Suppose that you are given a decision situation with three possible state of nature: s1, s2, and s3. The prior probabilities are P(s1) =.1, P(s2) = .6, and P(s3) = .3. With sample information I, P(I|s1) =.15, P(I|s2) = .2, and P(I|s3) = .1.
Compute P(s1|I)=
Compute P(s2|I)=
Compute P(s3|I)=
To compute the posterior probabilities P(s1|I), P(s2|I), and P(s3|I), we use Bayes' theorem:
P(s1|I) = P(I|s1) * P(s1) / P(I)
P(s2|I) = P(I|s2) * P(s2) / P(I)
P(s3|I) = P(I|s3) * P(s3) / P(I)
First, let's find the probability of I, P(I):
P(I) = P(I|s1) * P(s1) + P(I|s2) * P(s2) + P(I|s3) * P(s3)
P(I) = (.15)(.1) + (.2)(.6) + (.1)(.3)
P(I) = .015 + .12 + .03
P(I) = .165
Now let's calculate P(s1|I):
P(s1|I) = (.15)(.1) / .165
P(s1|I) ≈ 0.0909
P(s2|I):
P(s2|I) = (.2)(.6) / .165
P(s2|I) ≈ 0.7273
P(s3|I):
P(s3|I) = (.1)(.3) / .165
P(s3|I) ≈ 0.1818
So the computed probabilities are:
P(s1|I) ≈ 0.0909
P(s2|I) ≈ 0.7273
P(s3|I) ≈ 0.1818
To compute P(s1|I), we can use Bayes' theorem:
P(s1|I) = (P(I|s1) * P(s1)) / P(I)
Given:
P(s1) = 0.1
P(I|s1) = 0.15
To compute P(I), we can use the law of total probability:
P(I) = P(I|s1) * P(s1) + P(I|s2) * P(s2) + P(I|s3) * P(s3)
Given:
P(s2) = 0.6
P(I|s2) = 0.2
P(s3) = 0.3
P(I|s3) = 0.1
Substituting these values into the equation, we get:
P(I) = (0.15 * 0.1) + (0.2 * 0.6) + (0.1 * 0.3)
P(I) = 0.015 + 0.12 + 0.03
P(I) = 0.165
Now, we can compute P(s1|I) using Bayes' theorem:
P(s1|I) = (0.15 * 0.1) / 0.165
P(s1|I) = 0.015 / 0.165
P(s1|I) = 0.0909
Similarly, we can compute P(s2|I) and P(s3|I) using the same process:
P(s2|I) = (0.2 * 0.6) / 0.165
P(s2|I) = 0.12 / 0.165
P(s2|I) = 0.7273
P(s3|I) = (0.1 * 0.3) / 0.165
P(s3|I) = 0.03 / 0.165
P(s3|I) = 0.1818
Therefore:
P(s1|I) = 0.0909
P(s2|I) = 0.7273
P(s3|I) = 0.1818
To compute the posterior probabilities P(s1|I), P(s2|I), and P(s3|I), we can use Bayes' Theorem. Bayes' Theorem allows us to update our prior probabilities based on new information.
Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, A represents the state of nature (s1, s2, s3) and B represents the sample information (I).
Let's calculate the values:
P(s1|I) = (P(I|s1) * P(s1)) / P(I)
P(s2|I) = (P(I|s2) * P(s2)) / P(I)
P(s3|I) = (P(I|s3) * P(s3)) / P(I)
We need to find P(I), which is the probability of observing the sample information regardless of the state of nature. To calculate P(I), we can use the law of total probability:
P(I) = P(s1) * P(I|s1) + P(s2) * P(I|s2) + P(s3) * P(I|s3)
Substituting the given probabilities, we have:
P(I) = (0.1 * 0.15) + (0.6 * 0.2) + (0.3 * 0.1) = 0.015 + 0.12 + 0.03 = 0.165
Now, we can substitute this value into the equations for P(s1|I), P(s2|I), and P(s3|I):
P(s1|I) = (0.15 * 0.1) / 0.165
P(s2|I) = (0.2 * 0.6) / 0.165
P(s3|I) = (0.1 * 0.3) / 0.165
Calculating each equation:
P(s1|I) = 0.015 / 0.165 ≈ 0.0909
P(s2|I) = 0.12 / 0.165 ≈ 0.7273
P(s3|I) = 0.03 / 0.165 ≈ 0.1818
Therefore, the computed probabilities are:
P(s1|I) ≈ 0.0909
P(s2|I) ≈ 0.7273
P(s3|I) ≈ 0.1818