Suppose one were to draw a point P at random from a uniform distribution over the surface of a sphere equipped with lines of longitude and latitude with a circumference of 360 miles. Consider following pair of hypotheses:
- $H_1$: P lies on the equator (0 latitude), omitting the point at which the equator intersects the line of 180 degrees longitude.
- $H_2$: P lies on the “prime meridional circle,” that is, the great circle consisting of the prime meridian (0 longitude) and its opposite (180 degrees longitude), omitting the point at which the prime meridian intersects the equator.
Now suppose one learns the following:
- E: P lies within a one-mile radius of the intersection of the equator and the prime meridian.
Intuitively, E is evidentially neutral between $H_1$ and $H_2$. After all, neither the datum E nor the setup of the problem distinguishes between the equator and the prime meridional circle. But according to the Law of Likelihood, E favors $H_2$ over $H_1$ to a degree that is small but not negligible: the relevant likelihood ratio is about 1.6. [Read more…]