Background

In a recent Slate Star Codex Post (http://slatestarcodex.com/2016/01/02/2015-predictions-calibration-results/), Scott Alexander made a number of predictions and presented associated confidence levels, and then at the end of the year, scored his predictions in order to determine how well-calibrated he is. In the comments, however, there arose a controversy over how to deal with 50% confidence predictions. As an example, Scott has these predictions at 50% confidence, among his others:

Proposition		Scott's Prior	Result
A	Jeb Bush will be the top-polling Republican candidate	P(A) = 50%	A is False
B	Oil will end the year greater than $60 a barrel	P(B) = 50%	B is False
C	Scott will not get any new girlfriends	P(C) = 50%	C is False
D	At least one SSC post in the second half of 2015 will get > 100,000 hits: 70%	P(D) = 70%	D is False
E	Ebola will kill fewer people in second half of 2015 than the in first half	P(E) = 95%	E is True

Scott goes on to score himself as having made 0/3 correct predictions at the 50% confidence interval, which looks like significant overconfidence. He addresses this by noting that with only 3 data points it’s not much data to go by, and could easily have been correct if any of those results had turned out differently. His resulting calibration curve is this:

Scott Alexander's 2015 calibration curve

However, the commenters had other objections about the anomaly at 50%. After all, P(A) = 50% implies P(~A) = 50%, so the choice of “I will not get any new girlfriends: 50% confidence” is logically equivalent to “I will get at least 1 new girlfriend: 50% confidence”, except that one results as true and the other false. Therefore, the question seems sensitive only to the particular phrasing chosen, independent of the outcome.

One commenter suggests that close to perfect calibration at 50% confidence can be achieved by choosing whether to represent propositions as positive or negative statements by flipping a fair coin. Another suggests replacing 50% confidence with 50.1% or some other number arbitrarily close to 50%, but not equa

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Underconfidence

Blog posts

See also