### More than one kind of information

#### (Jul 19, 2018)

This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics.  If we are fitting a parametric model of the hazard rate, with log-likelihood:

$\ell( \alpha_1, \ldots, \alpha_n )$

as a function of $$n$$ parameters $$\alpha_1, \ldots, \alpha_n$$, then the information matrix is the matrix of second-order partial derivatives of $$\ell$$. That is, the matrix $${\cal I}$$ with $$ij$$th component:

${\cal I}_{ij} = \frac{\partial^2 \ell}{\partial \alpha_i \partial \alpha_j}.$

It is important because $$-{\cal I}^{-1}$$ evaluated at the fitted maximum $$(\hat{\alpha}_1, \ldots, \hat{\alpha}_n)$$ approximates the variance-covariance matrix of…