Constrained gradient via the chain rule

Returns $$ \nabla_{\boldsymbol{\rho}}\ell = \bigl(\nabla_{\boldsymbol{p}}\ell\bigr)^{\mathsf{T}} \mathbf{J}_{\boldsymbol{\psi}}(\boldsymbol{\rho}), $$ i.e. first-order chain rule for \(\ell\circ\boldsymbol{\psi}\).

Usage

gradient_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

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