Segment 27. Mixture Models

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To Calculate

The file Media:Mixturevals.txt contains 1000 values, each drawn either with probability from the distribution (for some constant ), or otherwise (with probability ) from the distribution .

1. Write down an expression for the probability of the file's data given some values for the parameters and .

2. Calculate numerically the maximum likelihood values of and .

3. Estimate numerically the Bayes posterior distribution of , marginalizing over 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 . What if is itself drawn (just once for the whole data set) from a distribution , with unknown hyperparameters . How would you now estimate the Bayes posterior distribution of , marginalizing over everything else?