A talk I recently gave prompted Mike Titelbaum to develop a purported counterexample to the Law of Likelihood. In this post I present the counterexample with minimal comments for the sake of eliciting reactions to it that are not influenced by my thoughts. I plan to follow up with my comments next week.
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 Titelbaum’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 $\Pr(E|H_2)=1>\Pr(E|H_1)=12/13$.
Please provide your reaction to Titelbaum’s example in the poll below. I would love it if you would also explain your response by leaving a comment.
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