Problem 4.5 Flying Bombs, part II – EM for mixtures of Poissons

Let's reconsider Problem 2.1, where we fit a Poisson distribution to the numbers of flying bombs hitting different areas in London. If we assume that the Germans were indeed targeting specific areas, then the bomb hit rate $\lambda$ would be higher for some squares (the targets), and lower for others (not the targets). Hence, the distribution over all squares should be a mixture of Poissons, with each Poisson component corresponding to squares with a particular hit rate. For $K = 2$, the components would correspond to target squares and non-target squares. For $K > 2$, one component would correspond to the target hit rate, while the other (non-target) components would have some gradation of hit rates (with squares far away from the target squares having lower hit rates).

(b) Implement your algorithm and run it for different values of $K \in \{1, 2, 3, 4, 5\}$ on the following data obtained from 2 cities (learn a separate mixture model for each city):

city	number of hits ($k$)	0	1	2	3	4	5 and over
London	number of cells with $k$ hits	229	211	93	35	7	1
Antwerp	number of cells with $k$ hits	325	115	67	30	18	21

What conclusions can you make about the attacks on each city? Is there any evidence to suggest there is specific targeting of areas in London or Antwerp?