Segment 27. Mixture Models
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Problems
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?