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Suppose we have a matrix $V$ of size $p \times n$ where $p \gg n$. We want to factor $V$ into two non-negative matrices $W$ and $H$ where $W$ has size $p \times k$ and $H$ has size $k \times n$. Ultimately we won't get an exact factoring. In other words, $V \approx WH$ or $V = WH + \epsilon$ where $\epsilon$ is some noise. Suppose we want to determine the Poisson likelihood of generating $V$ from $WH$ ($\epsilon$ is Poisson noise). So the likelihood function would be $$L = \prod_{i=1}^{n} f(x_i)$$ where $f(x_i) = e^{-x_i} \frac{x_i^{k}}{k!}$. So $$L = \prod_{i=1}^{n} e^{-x_i} \frac{x_i^{k}}{k!}$$ or $$\log L = \sum_{i=1}^{n} \log e^{-x_i}+ \log \left(\frac{x_{i}^{k}}{k!} \right)$$

Then $$ \log L = -(x_1-x_2- \dots-x_n)+ \sum_{i=1}^{n}\log \left(\frac{x_{i}^{k}}{k!} \right)$$

We want to minimize $- \log L$. So $$- \log L = x_1-x_2 \dots-x_n- \sum_{i=1}^{n} \log\left(\frac{x_{i}^{k}}{k!} \right)$$

From here how would you get the desired result (i.e. to consider $\mu_1 \log \left(\frac{\mu_1}{\mu_2} \right)-\mu_1+\mu_2$)?

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Does anyone have any hints? Note that $\mu_{1} \log\left(\frac{\mu_1}{\mu_2} \right)-\mu_1+\mu_2$ is for 2 poisson random variables. –  Thomas Jan 8 '12 at 16:59

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