Here is my writeup on the "real" difference between frequentist and Bayesian methods: http://stats.stackexchange.com/a/2287/1122

Even more here: http://qr.ae/17BEW

The truth is they both make tradeoffs that can appear ridiculous. In fact, the criticisms of confidence intervals and p-values apply almost exactly, in transpose, to credibility intervals and posterior probabilities.

Confidence intervals and p-values are a worst-case technique. The p-value will always control the false positive rate below alpha, even in the worst input. Sometimes you do want this -- e.g. when we say that the worst-case runtime of QuickSort is O(n^2), that's useful,
even if we do have a prior distribution over the inputs and could also say that the expected runtime is O(n log n). But the errors are correlated across observations. You can have a valid "95%" confidence interval that always produces total nonsense when the experiment ends up with output x, as long as x happens <5% of the time for all possible inputs.

Credibility intervals and posterior probabilities are an average-case technique, where we integrate over the prior. Even if the prior is correct, the errors are correlated across inputs, which can be a problem. In the cookie-jar example at stackexchange, the 70% credibility interval is "wrong" 80% of the time when the jar is type B. That means if you send out 100 "Bayesian" robots to assess what type of jar you have, each robot sampling one cookie, you will expect 80 of the robots to get the wrong answer, each having >73% posterior probability in that wrong conclusion! That's a problem, especially if you want most of the robots to agree on the right answer. The two methods just make different tradeoffs in the way they quantify uncertainty.

My quibble with the cartoon, though, was that it's not really about the frequentist vs. Bayesian debate. If you want to decide whether to take action (like shuttering a satellite) in response to a "YES" output from the instrument, everybody will agree that you need to calculate {rate of events} * {false negative rate} * {cost of false
negative} and compare it with {1 - rate of events} * {false positive rate} * {cost of false positive}.

The frequentist agrees with this math, the Bayesian agrees with this math, and the math doesn't even use Bayes' rule. This is basic actuarial science or decision theory.

The frequentist might do the mechanics in a certain way. They may say they are first going to calculate a p-value, and then ask whether the p-value is less than a threshold alpha, where alpha was set based on the costs and rate of events in order to control the "false discovery rate." And then take action only on a "significant" result.

The Bayesian might do the calculation a little differently too; they could say they are first going to use the expected rate of events as a prior, then calculate the conditional probability that there has been an event (given the instrument's reading), and then multiply this posterior probability by the cost of false negative, and its complement by the cost of false positive, to decide which action has lower expected cost.

But both the frequentist and Bayesian will get the same answer and end up with the same result as somebody who evaluates the inequality above directly. I don't think any technique has a monopoly on the correct answer here.