Constrained Hessian of \(\ell\circ\boldsymbol{\psi}\)

Second-order chain rule $$ \mathbf{H}_{\boldsymbol{\rho}} = \mathbf{J}_{\boldsymbol{\psi}}^{\mathsf{T}} \mathbf{H}_{\boldsymbol{p}} \mathbf{J}_{\boldsymbol{\psi}} +\sum_{i=1}^{J} \frac{\partial\ell}{\partial p_i}\, \frac{\partial^{2}p_i}{\partial\boldsymbol{\rho}\partial\boldsymbol{\rho}^{\mathsf{T}}}. $$

Usage

hessian_loglik_constrained(rho, y, mean_signature_matrix, Sigma)

Arguments

rho

Numeric vector \(\boldsymbol{\rho}\in\mathbb{R}^{J-1}\).

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 matrix in \(\mathcal{M}_{(J-1)\times(J-1)}\).