Segment 27. Mixture Models

From Computational Statistics (CSE383M and CS395T)
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To Calculate

The file Media:Mixturevals.txt contains 1000 values, each drawn either with probability <math>c</math> from the distribution <math>\text{Exponential}(\beta)</math> (for some constant <math>\beta</math>), or otherwise (with probability <math>1-c</math>) from the distribution <math>p(x) = (2/\pi)/(1+x^2),\; x>0</math>.

1. Write down an expression for the probability of the file's data given some values for the parameters <math>\beta</math> and <math>c</math>.

2. Calculate numerically the maximum likelihood values of <math>\beta</math> and <math>c</math>.

3. Estimate numerically the Bayes posterior distribution of <math>\beta</math>, marginalizing over <math>c</math> as a nuisance parameter. (You'll of course have to make some assumption about priors.)

To Think About

1. In problem 3, above, you assumed some definite prior for <math>c</math>. What if <math>c</math> is itself drawn (just once for the whole data set) from a distribution <math>\text{Beta}(\mu,\nu)</math>, with unknown hyperparameters <math>\mu,\nu</math>. How would you now estimate the Bayes posterior distribution of <math>\beta</math>, marginalizing over everything else?