Let’s assume there some people in a group.
20% are from France=
30% are from UK
50% are from Canada
50% of french are boys=
30% of English are boys
25% of Canadians are boys
Obviously If we choose a person randomly the probability of choosing a french person is 20%.
If I tell you that the chosen person is a boy, What would be the probability of the chosen to be french.
Posterior probability is a revised probability that takes into account new available information.
From the symmetry we know
We can also interpret that as
So we can consider as the correction for prior probability which is called “Prior probability”.
as the result of the other information we have about a condition which is known to be the case.
In the example above we are interested in
If we knew the total proportion of boys the answer would be easy but we don’t.
therefore even without knowing the Probability of B, if we know about B’s posterior probabilities, we can answer the question.
2013 (c) Amir H. Ghaseminejad
Here is a nice proof for the fundamental conditional probability formula from:
Hypothesis testing for Bayes:
what is the probability that we can reject H0 and accept H1 at some level of significance (alpha, P)
We get some evidence for the model (“likelihood”) and then can even compare “likelihoods” of different models
We want to know the probability of Hypothesis to be true if Observation is observed.
But what is the probability of H0 and what is the probability H1
So the classical P-Value can be interpreted as probability of H0 if O.
If H0 is 80% probable (or anything less than 1), while Observation has close to 0.05 chance of happening if H0
This means that if H0 is more likely then a more significant observation is necessary.
This means that if H0 is highly probable then we need a very significant observation to refute it.
If H0 is .85 probable in my mind and an observation is only 1% probable based on it, then the probability of its truth is about 5%.