Hey everyone,
In the linked exercise I was asked to prove Bayes Theorem for a finite sequence by taking the case i=1 and prove via induction.
As a first step I already proved the case i=1, which I did in the "classical way":
$$P(A \cap B) = P(B \cap A) = P(B|A)P(A) = P(A|B)P(B)$$ which leads to $$P(A | B) = \frac{P(B|A)}{P(B)}P(A)$$
My intuition is as follows: We can think of $$P(A | B) = \frac{P(B|A)}{P(B)}P(A)$$ as P(A) multiplied by a factor which makes A more or less likely depending on the relative probability of B|A with regards to B.
For multiple signals this intuition naturally extends to the above given formula. What I have problems with is a formal way of how to prove the question given induction.