Gradient \(\nabla_{\boldsymbol{p}}\ell\) of the unconstrained log-likelihood

Analytic gradient of loglik_multivariate() with respect to \(\boldsymbol{p}\). Writing \(\boldsymbol{\Theta}=\boldsymbol{\Sigma}(\boldsymbol{p})^{-1}\) and \(\boldsymbol{r}=\boldsymbol{y}-\boldsymbol{\mu}\boldsymbol{p}\), the \(j\)-th coordinate is $$ \frac{\partial\ell}{\partial p_j} = -2p_j\,\mathrm{Tr}\!\bigl(\boldsymbol{\Theta}\boldsymbol{\Sigma}_j\bigr) +\boldsymbol{r}^{\mathsf{T}}\boldsymbol{\Theta}\boldsymbol{\mu}_{\cdot j} +p_j\,\boldsymbol{r}^{\mathsf{T}} \boldsymbol{\Theta}\boldsymbol{\Sigma}_j\boldsymbol{\Theta}\boldsymbol{r}. $$

Usage

gradient_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

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

Details

Unit tests compare this analytic gradient to a numerical reference from numDeriv::grad() applied to loglik_multivariate(). For that check the Richardson method is preferred; main method.args knobs:

eps

Initial finite-difference step (default 1e-4).

r

Number of Richardson extrapolations (default 4; tests use 6). Raising r usually improves accuracy more safely than shrinking eps alone.

d, v

Relative step factor and geometric reduction between extrapolations (default v = 2).

zero.tol, show.details

See ?numDeriv::grad.

Alternative method values: "simple" and "complex".

See also

numDeriv::grad(), hessian_loglik_unconstrained()