In posts here, here, and here, I discuss possible responses to a counterexample to the Law of Likelihood based on Borel’s paradox that stay within Kolmogorov’s theory of regular conditional distributions. In this post I discuss the alternative option of abandoning the Kolmogorov theory in favor of the theory of coherent conditional probabilities developed by de Finetti (1974) and Dubins (1975). [Read more…]

## Remedy 1 to Borel’s Paradox: Restrict the Law of Likelihood

In “Borel’s Paradox as a Counterexample to the Law of Likelihood,” I present a new counterexample to the Law of Likelihood and briefly describe three possible remedies for it. Here I motivate the first of those remedies and raise a few worries for it. Those worries do not seem to be fatal, but they provide some motivation for considering other possible responses. [Read more…]

## An Explanation of Borel’s Paradox That You Can Understand*

*…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!

## Borel’s Paradox

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. [Read more…]