#### Subject 9. Bayes' Formula

Bayes' formula ties in nicely with the total probability rule. This rule is:

E(X) = E(X|S)P(S) + E(X|SC)P(SC)

Thus, with the total probability rule, we calculate an unconditional probability (in this case, the probability of event X) by using the fact that we know conditional probabilities of X given other events (S and SC) and their respective unconditional probabilities.

Bayes' formula is used when we know that the event whose probability we have just calculated (i.e., event X) has occurred, and we wish to evaluate conditional probabilities based on this fact.

Thus, Bayes' formula allows us to calculate P(S|X) and P(SC|X), (i.e., conditional probabilities) another way from how they appear in the above formula.

Effectively, we are using Bayes' formula to update our knowledge of a specific event occurring in light of new information received (an event has occurred).

The general formula is:

Updated probability = (probability of the new information given event / unconditional probability of the new information) x prior probability of event.

Typical exam question

An analyst has developed a ratio to identify a company's expectation of experiencing declining PE multiples over time. Research shows that 55% of firms with declining PEs have a negative ratio, while only 25% of firms not experiencing a decline in PEs have a negative ratio. The analyst expects that 15% of all publicly traded companies will experience a decline in PE next year. The analyst randomly selects a company and its ratio is negative. Based on Bayes' theorem, compute the probability that the company will experience a PE decline next year.

This question deals with applying Bayes' theorem to determine the probability that a company will experience a decline in PE.

We need to define the notation to begin with.
X1: PE will decline. P(1) = 0.15
X2: PE will not decline. P(X1) = 0.85
P (B|X1) = 0.55: ratio is negative when PE declines
P (B|X2) = 0.25: ratio is negative when PE does not decline

We are looking for the probability that PE will decline given a negative ratio.

Bayes' theorem can be applied as follows:
P (X1|B) = [P(X1) x P (B|X1)] / Unconditional probability of the new information
= [P(X1) x P (B|X1)] / [P(X1) x P (B| X1) + P(X2) x P (B| X2)]
= [0.15 x 0.55] / [(0.15 x 0.55 + 0.85 x 0.25] = 28%