probability. So I have worked out a concrete math model to explore our dispute. I suggest that if you are math averse you stop reading this post now.

Consider the example of power law disasters, where the chance of an event with severity greater than *x* goes as a negative power *x*^-*v*. If we talk in terms of the magnitude *y* = log(*x*) of the disaster, then our prior on y is distributed exponentially as exp(-*v*y*).

Imagine that a claimer gets a signal *s *about our next disaster magnitude *y*, a signal normally distributed with mean *y* and standard deviation *d*. His maximum posterior estimate (MPE) of *y* should rationally be *z = s – d*v*, and his posterior for *y* should be normally distributed with mean* z* and standard deviation *d*. The claimer will claim to us either that his signal was *s’*, or equivalently that his MPE was *z’*. We must then decide what to believe about the distribution of *y* (or *x*) from this claim.

Let us assume that the payoff *U* of this claimer is the sum of two parts: attention *U1* and truth *U2*. First, he gets an attention payoff *U1* that goes as *a*z’*, rewarding him more for claiming that a bigger disaster looms. Second, he gets a truth payoff* U2* that is a logarithmic proper scoring rule. That is, he must declare a probability distribution* p*(*y*) and will be paid as log(*p*(*y*)), using the chance he assigned to the actual disaster magnitude y that occurs. (So *U *= *a*z*(*s’*) + Int_*y* *p*(*y*|*s*) log(*p*(*y*|*s’*)) d*y*.)

Let us also assume that the claimer is not an exact rational agent; the chance he makes any claim *z’ *(or *s’*) is exponential in his total payoff, as in exp(*r*U*), where *r* is his rationality. Finally, assume that we know parameters like *a,d,r,v,* and that the claimer’s signal *s* is well away from the lower boundary of possible *y*, so we can treat integrals over normal distributions as if they went to infinity.

Putting this all together and turning the crank, we find that a claimer with zero attention payoff will claim a *z’* that is normally distributed with a mean of the infinitely rational *z* (or equivalently a *s’* that is normally distributed around a mean of the actual *s*). The standard deviation *c* of this distribution depends on his rationality *r *and the strength of his payoffs *U*. For a claimer with a positive attention payoff, his *z’* (or *s’*) is also normally distributed with the same standard deviation *c*, but the mean of this distribution is biased toward larger disasters, with a bias *b* that is proportional to *a*.

When we hear the claimer’s claim* z’ *(or *s’*), we make an inference about *y*. Our rational posterior for *y* should have a mean *z = z’-b-c*v* and a standard deviation *c*, which depends on both *d* and *r*. That is, we are skeptical about the claim for two reasons. First, we correct for the bias* b* due to the claimer distorting his claims to gain more attention. Second, we correct for the fact that larger disasters are a priori less likely, a correction that rises as the claimer’s rationality falls.

So this model contains effects similar to those Eliezer and I were discussing. My claim was that in practice the attention payoff effect is larger than the irrational claimer effect, but I have not shown that yet. Note, however, that the correction, and the relative size of the two effects, does *not* depend on how a priori unlikely is the claim *z’*.

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