How explanation guides confirmation

Where E is the proposition that [If H and O were true, H would explain O], William Roche and Elliot Sober have argued that P(H | O&E) = P(H | O). In this article I argue that not only is this equality not generally true, it is false in the very kinds of cases that Roche and Sober focus on, involving frequency data. In fact, in such cases O raises the probability of H only given that there is an explanatory connection between them.