Four Cases and Two Meanings of a Likelihood Ratio

In Transposing Likelihoods, I quoted descriptions of likelihood ratios in four recent cases. The opinion in Commonwealth v. McClellan, No. 2014 EDA 2016, 2018 WL 560762 (Pa. Super. Jan. 26, 2018), is the only one that presents a likelihood as a probability of the evidence given a hypothesis. The opinion refers to “the conclusion that the DNA sample was at least 29 times more probable if the sample originated from Appellant ... than if it originated from a relative to Appellant and two unknown, unrelated individuals.” Evidence that is 29 times more probable under one hypothesis than under some rival hypothesis favors the former hypothesis over the latter.

The other three opinions mischaracterize the likelihood ratio by treating it as a statement about the relative probabilities of the hypotheses themselves. ^1/

To give a simple example of the conceptual difference between a likelihood and a probability, suppose a card is drawn from a well shuffled deck, and ♦ is the hypothesis that it is a diamond. A witness who never makes a mistake informs us that the card is red. The witness’s report certainly is relevant evidence. It is more probable to occur when the card is a diamond than when it is not. The likelihood ratio simply tells us how many times more probable the evidence is for the hypothesis ♦ than for the rival, composite hypothesis of ♥ or ♠ or ♣.

The value of the likelihood ratio in this case is 3. To see this, first consider the probability of this evidence E if ♦ is true. Because all diamond cards are red, this conditional probability is P(E|♦) = 1.

The probability of E if ♥ or ♠ or ♣ is true is the proportion of red cards among the hearts, spades, and clubs. This proportion is 1/3.

The likelihood ratio for diamonds therefore is P(E | ♦) / P(E | ♥ or ♠ or ♣) = 1 / (1/3) = 3. It is three times more probable to learn that the randomly drawn card is red if it is a diamond than if it is not.

It does not follow, however, that the probability that the card is a diamond given that it is red is three times the probability that it is not a diamond given that it is red. Because half of the red cards are diamonds, the conditional probability of a diamond is P(♦ | E) = ½. Thus, the ratio of the probability of the hypotheses of ♦ to non-♦ is only ½ / ½ = 1.  ^2/

Notes

1. The magnitude of the ratio in these cases makes the error resulting from the transposition somewhat academic. See False, But Highly Persuasive: How Wrong Were the Probability Estimates in McDaniel v. Brown?, 108 Mich. L. Rev. First Impressions 1 (2009).
2. Before learning the card's color, the probability it was a diamond was 1/4 (odds of 1:3). After the witness's report, the probability became 1/2 (odds of 1:1). In the terminology of Bayesian inference, the posterior odds of 1:1 are the Bayes factor of 3 times the prior odds of 1:3. That is, 3 x 1:3 = 3:3 = 1:1.