A Bayesian fragment of little value:
What if P(B|not A) ≠ 0?
Now we CANNOT just say A exists given B.
What we'd like is P(A|B): the probability of A given B.
This leads us to the important subject of Bayesian analysis.
Let's make A and B general for this subsection and return in the example subsections below to our special meanings: A (something we may need to exist) and B (we exist):
Bayes' theorem gives a way to find P(A|B) in principle and its really simple in principle.
Say that P(AB) is the joint probability of events A and B. Then "obviously", we get Bayes' theorem
where P(A) and P(B) are the probabilities, respectively, of A and B, P(A|B) is the conditional probability of A given B, and P(B|A) is the conditional probability of B given A.
If Bayes' theorem still looks incredible, just imagine pulling all the B's out of statistical population, then all the A's out of the statistical sub-population of B's, then clearly P(AB) = P(A|B)P(B) and, mutatis mutandis, P(AB) = P(B|A)P(A).
Bayes' theorem is often written in the forms
Bayes' theorem is an exact general probability result. The application of Bayes' theorem is Bayesian analysis.
In nontrivial cases of Bayesian analysis, P(A) and P(B) are NOT as simple as they look.
Really they should be written P(AK) = P(A|K)P(K) and P(BK) = P(B|K)P(K) where K is all your background knowledge. So with K we have for Bayes' theorem
Now what if A is theory? It does seem weird to calculate the probability of a theory since you might say a theory should be just true or false.
The weirdness goes away when you understand that Bayes' theorem (as usually applied) concerns what we know about reality and NOT what reality is.
So P(A|BK) is the conditional probability of theory A given the everything we know about reality (i.e., BK)
In fact, Bayesian analysis is what everyone has done forever qualitatively---"This is probably true based on everything we know now." etc.