This preview shows pages 1–3. Sign up to view the full content.
1. Bayes theorem: (a) What is the purpose of Bayesian analysis? (b)
What is the relationship between prior probabilities and posterior
probabilities?
The purpose of Bayes’s theorm is to determine what the probability is that some
hypothesis is true given that we have observed some data. Bayes’s theorem is especially
useful since we do not have perfect tests. Therefore, the data will always be flawed in
some way – perfect information does not exist in reality, as much as we may think we can
approach the limit of the truth. So how do we get to the probability that the test is
representative of reality? As an example, let us consider a medical test in which the test is
not 100% accurate. A rational doctor would ask him or herself, given that I have a
positive test result for sickness, what is the probability that the person is
actually
sick? If
we say that p(H) is the prior probability, then p(DH) is the posterior probability. If we
look at p(DH), this is will help us calculate the probability of health given some certain
datum, or rather, given the probability of a positive test result, what is the probability that
the test is true. Here, the doctor is given some
prior
probability of health for this patient
(e.g. this person’s age, weight, sex, etc means a 30% chance of sickness). Using this
information, the doctor can observe some data (D) to calculate the posterior probability
p(DH) to determine the appropriate decision to be made in regards to this patient
Human’s are not good at calculating the prior data
The form of Bayes’s theorem is:
p(HD) = p(DH)p(H) / {[p(DH)p(H)] + [p(D notH)p(notH)]}

 prior probabilities to calculate posterior probability
o
p(H) is the prior probability
o
p(DH) is the posterior probability


Medical test, given sickness is not 100%
o
Given that you have a positive test result for sickness, what is the
probability that the person is
actually
sick
p(HD) – probability of health given the data – given the probability of a positive test
result, what is the probability that the test is true
p(HD) = p(D+H) / [p(D+H) + p(D+ notH)]
p(HD) = p(DH)p(H) / {[p(DH)p(H)] + [p(D notH)p(notH)]}
use this for the
homework
Bayes’s Theorem
d = data
h = hypothesis
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentp(D) = p(notH + D) + p(H + D)
P(notH + D) = p(D  notH) x p(notH)
P(H + D) = p(D  H) x p(H)
Thus p(H  D) = p(H + D) / [p(notH + D) + p(H + D)] = P(D  H) x p(H)
P(H D) = [p(DH) x p(H)] / [[P(DH)x(p(H)] + [p(D  notH) x p(notH)]]
There is a strong relationship between Bayes’s theorem and conditional assessment
Ratio form of Bayes’s theorem
p(D  H) × p(H)
p(H  D) = 
[ p(D  H) × p(H) + p(D  notH) × p(notH) ]
p(D  notH) × p(notH)
p(notH  D) = 
[ p(D  H) × p(H) + p(D  notH) × p(notH) ]
p(H  D)
p(D  H) × p(H)
p(D  H)
p(H)
 =
 =
 × 
p(notH  D)
p(D  notH) × p(notH)
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '11
 Mellers

Click to edit the document details