### Measuring the Strength of Evidence

All of the probabilities on this page are prior credences. As done on the preceding page, we’ll drop the subscript ‘1’ to simplify notation.

The non-comparative version of the likelihood principle offers another unique advantage. We can use its two conditional probabilities, pr(E|H) and pr(E|~H), to define a measure of evidential strength. Specifically, the ratio of these two conditional probabilities provides a measure of the degree to which E supports H. This ratio is called the Bayes factor of E relative to H:

 Bayes factor  = pr(E|H) pr(E|~H)

For example, if the evidence is 10 times more likely given H than given ~H, the Bayes factor is 10. The greater the Bayes factor, the stronger the evidence: a proposition with a Bayes factor of 100 (relative to hypothesis H) provides stronger evidence for H than a proposition with a Bayes factor of only 10, for instance. Similarly, propositions with a Bayes factor less than 1 count as evidence against H; and the closer the Bayes factor is to zero, the stronger the evidence is against H.

Be careful not to confuse the Bayes factor with the Bayesian multiplier, defined previously. For comparison, here are both formulas:

 Bayes factor  = pr(E|H) pr(E|~H)
 Bayesian multiplier  = pr(E|H) pr(E)

Only the Bayes factor, not the Bayesian multiplier, provides a measure of evidential strength. The Bayesian multiplier is the number by which you should multiply your prior credence in H upon learning E. Since a large Bayesian multiplier means a large increase in your credence, it might seem to indicate strong evidence. However, the size of the Bayesian multiplier is not a good measure of the strength of the evidence. It is possible for weak evidence to result in a large shift in your credence; and conversely, it’s also possible for strong evidence to result in a small shift in your credence.

To see why, consider an extreme case: suppose evidence E logically entails hypothesis H. Logical entailment is the strongest possible evidential relationship. No evidence can support H more strongly than E does, since E literally guarantees the truth of H. Nevertheless, the Bayesian multiplier might be small, depending on your credence in H prior to learning E. If you were already 95% confident in H before you learned E, your credence can only shift upward by 5%, and the Bayesian multiplier is just barely more than 1. That’s why the Bayesian multiplier isn’t a good measure of evidential strength. The Bayes factor, in contrast, correctly indicates that logical entailment is the strongest possible evidential relationship. If E entails H, then pr(E|~H) is zero, so the Bayes factor is infinite!