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.
“An Explanation of Borel’s Paradox That You Can Understand” explains how this result arises. To my knowledge, it has not been presented as a counterexample to the Law of Likelihood before.
Three Red Herrings
There are three seemingly promising places to look for a response to this counterexample that in fact lead nowhere.
Redding Herring 1: The Omitted Points
Contrary to what one might suspect, the fact that one point is omitted from each hypothesis plays no role in generating the likelihood ratio of 1.6. Because those points have no length, the same likelihood ratio arises no matter how they are handled.
If anything, one would think that the omitted points would lead to the result that E favors $H_1$ over $H_2$ because $H_2$ omits the point at the center of the region in which E says that P lies, while $H_1$ omits the farther point from that region on the surface of the sphere. But the Law of Likelihood says just the opposite.
I omit one point from each hypothesis only so that one cannot avoid the counterexample by restricting the Law of Likelihood to mutually exclusive hypotheses as I advocate in “New Responses to Three Counterexamples to the Likelihood Principle.”
Red Herring 2: The Small Likelihood Ratio
The misleading likelihood ratio is only 1.6, which is far short of the value of 8 that likelihoodists conventionally require in order to declare a result “fairly strong” evidence (Royall 2000, 761). One might think that this fact somehow excuses the Law of Likelihood. But the fact that the likelihood ratio is small does not help, for two reasons:
- Regardless of what the Law of Likelihood says about the degree of evidential favoring in this case, it still implies the incorrect qualitative claim that the result favors the prime meridional circle hypothesis over the equator hypothesis.
- One can produce an analogous but more dramatic result by using a strange coordinate system. As I explain in “An Explanation of Borel’s Paradox That You Can Understand,” the fact that likelihood ratio in this case is greater than one comes from the fact that lines of longitude are farther apart at the equator than near the poles, while lines of latitude are spaced equally all the way around the sphere. One could get a larger likelihood ratio (as large as one likes, I suspect) by using a system of “pseudo-longitudes” that exaggerates this effect around the prime meridian.
Red Herring 3: Circles with No Width and the Finite Precision of Measurement Techniques
Borel himself points out that actual methods of observation do not allow you to learn that a particular point on a sphere lies on a particular great circle (1909, 102-3). From a position on the prime meridian, you might be able to use astronomical observations and a chronometer to determine that your longitude is between 0.1″ East and 0.1″ West, but you would not be able to determine that your longitude is exactly 0.
In the standard (Kolmogorov) theory, the conditional probability for the location of P given that it lies on the prime meridional circle is the just the conditional probability for the location of P given that the point lies with ε degrees longitude of that circle in the limit as ε goes to 0. This distribution is thus appropriate if it is supposed to reflect one’s uncertainty about the point’s latitude upon learning that it lies on the prime meridional circle from a technique that has a small but finite margin of error.
This line of reasoning is sufficient to defend Kolmogorov’s theory of conditional probability as a suitable tool for updating one’s uncertainty about P‘s latitude given any real measurement of its longitude. But it is not sufficient to defend the use of Kolmogorov’s theory of conditional probability in the Law of Likelihood. There the probability-zero hypotheses upon which one conditions are not idealizations of facts learned by observation, but rather hypotheses one wishes to evaluate in light of some other fact (such as E) learned by observation. Those hypotheses are not associated with any particular measurement technique, so it is not appropriate to regard them as limits of latitude or longitude measurements, but instead to regard them simply as great circles.
The Real Culprit: The Standard Theory of Conditional Probability for Events of Probability Zero
The likelihood ratio of 1.6 arises from the fact that in the standard (Kolmogorov) theory, probability conditional on an event of probability zero is relative to the σ-field in which that hypothesis is embedded (see “An Explanation of Borel’s Paradox That You Can Understand”). To characterize the blue circle in the figure above as an equator and the red circle as a union of two meridians is to impose on the sphere a system of latitudes and longitudes that encodes a particular σ-field that is not given by the problem itself. One would get the opposite result by reversing those characterizations. This feature of the Kolmogorov theory is the source of the counterexample.
I can see three possible remedies for the Law of Likelihood:
- Deny that the Law of Likelihood applies to $H_1$ and $H_2$.
- Maintain that evidential favoring in cases in which Borel’s paradox arises is relative to the σ-field imposed on the hypothesis space.
- Deny that the likelihoods that appear in the Law of Likelihood are conditional probabilities in Kolmogorov’s sense.
None of these remedies provides a cheap and easy fix.
- One challenge for Remedy 1 is to provide a principled basis for excluding $H_1$ and $H_2$ from the scope of the Law of Likelihood without excluding too much.
- One challenge for Remedy 2 is that it’s not clear what the Law of Likelihood is supposed to be doing if we adopt it: I for one do not seem to have an informal notion of “evidential favoring relative to a σ-field” for it to explicate.
- One challenge for Remedy 3 is to explain what the likelihoods that appear in the Law of Likelihood are, if not conditional probabilities in Kolmogorov’s sense. One option is to claim that they are conditional probabilities in the sense of de Finetti’s alternative to Kolmogorov’s theory (1974), but this approach gives rise to other serious difficulties that would need to be addressed (see e.g. this blog post by Larry Wasserman and Chapter 3 of Kadane 2011).1
I will discuss these remedies further in future posts.
Thanks to Teddy Seidenfeld for suggesting that Borel’s paradox might be a problem for likelihoodism. Thanks to Branden Fitelson for directing my attention to Kenny Easwaran treatment of Borel’s paradox in Chapter 8 of his dissertation. The idea for Remedy 2 came from Easwaran’s thesis that conditional probability is relative to a partition.
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