gradient_log_likelihood2: Analytic gradient of the negative log-likelihood (mode 2)
In cNORM: Continuous Norming

gradient_log_likelihood2

R Documentation

Analytic gradient of the negative log-likelihood (mode 2)

Description

Computes the exact gradient of log_likelihood2 with respect to all regression coefficients. With the per-observation log-probability

\log p = \mathrm{lchoose}(n, y) + \mathrm{lbeta}(y+\alpha, n-y+\beta) - \mathrm{lbeta}(\alpha, \beta),

the partial derivatives on the natural scale are (with digamma function \psi):

\partial \log p/\partial\alpha = \psi(y+\alpha) - \psi(\alpha) + \psi(\alpha+\beta) - \psi(n+\alpha+\beta)

\partial \log p/\partial\beta = \psi(n-y+\beta) - \psi(\beta) + \psi(\alpha+\beta) - \psi(n+\alpha+\beta)

The chain rule through the log link multiplies by \alpha resp. \beta; coefficient gradients follow as -X^\top(w \cdot \partial\ell/\partial\eta) for the negative log-likelihood. Contributions of observations whose linear predictor is clamped (see log_likelihood2) are zeroed, which is the exact subgradient of the clamped objective.

Usage

gradient_log_likelihood2(params, X, Z, y, n, weights = NULL, lch = NULL)

Arguments

`params`	A numeric vector containing all model parameters. The first n_alpha elements are coefficients for the alpha model, and the remaining elements are coefficients for the beta model.
`X`	A matrix of predictors for the alpha model.
`Z`	A matrix of predictors for the beta model.
`y`	A numeric vector of response values.
`n`	The maximum score (number of trials).
`weights`	A numeric vector of weights for each observation (NULL = equal weights).
`lch`	Optional precomputed `lchoose(n, y)`. Since this term does not depend on the parameters, passing it once avoids recomputation in every optimizer iteration.

Details

Supplying this gradient to optim (L-BFGS-B) replaces the finite-difference approximation, which required length(params) + 1 likelihood evaluations per gradient.