In this paper, we show that the conditional frequentist method of
testing a precise hypothesis can be made virtually equivalent to Bayesian
testing. The conditioning strategy proposed by Berger, Brown and Wolpert in
1994, for the simple versus simple case, is generalized to testing a precise
null hypothesis versus a composite alternative hypothesis. Using this strategy,
both the conditional frequentist and the Bayesian will report the same error
probabilities upon rejecting or accepting. This is of considerable interest
because it is often perceived that Bayesian and frequentist testing are
incompatible in this situation. That they are compatible, when conditional
frequentist testing is allowed, is a strong indication that the "wrong"
frequentist tests are currently being used for postexperimental assessment of
accuracy. The new unified testing procedure is discussed and illustrated in
several common testing situations.