In

three previous posts I discussed Martin Smith's paper "Why throwing 92 heads in a row is not surprising" [

1]. I attempted a Bayesian interpretation of the concept of

**surprise**, but I was sure that this had already been done before; a cursory literature search confirmed this impression (see below). Can one go further? Martin relates surprise to the more general concept of

**normic support**, and the obvious question is whether the latter can also be interpreted in Bayesian terms.

I'll use the example of judicial evidence, that Martin treats in Ref. [

2], where normic support is defined as follows :

a body of evidence E *normically supports* a proposition P just in case the circumstance in which E is true and P is false would be less normal, in the sense of requiring more explanation, than the circumstance in which E and P are both true.

The Blue Bus paradox can then be solved by arguing that testimonial evidence has normic support, while statistical evidence does not ([

2], page 19). To put it in the terms above, finding out that the testimonial evidence is false would

**surprise **us, while the failing of statistical evidence would not.

Can we restate this idea in terms of belief update, as I tried to do for surprise and, in particular, is the distinction between testimonial and statistical evidence similar to that between the coin throw and the lottery examples I drew

here? The proposition P being in both cases "the bus involved was a Blue-Bus bus", we need to identify the evidence E of each type.

**testimonial:** the witness can identify the color of the bus with 90% accuracy.
**statistical: ** 90% of the buses operating in the area on the day in question were Blue-Bus buses.

By quick analogy with the respective coin throw and the lottery examples, respectively, we can then say:

- If the witness is wrong, the result
- calls into question his/her priorly presumed accuracy and prompts us to revise our estimate (Bayesian interpretation.)
- surprises us and requires more explanation (normic support perspective.)

- If the bus is not blue then, although non-blue buses only account for 10% of the total,
- since the Blue-Bus deduction was merely based on the proportion of each type of bus there is no prior knowledge to revise (Bayesian interpretation.)
- the result is unlikely but not abnormal, and thus it does not call for further explanation (normic support perspective.)

I'll discuss in future posts how similar the two interpretations are and whether they solve the paradox (right now my feeling is that they don't, but I need to think about it some more.)

####
Bayesian surprise

A reference on the Bayesian treatment of surprise, defined as the Kullback-Leibler divergence of the posterior distribution with respect to the prior one.

Itti, L., & Baldi, P. F. (2006).

Bayesian surprise attracts human attention. In

* Advances in neural information processing systems* (p. 547–554).

^{1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017.↩}

^{2. Martin Smith, When does evidence suffice for conviction?, Mind (forthcoming.)↩}