The equations to solve

Our compact equations used a definition of a vector $\hat{y}$ with $n$ elements $y_i$, an $n\times p$ matrix $\hat{X}$ which contains the $x_i$ values and a vector $\hat{p}$ of fitted probabilities $p(y_i\vert x_i,\hat{\beta})$. We rewrote in a more compact form the first derivative of the cost function as $$\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).$$

If we in addition define a diagonal matrix $\hat{W}$ with elements $p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})$, we can obtain a compact expression of the second derivative as $$\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.$$ This defines what is called the Hessian matrix.

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