Confidence Confusion

by radimentary

“Captain, if you steer the Enterprise directly into that black hole, our probability of surviving is only 2.234%”  Yet nine times out of ten the Enterprise is not destroyed.  What kind of tragic fool gives four significant digits for a figure that is off by two orders of magnitude?

~Why truth? And…

This post poses a basic question about probabilities that I’m confused about. Insight would be appreciated.

It’s inspired by a quick skim I gave to Proofiness, which argues that precise numbers are a powerful persuasion technique.

The Dinosaur Market

One of the CFAR prediction markets was: A randomly selected participant will correctly name seven real species of dinosaurs.

I made a series of suspect Fermi estimate as follows:

Extrapolating from the current bids, half the participants will not bid in this market. If selected, they will just try to name as many dinosaurs as they can. 20% of participants will be able to. 

Half the participants do bid in this market. Among those that bid high on the market, 70% will take the time to study dinosaurs and memorize seven. The other half who bid low will intentionally or unintentionally fail if they’re paying attention. I give them a 10% chance of success.

That comes out to a total of 30%.

There’s a 5% chance of anomalous situations such as one person caring enough to teach people or publicly post dinosaur names. In this case much higher chances of market evaluating to True, say 60%.

I arrived at a probability estimate of .95 * .3 + .05 * .6 = .315.

At this point, I felt obligated to round 31.5% to 30%, and so I bid 30 on the market instead of 31 or 32. Is there a valid reason to do so?

Precision is Confidence

I’ve been focusing on my aversion to report high-precision numbers, even if I believe them to be closer to the truth. When I report 31.5%, I feel more confident than I am.

Teasing out what it means for 31.5% to be more confident 30% the issue is that any number comes with an implicit confidence interval based on the number of significant figures. 30 really means 30 \pm 5, whereas 31.5 really means 31.5 \pm 0.05.

In the absence of explicitly reporting confidence intervals around every probability estimate, 30 thus feels like a more honest report of my actual beliefs. Despite the simultaneous fact that I would buy at any price less than 31 and sell at any price over 32.

While explicitly reporting confidence intervals solves the issue – I’d rather say 31.5 \pm 5 instead of 30 \pm 5 – this strategy seems impractical and carries its own signalling problems.

A Thought Experiment

A large part of my aversion to putting precise numbers on beliefs is a result of the type of error above: in a social setting you cannot just say what you mean, and in particular the number of significant figures also signifies a level of confidence/information. This seems to be the norm: almost every prediction Scott Alexander makes is a multiple of 5 or 10.

Here’s a thought experiment:

Albert and Betty are astronauts sent to study a mysterious coin on Europa and independently transmit short messages back to Earth about the coin’s bias. Albert finds the coin successfully and flips it a thousand times, seeing 531 heads and 469 tails. He concludes the coin is fair and reports 50%.

Betty’s landing capsule collides with a giant teapot in upper orbit and lands several hundred miles away from target. She still has to report her beliefs about the mysterious coin. She has two choices:

  1. Report 50%, because that’s her prior in the absence of information. After all, probability is in the mind. If she does this, however, Albert’s higher confidence is lost in transmission.
  2. Report nothing to convey the fact that she has no information. Any other rationalist can automatically compute her odds are 50-50 anyway – that’s a shared prior.

What would you do? What are the correct norms around sharing probabilities in conversation?

If social norms indeed dictate that significant figures transmit confidence, might it be deceptive to report 31.5 instead of 30 in conversation about the dinosaur market?