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?
