Logical counterfactuals using optimal predictor schemes
post by Vanessa Kosoy (vanessa-kosoy) · 2015-10-04T19:48:23.000Z · LW · GW · 0 commentsContents
Definition Theorem Note Proof of Theorem None No comments
We give a sufficient condition for a logical conditional expectation value defined using an optimal predictor scheme to be stable on counterfactual conditions.
Definition
Given an error space of rank , the stabilizer of , denoted is the set of functions s.t. for any we have .
Theorem
Consider an error space of rank 2, , a word ensemble and a -optimal predictor scheme for . Assume is s.t.
(i)
(ii)
Consider and , -optimal predictor schemes for . Then, .
Note
This result can be interpreted as stability on counterfactual conditions since the similarity is relative to rather than only relative to . That is, and are similar outside of as well.
Proof of Theorem
We will refer to the previously established results about -optimal predictor schemes by L.N where N is the number in the linked post. Thus Theorem 1 there becomes Theorem L.1 here and so on.
By Theorem L.A.7
On the other hand, by Lemma L.B.3
Combining the last two statements we conclude that
It follows that
0 comments
Comments sorted by top scores.