Hessian \(\mathbf{H}\) of the unconstrained log-likelihood

Analytic Hessian \(\mathbf{H}\in\mathcal{M}_{J\times J}\) with entries \(\mathbf{H}_{i,j}=\partial^{2}\ell/(\partial p_i\partial p_j)\), matching the matricial formulae of the article (quadratic forms in \(\boldsymbol{\Theta}\), \(\boldsymbol{\Sigma}_i\), \(\boldsymbol{\mu}_{\cdot i}\) and residual \(\boldsymbol{r}=\boldsymbol{y}-\boldsymbol{\mu}\boldsymbol{p}\)).

Usage

hessian_loglik_unconstrained(p, y, mean_signature_matrix, Sigma)

Arguments

p

Numeric vector \(\boldsymbol{p}\in\mathbb{R}^{J}\).

y

Numeric vector (or one-column matrix) \(\boldsymbol{y}\in\mathbb{R}^{G}\).

mean_signature_matrix

Numeric matrix \(\boldsymbol{\mu}\in\mathcal{M}_{G\times J}\) (plug-in means).

Sigma

Array of cell-type covariances in \(\mathcal{M}_{G\times G\times J}\).

Value

Symmetric numeric matrix \(\mathbf{H}\).