Solution 4 Computational Intelligence Lab 2012

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Problem 1

1.1 Write down the log of the likelihood function

(slides)

1.2 Give the log of the likelyhood of this datapoint

Given:

1.3 Calculate (...)

Adding the criterion :

1.4 Limit to 0, discuss the influence of this issue on the maximization of the likelihood function.

The result makes sense since in this case the function is a probability density function!

Discussion: According to http://www.cedar.buffalo.edu/~srihari/CSE574/Chap9/Ch9.2-MixturesofGaussians.pdf (Slide 12) the maximization of log-likelihood is not well posed.

1.5 Can this situation occur in the case of a single Gaussian distribution

No, because:

Here we insert the condition: :

And add the condition:

(for N > 0)

So the probability density is which makes this event impossible.

I put the question in the VIS forum: https://forum.vis.ethz.ch/showthread.php?15350

1.6

We can detect if clusters come near any datapoint and reset it.

Problem 2

2.1 Suppose that we have solved a mixture of K Gaussians problem, and have obtained the values of the parameters. For this given solution, how many equivalent solutions do exist?

2.2 This problem is known as identifiability. Explain why this problem is irrelevant for the purpose of data clustering.

We are only interested which points can be grouped together.