= Poisson slope information after eliminating an intercept
{c}
{title2=$\operatorname{Var}(\widehat\beta)\approx[\sum_i\mu_i(x_i-\bar x_\mu)^2]^{-1}$}
In <Poisson regression> with log means $a+\beta x_i$, the <Fisher information matrix> is $\begin{pmatrix}S_0&S_1\\S_1&S_2\end{pmatrix}$, where $S_j=\sum_i x_i^j\mu_i$. Inverting this matrix gives slope <variance> $S_0/(S_0S_2-S_1^2)$. Writing $\bar x_\mu=S_1/S_0$ reduces it to the displayed reciprocal weighted sum of squares. Thus a common intercept consumes information about the overall rate, while slope information comes from the spread of the covariates under the mean-count weights. Constant covariates make the two parameters unidentifiable.
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