A Bayes factor of 3 indicates what about the relative likelihood of the hypotheses given the data?

• A. The data are three times more likely under the null hypothesis than the alternative.
• B. The data are equally likely under both hypotheses.
• C. The data are three times more likely under the alternative hypothesis than the null.
• D. The alternative hypothesis is impossible.

Answer: (C)

Think of a Bayes factor as a measure of how the data shift support between two competing hypotheses by comparing how likely the observed data are under each one. It is computed as the likelihood of the data under the alternative hypothesis divided by the likelihood of the data under the null hypothesis.

A Bayes factor of 3 means the data are three times more likely if the alternative hypothesis is true than if the null hypothesis is true. In other words, the observed data favor the alternative by a factor of three, but the evidence is still modest rather than overwhelming.

This is not saying the hypotheses themselves have threefold probabilities; it’s about the relative likelihood of the data under each hypothesis. Other interpretations would correspond to a Bayes factor of 1/3 (data three times more likely under the null), 1 (equal likelihood), or 0 (the alternative has zero likelihood).