I see that with logistic regression that the standard error can be computed as in How to compute the standard errors of a logistic regression's coefficients
which amounted to taking
$\sqrt{(X^TVX)^{-1}}$ where V is a diagonal matrix where the diagonal entries was probability of being in class A, $\pi_a * (1-\pi_a)$
Looking at the standard error of coefficients for linear regression, it seems like $\sqrt{\sigma^2(X^TVX)^{-1}}$ where s is the variance of the residuals (as per my understanding of https://stats.stackexchange.com/a/27920/239020 )
From the above, it seems like we want to encompass some notion of "how wrong" our prediction is compared to a label. So, in this case I'm wondering what it might be for a poisson regression? Am I on to something or am I seeing a pattern where there is none?
