# What is the probability of a null-hypothesis being True?

Suppose we are given:
P(reject H0 | H0 is true) = $$\alpha$$ (probability of type I error)
P(don't reject H0 | H1 is true) = $$\beta$$ (probability of type II error)

Assume that we know the power of the test to be
P(reject H0 | H1 is true) = 0.99

Let's say that we didn't reject H0. Can we say anything about the probability of H0 being true in such case? Since H0 and H1 are mutually exclusive, and in the case of not rejecting H0, the probability of H1 is very small (since the power of the test is very large), can we say with 0.99 (99%) probability that H0 is true? Or taking into account the possibility of type I error (with probability $$\alpha$$), can we say with probability $$0.99-\alpha$$ that H0 is true?

Or if the previous is incorrent, is it possible to calculate the probability
P(H0 is true | we didn't reject H0) = ??

Using Bayes formula gives
P(H0 is true | we didn't reject H0) = P(we didn't reject H0 | H0 is true) * (P(H0 is true)/P(we didn't reject H0)).

The probability of complementary event P(¬A|B) = 1 - P(A|B)
So the likelihood on the right side of the bayes formula
P(don't reject H0 | H0 is true) = 1 - P(reject H0 | H0 is true) = $$1- \alpha$$.

Substituting this into right side of the Bayes formula we get
P(H0 is true | we didn't reject H0) = ($$1-\alpha$$) * (P(H0 is true)/P(we didn't reject H0)).
Can we say anything about P(H0 is true) ja P(we didn't reject H0)?

• You can't apply Bayes' formula until you have provided a prior distribution on the hypothesis space.
– whuber
Feb 11, 2021 at 20:53
• Are there recommandations/research on the types of priors one should set to transform hypothesis testing into the probability of the hypothesis given the observed? Feb 12, 2021 at 7:56