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In Elements of Statistical Learning, page 125 it is written that the coefficients of a logistic regression converges to $\mathcal{N}(\beta, (X^T W X)^{-1})$ with:

  • $X$ the $N \times (p+1)$ matrix of data
  • $W$ a $N \times N$ diagonal matrix of weights with $i$th diagonal element $p(x_i;\beta)(1-p(x_i; \beta))$ where by definition $p(x_i;\beta) = \exp(\beta^T x_i)/(1+\exp(\beta^T x_i))$.

Is there a proof/intuition for the $(X^T W X)^{-1}$?

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