A classical statistical test controls the probability P(R|H0) that H0 will be falsely rejected. Conventionally this is controlled to be α=0.05 or α=0.01.
( As a small complication, H0 may be a set of hypotheses. α is the probability of rejection of the hypothesis in the set that is most likely to be rejected. A Bayesian may believe some of the hypotheses are more likely than others, but whatever they believe they will still have that P(R|H0) ≤ α. )
A Bayesian, in order to update their belief in the odds of H1 as opposed to H0, P(H1)/P(H0), also needs to know the statistical power, P(R|H1), the probability that H0 will be rejected if it is actually false. The Bayes Factor is then at least P(R|H1)/P(R|H0), and the odds of the competing hypotheses can be conservatively updated as:
P(H1|R)/P(H0|R) = P(R|H1)/P(R|H0) * P(H1)/P(H0)
So we have the curious conclusion that a Bayesian will pay more heed to a statistical test they believe to have high statistical power. If the Bayesian believes that H0 had little chance of being correctly rejected if false, they will be surprised by the rejection but it will not update their beliefs much.
( The classical test sought only to reject H0, not to confirm H1. If the test is rejects H0, perhaps the Bayesian should consider that their H1 was not sufficiently broad an alternative hypothesis. )
Confidence intervals are increasingly accepted as a superior alternative to p-values. A Bayesian argument for this can be given: A confidence interval gives an indication of the accuracy to which an effect has been measured, and hence the statistical power. A Bayesian may use a confidence interval to update their beliefs immediately, whereas they would require further information if only provided with a p-value. ( Leaving aside the technical distinction between confidence intervals and credible intervals, which does not exist for simple tests such as the t-test. )
( The probabilities above are Bayesian -- personal and subjective quantities used to make decisions. If we were talking frequency probabilities, the above would earn a "No!" )