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STATS 731, 2022, Semester 2

Tutorial 1

Question 1: Medical Example from Lectures

A patient goes to the doctor because he has some symptoms.  Consider the following two hypotheses:

H :   The patient has cancer                                                    (1)

¬H :   The patient does not have cancer.                                      (2)

Let the prior probabilities be P(H) = 0.01 and P(¬H) = 0.99.

The patient is tested, and the result of the test will be one of the following two mutually exclusive results:

D :   The test says the patient has cancer                                         (3)

¬D :   The test says the patient does not have cancer.                          (4)

However, the test isn’t perfect — there’s a 5% probability it simply gives the wrong answer:

P(D |H)   =   0.95

P(¬D |H)   =   0.05

P(D|¬H)   =   0.05

P(¬D | ¬H)   =   0.95

The test comes back positive — i.e., D is found to be true, and ¬D false.

(a) Use Bayes’rule twice, or a Bayes Box (from STATS 331), to find the posterior probabilities

P(H |D) and P(¬H |D). and the marginal likelihood P(D).

(b) Two of the six probabilities in the question did not appear in the calculations in part (a). Which two? What is special about them? Note: The fact that the answer doesn’t depend on these probabilities is called the likelihood principle.

(c) Draw a Hasse diagram of the lattice of statements in this problem.  On your diagram, circle H , ¬H , D, and ⊤. Hint: There are initially four mutually exclusive propositions here: H V D , H V ¬D , ¬H V D, and ¬H V ¬D .

Question 2: Probability Distribution Brush-ups

p(x) ∝ x2 .                                                               (5)

(a) Let y = x2 . Find the probability density function for y .

(b) Find the posterior density for x given that x  <  1/3.  Write the answer so that it is

normalised, i.e., don’t use ∝ in your result. You can use common sense to write down the answer — you don’t have to use Bayes’rule (though it is possible to do so).