A draft of my paper “What Likelihoodists Should (and Should Not) Say About Borel’s Paradox” is here. I am not aware of any previous treatments of this issue. Addressing it is not necessary for the day-to-day use of likelihoodist methods, but it is necessary for giving a completely correct statement of the likelihoodist position. [Read more…]

## The Difference Between Substantive and Statistical Hypotheses

I have found it useful at times to distinguish between “substantive hypotheses” and “statistical hypotheses.” For instance, in this post I consider a proposal to restrict the Law of Likelihood to statistical hypotheses. **But what is the distinction between substantive and statistical hypotheses?** Can it be made precise? Can it serve the purposes for which I would like to use it?^{1} [Read more…]

## Why a Likelihoodist Would Not Want to Use Coherent Conditional Probabilities

Likelihoodists intend for the Law of Likelihood to provide an objective measure of the degree to which a given datum $E$ favors one hypothesis over another. When $H_1$ and $H_2$ are probability-zero hypotheses, the theory of coherent conditional probabilities places no constraints at all on the likelihood ratio $\mbox{Pr}(E|H_1)/\mbox{Pr}(E|H_2)$ that the Law of Likelihood proposes to use for this purpose. Thus, **the Law of Likelihood cannot serve its intended purpose for probability-zero hypotheses if the likelihoods it contains are taken to be coherent conditional probabilities.**

I suspect that this conclusion would lead most likelihoodists either to restrict the Law of Likelihood to hypotheses with positive probability (as I discuss here) or to maintain that the likelihoods that the Law of Likelihood contains come from regular conditional distributions and that evidential favoring for probability-zero hypotheses is relative to a $\sigma$-field (as I discuss here).

More generally, **a likelihoodist might wish to say that the Law of Likelihood provides an appropriate measure of evidential favoring only when (and perhaps relative to a structure in which) the relevant likelihood ratio has a definite, objective value.** This move seems to me rather sensible.

To **share your thoughts about this post**, comment below or send me an email. To use $\LaTeX$ in comments, surround mathematical expressions with single dollar signs for inline mode or double dollar signs for display mode.

## Remedy 4 to Borel’s Paradox: Appeal to Symmetry Considerations

This post continues a series on possible remedies to a counterexample to the Law of Likelihood based on Borel’s paradox. In this post I discuss the possibility of avoiding the counterexample by using symmetry principles to restrict the conditional probabilities that appear in the Law of Likelihood. **This remedy is prima facie appealing, but it runs afoul of an ingenious version of Borel’s paradox due to Kenny Easwaran** (2008, Ch. 8). [Read more…]

## Remedy 3 to Borel’s Paradox: Adopt the Theory of Coherent Conditional Probability

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 to Simple Statistical Hypotheses

One possible response to “Borel’s Paradox as a Counterexample to the Law of Likelihood,” that I did not mention in my post introducing the counterexample is to **restrict the Law of Likelihood to simple statistical hypotheses**, as opposed to substantive hypotheses that merely give rise to simple statistical hypotheses when conditioned upon. I’ll call this response **Remedy 1′** because it is similar to Remedy 1 but proposes a stronger restriction.

An obvious objection to Remedy 1′ is that it is **unnecessary,** at least for the purpose of addressing Borel’s paradox: it proposes a stronger restriction than Remedy 1, which is sufficient to block the counterexample. Moreover, its additional strength is apparently costly because it would prevent us from applying the Law of Likelihood in many cases in which we would like to be able to evaluate the degree to which some data favor one hypothesis over another. For instance, it would prevent Sober from applying the Law to the debate between advocates of evolutionary theory and advocates of intelligent design (see Sober 2008, Ch. 2 and Sober 2004). [Read more…]

## Why Sigma Fields?

The notion of a $\sigma$-field is easy to understand given basic knowledge of elementary set theory. Unfortunately, many texts formally characterize σ-fields without motivating the characterization they give, which makes the notion harder to grasp that it needs to be.

A $\sigma$-field $\mathcal{F}$ is simply a class of sets with **three properties**. $\sigma$-fields are important for probability theory because **those properties ensure that $\mathcal{F}$ contains all of the sets to which the axioms of probability refer** when applied to elements of $\mathcal{F}$.

## Remedy 2 to Borel’s Paradox: Evidential Favoring is Relative to a Sigma Field

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.*

## 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…]

## Borel’s Paradox as a Counterexample to the Law of Likelihood

## The Counterexample

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,**to a degree that is small but not negligible: the relevant likelihood ratio is about 1.6. [Read more…]

*E*favors $H_2$ over $H_1$