5
$\begingroup$

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 $X_i.$

$\endgroup$
12
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I see. This cleared it up. I appreciate your help sir! $\endgroup$ – Benj Cabalona Jr. Nov 8 at 6:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.