# Segment 27. Mixture Models

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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 **Failed to parse (unknown error): \text{Exponential}(\beta)**
(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.)

#### To Think About

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 **Failed to parse (unknown error): \text{Beta}(\mu,\nu)**
, 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?