Constrained gradient via the chain rule

Returns $$ \nabla_{\boldsymbol{z}}\ell = \mathbf{J}_{\boldsymbol{\psi}}^{\mathsf{T}} \nabla_{\boldsymbol{p}}\ell = \mathbf{V}^{\mathsf{T}} \mathbf{S}(\boldsymbol{p}) \nabla_{\boldsymbol{p}}\ell, $$ i.e. the first-order chain rule for \(\ell\circ\boldsymbol{\psi}\) in ILR coordinates.

Usage

gradient_loglik_constrained(z, y, mean_signature_matrix, Sigma, V = NULL)

Arguments

z

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

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 \(\mathcal{M}_{G\times G\times J}\).

V

Optional ILR basis; see isometric_logistic().

Value

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

Examples

mu <- matrix(c(20, 22, 22, 20), 2)
Sigma <- array(c(diag(2), diag(2)), dim = c(2, 2, 2))
p <- c(0.6, 0.4)
y <- drop(mu %*% p)
gradient_loglik_constrained(isometric_log_ratio(p), y, mu, Sigma)
#> [1] -0.2610856