In textbook statistical tests, we usually calculate the probability of observing the data we observed given that the null hypothesis is true, i.e. $P[D|H_0]$. If this probability is small (e.g. $<0.05$), we claim that the null hypothesis is unlikely given the data, i.e. we reject the null hypothesis. That is, we claim that because $P[D|H_0]$ is small, $P[H_0|D]$ is also small. But that is not generally true.
As an example, this is similar to the following line of reasoning that leads to an incorrect conclusion: If a person is an American, he is probably not a member of Congress. This person is a member of Congress. Therefore, he is probably not an American. (Pollard & Richardson, 1987)
And yet, rejection of the null hypothesis based on small p-values appears to be both widely taught and widely used. Why? What are the assumptions on the Bayesian priors that allow us to reject the null hypothesis based on the low likelihood of the data under the null hypothesis?