Bayes Rule is uncontroversial and unobjectionable to Frequentists.
What the latter object to is applying the Rule to the parameters of a probability distribution so that, say, the mean of a normally distributed random variable now becomes a random variable itself rather than a real albeit unknown constant. Welcome to Bayesianism.
Then the prior can only be interpreted as degrees of belief in its different values; a deviation from objectivism and pure empiricism represented by the undiluted information contained in a study sample. Since many forms of prior make the maths intractable, it is traditional to use a conjugate prior so that the posterior distribution can be more easily calculated, In other words a fudge on top of a fudged method.
Of course we only hear of Bayesians successes though Nate Silver's recent poll forecasting would suggest that what used to be called inverse probability is due a more sober press.
Biting the hand that feeds IT
