At the end of “Borel’s Paradox as a Counterexample to the Law of Likelihood”, I briefly describe three possible remedies to Borel’s paradox as a counterexample to the Law of Likelihood.

In “Remedy 1 to Borel’s Paradox”, I argue that restricting the Law of Likelihood so that it does not apply to the kinds of hypotheses for which Borel’s paradox arises yields a position that seems to be tenable but may not be the best available.

In this post, I consider Remedy 2: where Borel’s paradox arises, evidential favoring is relative to a $\sigma$-field. **Remedy 2 is preferable to Remedy 1** in that it promises to give us *something* defensible that we can say about the evidential bearing of a datum on a pair of probability-zero hypotheses. Moreover, if the proposal to relativize the notion of evidential favoring can be motivated independently of Borel’s paradox, then Remedy 2 is safer than Remedy 1 against the accusation of being *ad hoc.*

However, **Remedy 2 faces at least one objection that Remedy 1 avoids:** it’s not clear what it *means* to say that *E* favors $H_1$ over $H_2$ relative to a σ-field.

**In addition, both Remedy 1 and Remedy 2 face the objection** that, all else being equal, a theory that allows one to evaluate data as evidence with respect to a pair of probability-zero hypotheses would be preferable to a theory that does not. This fact provides motivation for considering Remedy 3, which appeals to an alternative theory of conditional probability.

## Recap of the Counterexample

Take a sphere of circumference 360 miles equipped with an arbitrary system of latitudes and longitudes. Let *E* be the datum that a point *P* randomly selected from a uniform distribution on the surface of the sphere lies within one mile of the intersection of the equator and the prime meridian. Let $H_1$ be the hypothesis that *P* lies on the equator. Let $H_2$ be the hypothesis that *P* lies on the union of the prime meridian and its opposite (the “prime meridional circle”).^{1}

The Law of Likelihood implies that *E* favors $H_2$ over $H_1$, regardless of which great circle is designated the equator and which the prime meridional circle. ^{2}

**This result is wrong** because the distinction between the equator and the prime meridional circle is arbitrary.

## Independent Motivation for Remedy 2

Remedy 2 for this counterexample is to relativize the notion of evidential favoring to a σ-field. **This proposal is safer than Remedy 1 against the accusation of being ad hoc to the extent that it can be motivated independently of Borel’s paradox.**

**Here are two arguments that attempt to provide such motivation.** They are rather weak, but they may be worth something.

**First, Remedy 2 is in some sense already built into the Law of Likelihood.** It is reasonable to assume that the likelihood functions to which the Law of Likelihood refers are to be understood according to Kolmogorov’s theory of regular conditional distributions because that theory is the standard one taught to graduate students and presupposed in most theoretical and applied work. On that theory, probabilities conditional on probability-zero hypotheses are defined only relative to a σ-field. Thus, it is reasonable to assume that the Law of Likelihood is to be understood in such a way that it has a definite application to probability-zero hypotheses only relative to a σ-field.

**This argument has a major weakness.** The major expositors of the Law of Likelihood do not seem to have considered the difficulties that arise in cases involving probability-zero hypotheses. Thus, Remedy 2 was built into the Law of Likelihood *accidentally.* The claim that this fact should give us confidence in Remedy 2 is far from compelling.

**A second argument for relativizing the notion of evidential favoring to a σ-field alleges that in the absence of a σ-field, probability-zero hypotheses are not even toy scientific theories in a proper sense.**^{3} Many philosophers of science have argued that scientific theories are better thought of as “models” than as mere sets of sentences. Models include methods for generating predictions from data. A probability-zero hypothesis does not imply definite probabilities for observations until it is embedded in a σ-field. Thus, according to our best accounts of what it is to be a scientific theory, probability-zero hypotheses in the absence of a σ-field do not qualify.^{4}

**This argument is weak** because it assumes that the Law of Likelihood should apply only to “proper” scientific theories according to a contentious account of what it requires to be such a theory.

Having considered two weak arguments in favor of Remedy 2, let us now consider **two objections** to it.

## Objection 1

Objection 1 to Remedy 2 is that **it is not clear what the Law of Likelihood is supposed to do for us if we adopt it.**

I argue in “Why I Am Not a Likelihoodist” that the Law of Likelihood is not a viable alternative to Bayesian and frequentist methods. Its only value for purposes other than Bayesian updating derives from the fact that it at least renders precise and systematic the informal notion of incremental evidential favoring (see “New Responses to Three Counterexamples to the Likelihood Principle”). I have suggested (e.g. here) that it may not even be compelling as a constraint on methodology. If this claim is correct, then the Law of Likelihood lacks genuine epistemological significance despite being plausibly regarded as true. **It is nothing more than a well-conceived dictionary entry.**

From a likelihoodist perspective, **relativizing the notion of evidential favoring makes this problem even worse.** Even if the Law of Likelihood is nothing more than a well-conceived dictionary entry, it is at least potentially useful to science journalists and historians of science who want to be precise. It’s far from clear that science journalists and historians of science will ever have a use for the notion of “evidential favoring relative to a σ-field.”

## Response to Objection 1

Suppose I am wrong in my general claim that the Law of Likelihood lacks epistemological significance. In that case, there is a reasonable response to Objection 1. An advocate of Remedy 2 can concede that the notion of the degree to which *E* favors $H_1$ to $H_2$ relative to a σ-field Θ has little or no immediate significance. However, it takes on significance as the degree to which *E* favors $H_1$ to $H_2$ *simpliciter* in a context in which Θ is in some sense the “relevant” or “preferred” σ-field. Thus, **while Proposal 2 makes the Law of Likelihood more or less idle for probability-zero hypotheses in the absence of a preferred σ-field, it allows for a sense in which the principle nevertheless applies in such cases and is “ready and waiting,” so to speak, for a preferred σ-field to be specified**.

For what it is worth, **this view seems tidier** than the view Remedy 1 proposes, namely that the Law of Likelihood simply does not apply to probability-zero hypotheses in the absence of a preferred σ-field.

## Objection 2

In my previous post, I discussed the objection that restricting the Law of Likelihood so that it does not apply to the kinds of cases in which Borel’s paradox arises seems to exclude cases that are of genuine scientific interest. Objection 2 is similar: **Remedy 2 makes the Law of Likelihood idle in cases that are of genuine scientific interest.**

## Response to Objection 2

Objection 2 cannot be fatal because a likelihoodist could always concede that the class of cases in which the Law of Likelihood yields useful conclusions is narrower than it is often taken to be. However, the more concessions of this kind he or she makes, the less interesting the Law of Likelihood becomes.

Probability-zero hypotheses often arise in science as point null hypotheses. In my previous post, I suggested that a likelihoodist can take any of the following three lines about tests of such hypotheses:

- The point null hypothesis has positive probability.
- The point null hypothesis is an oversimplified representation of what is implicitly a range null hypothesis (i.e., that the true parameter value lies within ε of the value given by the point null hypothesis for some ε).
- The scientists are doing something that is not of genuine scientific interest.

These responses work for Remedy 2 as well as for Remedy 1. They may not be sufficient to account for every case in which we would like the Law of Likelihood to apply, but they seem to be sufficiently comprehensive that **we need not worry that either Remedy 1 or Remedy 2 limits the effective scope of the Law of Likelihood so severely that it loses most of its interest.**

## Conclusion

For practical purposes, the choice between Remedy 1 and Remedy 2 is inconsequential: **Remedy 2 simply makes the Law of Likelihood idle where Remedy 1 makes it inapplicable.** Given that nothing much hangs on the choice between them, we are free to choose on aesthetic grounds, which seem to me to favor Remedy 2.

There is a concern that both Remedy 1 and Remedy 2 **“merely avoid the logical content of the problem”** (Hill 1970, 45) of comparing probability-zero hypotheses in the absence of a preferred coordinate system. Perhaps a better way of putting this objection is that Remedies 1 and 2 merely compensate for a strange feature of the Kolmogorov theory of conditional probability rather than addressing the problem at its root. This objection motivates Remedy 3, which proposes to use an alternative theory of conditional probability. I will discuss this remedy in a future post.

- As in “Borel’s Paradox as a Counterexample to the Law of Likelihood”, one should remove from each hypothesis one of the two points at which the two circles intersect so that one cannot avoid the counterexample by restricting the Law of Likelihood to mutually exclusive hypotheses. I omit this point here for ease of exposition.
- The Law of Likelihood says that
*E*favors $H_1$ over $H_2$ if and only if $\mbox{Pr}(E|H_1)/\mbox{Pr}(E|H_2)=k>1$, and $k$ measures the degree of that favoring. - Thanks to Alexander Pruss for a comment on my previous post that inspired this argument.
- One might prefer to say not that evidential favoring is relative to a σ-field, but rather that the Law of Likelihood only applies to probability-zero hypotheses
*qua*elements of σ-fields. I regard these views as mere notational variants of one another. I am not thereby committed to regarding Remedy 1 and Remedy 2 as mere notational variants of one another because I intend Remedy 1 to imply that the Law of Likelihood does not apply to $H_1$ and $H_2$ even*qua*embedded in some kind of structure on the hypothesis space.

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