From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Sat Feb 01 2003 - 16:10:10 MST
Okay, I've done some more math, and here's where things are currently:
Let's say have a group of mixed A and ~A, where A has real incidence P,
and you think the incidence of A is X: your knowledge for that group is:
(.5 - P)^2 - (P - X)^2
Let's say your priors are correct; you have a group with real incidence P
of A, and you know the incidence is P. There's an observation O which can
help distinguish between A and ~A, and two conditional probabilities M and
N for seeing O given the cases A and ~A.
How much will applying Bayes' Theorem increase your knowledge for the
entire group once the group has been subdivided into O and ~O, with the
associated Bayesian probabilities of A|O and A|~O?
The increase in knowledge - that is, the increase from the old knowledge
to the new knowledge - is:
[p(1 - p)(m - n)]^2/[pm + (1 - p)n][p(1 - m) + (1 - p)(1 - n)]
Again, does anyone know the name of this formula?
(Note that, using the updated definition of knowledge, a dangling factor
of two has dropped out of the formula.)
-- Eliezer S. Yudkowsky http://singinst.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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