Explain: Bay’s Theorem.

• Bay’s Theorem provides us with the best way to represent how probabilities can be reasoned.

• It allows us to use a supplied conditional probability in both directions.

• It is derived from the product rule, which is written as

                           P (a˄b) = P (a | b) P (b) … (1)
                           P (a˄b) = P (b | a) P (a) …(2)

• Equating RHS of both equation (1) and (2), and dividing by P (a), we get:

·         This equation is called Bay’s Theorem.

·         This rule is very useful in probabilistic inferences.

·         Generalized Bay’s rule is:

Probability of a patient having low sugar has high blood pressure is 50%.

• Here, let M be the proposition ‘Patient has low sugar’.

• Let S be the proposition ‘Patient has high blood pressure’.

• Suppose, we assume that, doctor knows the following 

unconditional fact –

            (i) Prior probability of M = 1/50,000

            (ii) Prior probability of S = 1/20 Then, we have

P (S|M)=0.5

P (M)=1/50,000 P (S)=1/20