counting_problem_poisson.m
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counting_problem_poisson.m
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			% Section 7.1.1: Counting problems with Poisson distribution
			% Boyd & Vandenberghe "Convex Optimization"
			% Joëlle Skaf - 04/24/08
			%
			% The random variable y is nonnegative and integer valued with a Poisson
			% distribution with mean mu > 0. In a simple statistical model, the mean mu
			% is modeled as an affine function of a vector u: mu = a'*u + b.
			% We are given a number of observations which consist of pairs (u_i,y_i),
			% i = 1,..., m, where y_i is the observed value of y for which the value of
			% the explanatory variable is u_i. We find a maximum likelihood estimate of
			% the model parameters a and b from these data by solving the problem
			% maximize sum_{i=1}^m (y_ilog(a'u_i + b) - (a'*u_i + b))
			% where the variables are a and b.

			% Input data
			rand('state',0);
			n = 10;
			m = 100;
			atrue = rand(n,1);
			btrue = rand;

			u = rand(n,m);
			mu = atrue'*u + btrue;

			% Generate random variables y from a Poisson distribution
			% (The distribution is actually truncated at 10*max(mu) for simplicity)
			L = exp(-mu);
			ns = ceil(max(10*mu));
			y = sum(cumprod(rand(ns,m))>=L(ones(ns,1),:));

			% Maximum likelihood estimate of model parameters
			cvx_begin
			variables a(n) b(1)
			maximize sum(y.log(a'u+b) - (a'*u+b))
			cvx_end