*…at least, you can understand it if you understand the notion of a limit. Or so I think. If not, then I’m to blame!
Consider a sphere equipped with lines of latitude (red) and longitude (blue):
Suppose we take a point at random from a uniform distribution over the surface of that sphere (i.e., a distribution that makes the probability that the point lies within a particular region proportional to that region’s area).
Now suppose we learn that the point lies on the union of the “prime meridian” (0 longitude) and its opposite (180 degrees longitude):
You might think that the probability distribution for the point over this circle would be uniform, but in fact it is greater around the equator than around the poles.
This result comes from a procedure that amounts to conditioning on the statement that the point’s longitude is between -ε and ε for some ε (illustrated for the case ε=π/36)…
…and taking the limit as ε goes to zero.
The resulting region is wider, and thus bigger, near the equator…
…than near the poles…
…and this inequality persists in the limit:
Now suppose that instead of learning that the point lies on the prime meridian, we had learned that it lies on the equator:
The probability distribution for the point over this circle is in fact uniform.
This result arises from the same kind of asymptotic reasoning as above:
Why It’s a Paradox
The fact that the conditional probability distribution over the prime meridian is different from the conditional probability distribution over the equator is puzzling because because a prime meridian in one coordinate system is an equator in another coordinate system, and vice versa:
Borel’s paradox also gives rise to a previously overlooked counterexample to the Law of Likelihood that I will present in my next post.
We have to choose between learning to live with Borel’s paradox and developing an alternative theory of conditional probability on which it does not arise. There is such a theory, developed primarily by Bruno de Finetti (1974), but it gives rise to other serious difficulties (see e.g. this blog post by Larry Wasserman and Chapter 3 of Kadane 2011).1 I will discuss these issues more in future posts.
Thanks to Teddy Seidenfeld and Satish Iyengar for directing my attention to Borel’s paradox.
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