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I am watching this Bloomberg course on machine learning, and i need help on understanding the application of Poisson Regression. So it goes like this
Let $Y=\{1,2,3,...\}$ and let $X$ be a vector in $\mathbb{R}^d.$
We can then model $Pr(Y_i|X_i) = e^{W^TX}.$ The vector $W$ can then be found using an optimization algorithm. But, $e^{W^TX}$ is not always an integer. So, my question is how do we predict a particular $Y_i$ given $X_i.$
In Poisson regression we use exponential link function. This means that $$ \mathbb{E}[Y | X] = e^{W^TX}. $$ Note that the expression above contains the expectation, not a probability. The expression is known as intensity and is usually denoted with $\lambda(X)$. Conditional on $X$, variable $Y$ has Poisson distribution with parameter $\lambda(X)$. This means that $$ P(Y = k | X) = \frac{\lambda(X)^k}{k!} e^{-\lambda(X)} = \frac{e^{kW^TX}}{k!} e^{-e^{W^TX}},\ \ \ k = 0, 1, 2, ... $$ As you can see, $Y$ takes only non-negative integer values.
