Classical v/s Bayesian Hypothesis Testing

Question

This question has 2 parts:

(1) What is the fundamental difference between classical and bayesian hypothesis testing? How do I interpret this difference.

(2) Here is a paragraph quoted from Casella and Berger Statistical Inference (Section 8.2):

I don't understand:

(i) Why is P(H0 is True | X) = {either 0 or 1} ?? --- if I toss a coin I know that I'll get either heads or tails but I do not say that the probability of getting heads is 0 or 1 if the outcome is unknown.

(ii) Why do these probabilities not depend on X?

Answer 1

With the coin tossing, the quantity of interest in the distribution is the probability the outcome is heads/tails. This is a fixed and constant number in the classical paradigm. For a fair coin, this probability is $0.5$ irrespective of whether you get an actual heads or tails (i.e. irrespective of the data X) and $P(H_0: p=0.5|X)=1$.