###### Last week’s post in which I presented a purported counterexample to the Law of Likelihood (due to Mike Titelbaum) generated a lot of interest. In this post I comment on the counterexample. Next week I plan to “go meta” by commenting on the purpose of the Law of Likelihood and of counterexamples such as Mike’s.

# Recap of the counterexample

The Law of Likelihood says that evidence $E$ favors hypothesis $H_1$ over hypothesis $H_2$ if and only if $\Pr(E|H_1)>\Pr(E|H_2)$.

Here is Mike’s counterexample, as he presented it to me in personal correspondence (shared with permission).

We’re playing Hearts (with a standard deck). At the beginning of the game Branden [Fitelson, who was also involved in the discussion,] passes me one card face down. I hate scoring any points in Hearts. It turns out that the Two of Hearts hardly ever yields points for its bearer, so if I receive the Two of Hearts from Branden I’m mildly annoyed. However if I receive a different heart, or the Queen of Spades, I’m really pissed off.

Now suppose you catch a glimpse of the card Branden passes me, and see only that it’s a heart. That’s your evidence; the two (mutually exclusive) hypotheses are that I’m mildly annoyed or that I’m really pissed off.

Mike claims that, intuitively, the fact ($E$) that the card is a heart favors the hypothesis ($H_1$) that he is really pissed off over the hypothesis ($H_2$) that he is mildly annoyed. However, **the Law of Likelihood says the opposite:** it says that $E$ favors $H_2$ over $H_1$ because $\Pr(E|H_2)=1>\Pr(E|H_1)=12/13$.

# Poll Results

I ran polls on both my site and *Choice and Inference* to get a sense of how people react to Mike’s example. These polls are, of course, wildly unscientific. The readership of the two sites on which they were posted are far from representative of the general population, and those who responded to the polls are likely unrepresentative of the readership. I did not give my thoughts on the counterexample in my original post for the sake of not affecting the poll results. However, regular readers of my site will have been exposed to my assessment of the Law of Likelihood (namely that it holds up rather well as a conceptual analysis or explication of “evidential favoring” but does not function well as the basis of a self-contained methodology for science).

The question I asked was, “Does the Law of Likelihood give the correct result in Titelbaum’s example?” with possible answers “yes” and “no.” It occurs to me now that the poll is flawed in that it does not allow us to distinguish between “no” voters who think that the Law of Likelihood yields a determinately incorrect result and “no” voters who think that it fails to give a determinately correct result only because there is no determinately correct result.

Despite these flaws in the poll design, I think it fair to conclude from the results that **Mike’s intuition is widely shared and probably more common** than the intuition that agrees with the Law of Likelihood. On my site, five readers said “yes” (in agreement with the Law of Likelihood) and six said “no” (in agreement with Mike). On *Choice and Inference*, none said “yes” and five said “no.” Overall, five said “yes” and eleven said “no.”

Again, these results are to be taken with a grain of salt. Moreover, their relevance to the evaluation of the Law of Likelihood depends on one’s view about what the Law of Likelihood is supposed to do for us. I will discuss this issue further next week.

# Why arguments from the Likelihood Principle to the Law of Likelihood are inadequate

The Law of Likelihood is almost a corollary of the Likelihood Principle, for which there are what I take to be compelling proofs. Almost, but not quite. The Likelihood Principle says that the evidential meaning of a datum $E$ with respect to a partition **H** depends only on the likelihood function $\Pr(E;H)$ considered as a function of $H$ over **H**. Given this principle and the assumption that there is such a thing as evidential favoring, it does not take much to argue for the Law of Likelihood **as applied to a pair of hypotheses $H_1$ and $H_2$ such that $H=[H_1,H_2]$**. However, generalizing this argument to all **H** requires the following *Non-Contextuality Principle*: the probabilities that other hypotheses in **H** other than $H_1$ and $H_2$ ascribe to $E$ are irrelevant to the evidential meaning of $E$ with respect to $H_1$ and $H_2$.

There are plausible Bayesian rivals to the Law of Likelihood that are contextual (i.e., violate the Non-Contextuality Principle). Assume (controversially) that if $E$ confirms $H_1$ more than $H_2$, then it favors $H_1$ over $H_2$. Then, for instance, the likelihood-ratio measure of confirmation leads to a contextual account of favoring. It implies that $E$ favors $H_1$ over $H_2$ if and only if $\Pr(E|H_1)/\Pr(E|\sim H_1)>\Pr(E|H_2)/\Pr(E|\sim H_2)$. Whether or not this criterion is satisfied can depend on $\Pr(E|H’)$ for $H’\in\textbf{H}\setminus[H_1,H_2]$. (This particular criterion agrees with Mike’s intuition and disagrees with the Law of Likelihood on Mike’s example.)

**The Non-Contextuality Principle seems to me intuitively appealing, but not sufficiently compelling to warrant throwing out Bayesian rivals to the Law of Likelihood that violate it.**

# Why I don’t find the example persuasive

My immediate reaction to (an earlier version of) Mike’s example was that the Law of Likelihood does seem to get it wrong. On further reflection, however, I came to think that the Law of Likelihood’s verdict is at least defensible. Indeed, if I had to pick a side I would agree with the minority of respondents to my polls who claimed that the Law of Likelihood does give the correct verdict. Here are a few arguments for this claim.

## 1. Judgments of favoring that accord with LL have nice interpretations that apply in this case

When the prior odds for a pair of hypotheses $H_1$ and $H_2$ are well defined, **the Law of Likelihood says that $E$ favors $H_1$ over $H_2$ if and only if updating by Bayesian conditioning on $E$ raises the odds of $H_1$ to $H_2$**. In Mike’s example, the prior odds for $H_1$ and $H_2$ are well defined. The Law of Likelihood’s verdict that $E$ favors $H_2$ over $H_1$ reflects the fact that $\Pr(H_2|E)/\Pr(H_1|E)>\Pr(H_2)/\Pr(H_1)$.

Alternatively (as Jake Chandler pointed out), the Law of Likelihood says that $E$ favors $H_1$ over $H_2$ if and only if updating by Bayesian conditioning on $E$ raises the probability of $H_1$ conditional on the disjunction of $H_1$ and $H_2$.

It seems rather natural to identify favoring with odds-raising, at least in cases in which the odds are well defined. Indeed, it seems odd to say that $E$ lowers the odds of $H_1$ to $H_2$ yet favors $H_2$ over $H_1$ (perhaps more odd than to say that $E$ favors $H_1$ over $H_2$ yet confirms $H_2$ more than $H_1$). Moreover, identifying favoring with odds-raising keeps the notions of favoring and belief clearly distinct yet connected in a way that makes relatively clear what the notion of favoring is *for*.

## 2. Only possibilities consonant with $H_1$ are ruled out

As Jake Chandler pointed out, the datum $E$ that the card in Mike’s example is a heart **rules out one possibility consonant with Mike being really pissed off** (the possibility that the card is the queen of spades) **and no possibilities consonant with Mike being mildly annoyed**. All of the other possibilities it rules out are neither here nor there with respect to those hypotheses. How then could it favor “pissed off” over “annoyed?”

## 3. Non-LL intuitions can be explained away

There is an easy explanation for Mike’s intuition that $E$ favors $H_1$ over $H_2$: the posterior probability of $H_1$ is quite high, while that of $H_2$ is rather low. But evidential favoring is about the *change* in relatively believability (as measured by odds?) rather than posterior relatively believability. Perhaps, as Shivaram Lingamneni suggested, Mike’s intuition is an instance of some kind of cognitive bias.

# Next post: what’s the point?

Given these arguments, it seems to me quite plausible that the Law of Likelihood actually yields the correct verdict in Mike’s case. But I do feel the pull of Mike’s intuition to some extent. How are we to decide whether to accept or reject Mike’s intuition? What difference does it make? Are we just arguing about words? I’ll get into these issues next week. In the process, I’ll respond to Branden Fitelson’s and Jonathan Livengood’s suggestion that the way to make headway is to clarify the notion of evidential favoring that we are trying to analyze or explicate.

#### Is there a point to debating examples like Mike’s? If not, why not? If so, what is it?

Thanks to Mike Titelbaum for the example and permission to share it.

To share your thoughts about this post, comment below or send me an email.

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The image at the top of this post was created by **Liko81**. It is licensed under the Creative Commons Attribution 3.0 Unported license. Its use here does not suggest that its creator endorses any of the views expressed here.

Shivaram Lingamneni says

In passing, it seems that I. J. Good [suggested](http://en.wikipedia.org/wiki/Bayes_factor#Interpretation) that Bayes factors of less than half a deciban are undetectable by unaided human intuition:

>A deciban or half-deciban is about the smallest change in weight of evidence that is directly perceptible to human intuition. I feel that it is an important aid to human reasoning and will eventually improve the judgements of doctors, lawyers and other citizens.

although it doesn’t seem like this observation was made with the support of formal psychological investigation.

A deciban and a half-deciban correspond to likelihood ratios of 1.25 and 1.12 respectively, and the likelihood ratio in the example is only 1.08. So maybe this is additional evidence that the example induces a breakdown of intuition, or is outside the domain where intuition is reliable.

Greg Gandenberger says

Good point, Shivaram. It suggests that if the Law of Likelihood is true, then (barring cognitive illusion) one should expect people to have the intuition that the evidence is basically neutral between the two hypotheses, without necesssarily having the intuition that it (weakly) favors $H_2$ over $H_1$ (at least when they are relying on “direct intuition” as opposed to, say, the fact that the evidence only eliminates possibilities consonant with $H_1$).

Of course, this point doesn’t help against the intuition that the evidence somewhat strongly favors $H_1$ over $H_2$.

Shivaram Lingamneni says

Thanks, that helps. I think we’re working with different conceptions of what it would mean for the Law of Likelihood to be true. I don’t think it’s true in the sense that it accounts for all our pre-theoretic intuitions about evidential favoring. Rather, I think that (at least in paradigm cases like this one, where the priors, likelihoods, and posteriors all have unambiguous frequency interpretations) it is true in a way that outruns, and should normatively replace, our pre-theoretic intuitions. I may have been missing the point of much of the discussion 🙂

Greg Gandenberger says

That’s helpful. Some philosophers distinguish between “conceptual analysis” and “explication” in roughly the same way: conceptual analysis puts the emphasis on capturing pre-theoretic intuitions, while explication puts the emphasis on fruitfulness.

I care a lot more about fruitfulness than I do about capturing pre-theoretic intuitions. But if an explication says something wildly at odds with pre-theoretic intuitions, then that fact might lead you to worry about how fruitful it could be.