Bayes Theorem
Bayes Theorem (Thomas Bayes 1761) allows calculation of true probability based on a sample of prior knowledge and a base rate^{1}. Seeing evidence should update prior beliefs, i.e. it should restrict the space of possibilities and the ratio you need to consider after that ^{2}.
Think about a base rate for the Hypothesis P(H)
(e.g. what is the ratio of librarians to farmers, say 1:20). Now consider seeing evidence (e.g. the person is meek and reserved), in the case where the hypothesis is true, let’s say that among librarians that would be 40% or 4 out of 10 librarians this is P(EH)
^{1}. Among farmers the rate is lower say 1/10 this would be P(E¬H)
. In order to find P(HE)
or the probability of the hypothesis given that we see the evidence you have to consider what is the base rate times the rate of seeing the evidence (P(H)P(EH)
) divided by the total group that fits the evidence P(H)P(EH)
and P(¬H)P(E¬H)
.
\begin{equation} P(AB)=\frac{P(BA) * P(A)}{P(BA)P(A)+P(\neg{A})P(B\neg{A})} \end{equation}
Simplified:
\begin{equation} P(AB)=\frac{P(BA) * P(A)}{P(B)} \end{equation}
P(A)
= Prior (base rate, probability that the hypothesis is true)P(B)
= Total probability of seeing the evidenceP(AB)
Posterior: Given Hypothesis, and some New Evidence, what is the probability that the hypothesis is trueP(BA)
= Likelihood: Probability that we would see this evidence give that the hypothesis is true


means given that, so P(EH) is Probability of seeing the evidence given that hypothesis is trueReferences