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Comment by jmoss20 | Hacker News Reader

Thought experiment: suppose your friend drives 80 miles to visit you. They tell you the trip took between 2 and 4 hours. You have no further information. How confident are you the trip took less than 3 hours?

Now they tell you they maintained a constant speed throughout the trip, a speed somewhere between 20 and 40mph. How confident are you your friend was driving faster than 30mph?

The principle of maximum entropy, applied to each, gives you different answers. P(30mph) = 0.5 implies the trip takes 2hr40mins, not 3hrs. What gives? Which is the real way we should formulate travel times?

See: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) Credit for this example: Michael Titelbaum

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https://bayes.wustl.edu/etj/articles/ambiguity.pdf

This paradox is a good motivator for when Bayesian probability is a useful. Your confidence is a posterior probability which is conditioned on some prior information. Initially you have little prior information, except for an interval of time and distance. When you receive information about the derivative of speed throughout the trip, this meaningfully updates your priors, and so the posterior changes.

The upshot here is that choosing the max entropy distribution as your prior isn't enough, you also need to choose some particular way to formulate the problem. Particular formulations (travel time vs. speed, here) imply different max entropy priors, even though the formulations are equivalent. Worse, there are infinite equivalent formulations, all with different implied max entropy priors.

You can get around this by choosing a non-max entropy prior, like [1], or by deciding on the One True Formulation for your problem. But (Bayesian) updating on the other formulations of the problem won't do it, because there isn't any information in the other formulations -- they're equivalent (by def).

[1]: https://en.wikipedia.org/wiki/Jeffreys_prior

There are more information provided when you mentioned "they maintained a constant speed throughout the trip".

How would you update on it?

(But you could just as well s/constant speed/average speed/, I don't think it makes a difference.)

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