# Bayesian Statistics: Properly updating the Prior for new analysis

I have three tables of information about $$A$$ and $$B$$ (gray cells, black font), their row and column marginal totals (black cells white font), and the grand total (white cell black font). The first two tables have information independent from one another, while the cells in third table in the picture are the sums of the two other tables. To their right I have their respective probability tables. I've used Bayes theorem on the first table to get $$P(B|A) = 0.1267605634$$.

Here is my work

I have two questions:

• If I use Bayes theorem on the second table to also compute $$P(B|A)$$, and if I want to use the information from table 1, would the correct prior be $$0.1267605634$$ or $$0.32$$ (i.e. do I simply assign the prior to table 1's posterior)?

• If I use Bayes theorem on the third table alone (the combination of both tables) to compute $$P(B|A)$$, should this conditional probability be identical to the conditional $$P(B|A)$$ I computed for table 2? Why or why not?

In this case, the objective is to derive the joint probability P(A and B) from the conditional P(A|B). Consequently, the correct prior term of the Bayesian numerator is P(B) and NOT the conditional P(B|A) because P(A|B) X P(B|A) ≠ P(A and B). Rather, the correct P(B) to use as the prior for table two is the sum of the updated posterior conditional probability, P(B), from table 1: P(B|A) + P(B|~A). As such, in this case, P(B|A)=0.09/0.71 = 0.13, and P(B|~A) = 0.23/0.29 = 0.79. So, given that A OR $$`$$A occurs, B has an occurrence probability of .13 + .79 = .92.