### Theorem

Branden Fitelson recently brought the following result to my attention.

Theorem:At least one of the following claims is false.

Qualitative Law of Likelihood:$E$ favors $H_1$ over $H_2$ if and only if $\Pr(E|H_1)>\Pr(E|H_2)$.Likelihood-Ratio Measure of Confirmation:The degree to which $E$ confirms $H$ is $\Pr(E|H)/\Pr(E|\sim H)$.Favoring-to-Confirmation Bridge Principle:If $E$ favors $H_1$ over $H_2$, then it confirms $H_1$ more than $H_2$.

### Proof

Here is a case in which $E$ favors $H_1$ over $H_2$ according to the Qualitative Law of Likelihood but confirms $H_2$ more than $H_1$ according to the Likelihood-Ratio Measure of Confirmation:

- $\Pr(E|H_1)=3/4$
- $\Pr(E|H_2)=29/40$
- $\Pr(E|\sim H_1 \mbox{ & } \sim H_2)=1/8$
- $\Pr(H_1)=1/10$
- $\Pr(H_2)=2/5$

The existence of such a case entails that if the Qualitative Law of Likelihood and the Likelihood-Ratio Measure of Confirmation are true, then the Favoring-to-Confirmation Bridge Principle is false. Therefore, the three principles cannot all be true.

### Discussion

I think the arguments for the Qualitative Law of Likelihood are quite strong (as I’ve argued here and here), at least when $H_1$ and $H_2$ are simple statistical hypotheses with respect to $E$. It has good Bayesian credentials in that it says that $E$ favors $H_1$ over $H_2$ if and only if conditioning on $E$ causes the odds of $H_1$ against $H_2$ to increase.

I also like the Likelihood-Ratio Measure of Confirmation. It seems like the natural way to extend the quantitative Law of Likelihood (which says that $\Pr(E|H_1)/\Pr(E|H_2)$ is the degree to which $E$ favors $H_1$ over $H_2$) to provide an account of degree of confirmation. It is relatively popular among confirmation theorists. (See e.g. Fitelson 2001 for discussion.)

But the Favoring-to-Confirmation Bridge Principle also seems quite natural. Moreover, it is consistent with a two-way bridge principle that would allow one to reduce the notion of favoring to that of “confirming more than.”

**The following options are all more appealing to me than giving up the Law of Likelihood:**

- Deny (as many likelihoodists do) that there is such thing as degree of confirmation, thereby denying the Favoring-to-Confirmation Bridge Principle.
- Deny the Favoring-to-Confirmation Bridge Principle without denying that there is such a thing as degree of confirmation.
- Drop the Likelihood-Ratio Measure of Confirmation in favor of some other measure such as the Ratio Measure $\Pr(H|E)/\Pr(H)$ that is compatible with the Qualitative Law of Likelihood and the Favoring-to-Confirmation Bridge Principle.

**I’m open to all three of these options, with a tentative preference for Option 2.** The Favoring-to-Confirmation Bridge Principle sounds right, but the idea that our best account of favoring and our best account of confirmation would be inconsistent with it does not seem crazy. It would be nice to have a simpler and more unified approach in which favoring reduces to “confirms more than,” particularly if you think of this kind of project in terms of a “best systems” approach in which the goal is to find the system of concepts that achieves the best combination of simplicity and strength,^{1} but I’m not at present willing to give up either the Law of Likelihood or the Likelihood-Ratio Measure of Confirmation on those grounds.

**Question:** Which of the three claims under consideration would you deny?

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.

- Thanks to Branden Fitelson for this point. ↩

Branden Fitelson says

Thanks for another great post, Greg!

As you know, I’m inclined to reject the Law of Likelihood (LL), and retain all the other principles you mention.

I’m sure we’ll be talking more about this in the coming year. But, for now, I’d like to suggest a possible path toward détente.

I’m starting to suspect that one reason (LL) seems so attractive to people in certain cases (e.g., in cases involving a contrast of simple statistical hypotheses) is that these tend to be cases in which it is natural to (implicitly) presuppose something like a “principle of indifference” in the background. That is, these tend to be cases in which we want to “treat the hypotheses equally, a priori (i.e., prior to the observation of statistical evidence E)”. And, so long as hypotheses are (a) mutually exclusive, and (b) equiprobable a priori, there can be no conflict between (LL) and a likelih0od-ratio confirmation measure + Favoring-to-Confirmation Bridge Principle explication of “favoring”.

Be that as it may, I, for one, find the sorts of examples you cite above (which are the relevant kinds of cases for a *contrast* between LL and LR+FTCBP) much less compelling from an (LL) perspective. And, the cases that do seem compelling to me are given the same verdicts by LR+FTCBP.

Greg Gandenberger says

Thanks, Branden. That’s a good thought. I accept a similar claim, namely that the Law of Likelihood seems like (and in fact is) a useful method in particular cases because equal prior probabilities seem reasonable in those cases.

It might be helpful to look at the prior and posterior probabilities and the prior and posterior odds for the case I gave:

The fact that the odds for $H_1$ against $H_2$ decrease upon conditioning on $E$ makes me inclined to say that $E$ does favor $H_1$ over $H_2$. I don’t have a clear intuition about whether $E$ confirms $H_1$ more than $H_2$ or vice versa. I certainly don’t find it strongly counterintuitive to say that $E$ confirms $H_1$ more than $H_2$. The idea that favoring and comparative confirmation can come apart in this way is a bit surprising initially, but I find that I’m getting used to it.

At any rate, I tend to think that we should view testing explications against intuitions about cases as just one means toward the end of evaluating the usefulness of those explications for the purposes that we want use their explicanda to serve. (See this blog post I wrote for Deborah Mayo.)

Neither LL (the claim that $\Pr(E|H_1)/\Pr(E|H_2)$ measures evidential favoring) nor Ll (the claim that $[\Pr(E|H_1)/\Pr(E|\sim H_1)]/[\Pr(E|H_2)/\Pr(E|\sim H_2)]$ measures evidential favoring, in accordance with the likelihood-ratio measure of confirmation and a confirmation-to-favoring bridge principle) can do what we would really like it to do for us, namely to tell us what we ought to believe or do about a pair of hypotheses without reference to a prior belief state. But LL at least tells us in a straightforward way what we ought to believe given a prior belief state when Bayesian conditioning is appropriate (just multiply the prior odds by the degree of favoring). I think that it also suggests a reasonable constraint on frequentist testing methods, as I argue in a revised draft of "Why I Am Not a Likelihoodist" that I am currently developing.

What can Ll do for us? It fits nicely with the appealing likelihood-ratio measure of confirmation, but I’m not inclined to regard that fact by itself as a great benefit.