Question 1 (25 marks)
(a) Consider the following training set of heights, x (in centimetres) and gender, c
(F/M) of some university students:
x 179 168 169 183 172 201 173 154 191
c F F F F F M M M M
Note: The Gaussian probability distribution function is:
With maximum likelihood estimators:
Build a parametric classifier for this data, using Gaussian distributions for each
(i) Use maximum likelihood estimation to calculate the parameters of your
likelihoods (p(x|c=M) and p(x|c=F)) and the class priors, P(M), P(F).
(ii) Calculate p(c = F|x,θ), where x=210.
(iii) What class does your classifier predict for the test point from (ii) above?
Explain why this prediction is the logical one to make.
(b) Consider training a Gaussian mixture model using the EM algorithm, with three
components on a 2D dataset. The model was trained repeatedly from a
randomized initialization of parameter values and the algorithm run for a large
number of iterations until apparent convergence. Figure 1 shows the results of four
of the training runs. Give a brief possible explanation for the differences in the four
(c) Figure 2 (a) shows kernel density estimators fitted to blood pressure data from two
groups of patients (with chronic heart disease (CHD) and not). [To answer this
question it doesn’t matter which group is which, but the “no CHD” curve (from left
to right) starts above the “CHD” curve and crosses it at a blood pressure value
around 140. The sample for “no CHD” is shown on the top frame of the graph,
while the sample for “CHD” is on the bottom frame.]
(i) Explain what a kernel density estimator is. In particular, you should explain
how it uses the data and what are the parameters of the model?
(ii) Explain how Bayes rule is used to produce a classifier from probability density
estimation models given supervised data. From this, explain what is shown in
Figure 2 (b).
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