The Bayes' formula is extremely important in investment applications as it updates the probability when new information arrives in the market.
Let's try to understand it with very simple example. Suppose the probability of the equity market going up is dependent on the result of the elections. If Democrats come in power, then the probability of the market going up is 40 percent, and if Republicans come in power, then the probability of the market going up is 50 percent. If the probabilities of the winning of Democrats and Republicans are 40 percent and 60 percent at the beginning of the year, then the expected value of the probability of market going up is 0.40*0.40 + 0.60*0.50 = 0.46 = 46 percent.
Once the election result is out, and we know that the Republican have won the election, then we will update our probability of market going up to 50 percent.
Derivation of Bayes' formula:
From the multiplication rule of probability, we know that
P(AB) = P(A|B)*P(B)
Also, P(BA) = P(B|A)*P(A)
But P(AB) and P(BA) are the same.
Hence, P(B|A)*P(A) = P(A|B)*P(B)
P(B|A) = [P(A│B)*P(B)]/P(A)
P(A|B) = [P(B│A)*P(A)]/P(B)
Updated probability of event given the new information = [(Probability of new information given event)/(Unconditional probability of the new information)]*Prior probability of event
P(Event| Information) = [P(Information|Event)/(P(Information)]*P(Event)
The updated probability is called as a posterior probability because it reflects the new information.
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Example 8: Updating probabilities using Bayes' theorem
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An IT company is considering expansion into overseas markets. The expansion will occur if foreign policy allows that. If expansion occurs, then they are going to raise their servicing cost. The probability of overseas expansion and increase in price is 0.6 and 0.4 respectively. The probability of overseas expansion given an increase in price is 0.7.What is the probability of an increase in price given that the company expands overseas?
Solution:
We first need to identify and name the events. The best way to solve any probability question is first to see what is being asked in the problem.
Event OA = Overseas expansion
Event IP = Increase in price
We need to calculate P(IP|OA)
Using the Bayes' formula, we get:
P(IP|OA) = P(OA|IP)P(IP)/P(OA)
P(OA/IP) = 0.7
P(IP) = 0.4
P(OA) = 0.6
Therefore, the updated probability P(IP|OA) = (0.7)(0.4)/0.6 = 0.467.
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