# Segment 27. Mixture Models

## Contents

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### Problems

#### To Calculate

The file Media:Mixturevals.txt contains 1000 values, each drawn either with probability Failed to parse (unknown error): c from the distribution $Exponential$ (for some constant Failed to parse (unknown error): \beta ), or otherwise (with probability Failed to parse (unknown error): 1-c ) from the distribution Failed to parse (unknown error): p(x) = (2/\pi)/(1+x^2),\; x>0 .

1. Write down an expression for the probability of the file's data given some values for the parameters Failed to parse (unknown error): \beta and Failed to parse (unknown error): c .

2. Calculate numerically the maximum likelihood values of Failed to parse (unknown error): \beta and Failed to parse (unknown error): c .

3. Estimate numerically the Bayes posterior distribution of Failed to parse (unknown error): \beta , marginalizing over Failed to parse (unknown error): c as a nuisance parameter. (You'll of course have to make some assumption about priors.)

1. In problem 3, above, you assumed some definite prior for Failed to parse (unknown error): c . What if Failed to parse (unknown error): c is itself drawn (just once for the whole data set) from a distribution $Beta$ , with unknown hyperparameters Failed to parse (unknown error): \mu,\nu . How would you now estimate the Bayes posterior distribution of Failed to parse (unknown error): \beta , marginalizing over everything else?