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

Second-order chain rule in ILR coordinates $$ \mathbf{H}_{\boldsymbol{z}} = \mathbf{J}_{\boldsymbol{\psi}}^{\mathsf{T}} \mathbf{H}_{\boldsymbol{p}} \mathbf{J}_{\boldsymbol{\psi}} +\sum_{i=1}^{J} \frac{\partial\ell}{\partial p_i}\, \mathbf{H}_{\psi_i}. $$ The second summand cannot be dropped away from stationarity. At an interior KKT point \(\nabla_p\ell=\lambda\mathbf{1}\) it vanishes because \(\sum_i\mathbf{H}_{\psi_i}=\mathbf{0}\).

Usage

hessian_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

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

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)
hessian_loglik_constrained(isometric_log_ratio(p), y, mu, Sigma)
#>          [,1]
#> [1,] -2.51645