# Eleisha's Segment 27: Mixture Models

To Calculate:

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

1. Write down an expression for the probability of the file's data given some values for the parameters $\displaystyle \beta$ and $\displaystyle c$ .

2. Calculate numerically the maximum likelihood values of $\displaystyle \beta$ and $\displaystyle c$ .

3. Estimate numerically the Bayes posterior distribution of $\displaystyle \beta$ , marginalizing over 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 $\displaystyle c$ . What if $\displaystyle c$ is itself drawn (just once for the whole data set) from a distribution $\displaystyle \text{Beta}(\mu,\nu)$ , with unknown hyperparameters $\displaystyle \mu,\nu$ . How would you now estimate the Bayes posterior distribution of $\displaystyle \beta$ , marginalizing over everything else?