Welcome to this Decision Analysis Tutorial for calculating Posterior Probabilities.
Suppose, in recent years, economic growth occurs 40% of the time, and economic decline
occurs 60% of the time. That is, probability of Growth is 0.4
and probability of Decline is 0.6 You’re considering hiring a consultant
who could either give a positive or negative report.
In the past, when there was economic growth,
the consultant’s report was positive 65% of the time
And when there was economic decline,
the consultant’s report was negative 70% of the time.
Notice that we can replace the word “when” with “given” in this sentence.
That is, probability of “Positive given Growth” is 0.65
and Probability of “Negative given Decline” is 0.7
Here is a summary of the given information. The probabilities of the States of Nature
at the top are called Prior Probabilities. That is, the probabilities assessed initially
or at the beginning. The 0.65 and the 0.7 probabilities are called
“Conditional Probabilities”. They represent the success history of the consultant.
To identify them, note that these probabilities are “given” the outcomes, Growth and Decline.
Whereas the posterior probabilities are “given” the consultant’s report.
To calculate the posterior probabilities, we will create tables for the consultant’s
reports – one for Positive and one for Negative. These table have rows representing the outcomes
or states of nature. The prior probabilities are 0.4 for Growth
and 0.6 for decline, for both tables.
The probability of Positive given growth is 0.65
which goes into the conditional cell here.
Some students immediately assume that the probability
of positive given decline is 0.35.
But that is incorrect.
The complement of Positive given Growth is
Negative given Growth.
Therefore the correct cell for the 0.35
(that is, 1 – 0.65) is the conditional cell here.
Also, the probability of negative given decline,
0.7, goes to the conditional cell here.
And its complement, Positive given Decline,
is 1-0.7, which equals 0.3.
The joint probabilities are computed by multiplying
the Prior and the conditional probabilities.
Therefore, for Positive and Growth, the joint
probability is 0.4 times 0.65 which equals 0.26.
For Positive and Decline, it is 0.18.
For Negative and Growth, it is 0.14.
And for Negative and Decline, it is 0.42.
Next we add the joint probabilities to obtain
marginal probabilities. For Positive, the sum is 0.44,
and for Negative, the sum is 0.56.
That is, the probability of a Positive report is 0.44
and the probability of a negative report is 0.56.
Note that the probability of positive report and negative report add up to 1.
To obtain the posterior probabilities, we simply divide the joint probabilities by their total.
So probability of Growth given Positive is
0.26 divided by 0.44 which gives 0.59. For Decline given Positive, 0.18 divided by 0.44
which equals 0.41. In a similar fashion we obtain the result
for Growth given Negative and Decline given Negative.
Notice that the posterior probabilities add up to 1.
And that’s how to calculate posterior probabilities from conditional probabilities.
Please leave a comment or question below and have great day!
Thank you!