What does your accuracy tell you about your confidence interval?

by HonoreDB 1 min read2nd Nov 201120 comments

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Yvain's 2011 Less Wrong Census/Survey is still ongoing throughout November, 2011.  If you haven't taken it, please do before reading on, or at least write down your answers to the calibration questions so they won't get skewed by the following discussion.

The survey includes these questions:

In the comments, several people including myself wondered what our level of accuracy in the first question said about the calibration of our answer to the second question.  If your guess for the first question was really close to correct, but your probability for the second question was low, were you underconfident?  If you were far off, but your probability was high, were you overconfident?
We could test our calibration by simply answering a lot of these pairs of questions, then applying a proper scoring rule.  But that seems like throwing out information.  Surely we could calibrate faster if we're allowed to use our accuracy as evidence?
I suspect there are people on here with the tools to work this out trivially.  Here's my try at it:

Suppose you state a p-confidence interval of ±a around your guess x of the true value X.  Then you find that, actually, |X - x| = b.  What does this say about your confidence interval?

 

As a first approximation, we can represent your confidence interval as a claim that the answer is uniformly randomly placed within an interval of ±(a/p), and that you have guessed uniformly within the same interval.  If this is the case, your guess should on average be ±(1/3 * a/p) off, following a triangular distribution.  It should be in the range (1/3 ± 3/16)(a/p) half the time.  It should be less than 1/3(3 - sqrt(6)), or about .18, 1/3 of the time, and greater than 1-1/(sqrt(3), or about .42, 1/3 of the time.

 

So, here's a rule of thumb for evaluating your confidence intervals based on how close you're getting to the actual answer.  Again, a is the radius of your interval, and p is the probability you assigned that the answer is in that interval.

 

1.  Determine how far you were off, divide by a, and multiply by p.

2.  If your result is less than .18 more than a third of the time, you're being underconfident.  If your result is greater than .42 more than a third of the time, you're being overconfident.

 

In my case, I was 2 years off, and estimated a probability of .85 that I was within 15 years.  So my result is 2/15 * .85 = .11333...  That's less than the lower threshold.  If I find this happening more than 1/3 of the time, I'm being underconfident.

 

Can anybody suggest a better system?

 

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