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.

## Leave a Reply