quadDeriv: Gradient and Diagonal of Hesse Matrix of Quadratic...

In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

Gradient and Diagonal of Hesse Matrix of Quadratic Approximation to Log-Likelihood Function L

Description

Computes gradient and diagonal of the Hesse matrix w.r.t. to \eta of a quadratic approximation to the reparametrized original log-likelihood function

L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^{\infty} \exp(\phi(t)) dt.

where L is parametrized via

{\bold{\eta}}({\bold{\phi}}) = \Bigl(\phi_1, \Bigl(\eta_1+ \sum_{j=2}^i (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).

{\bold{\phi}}: vector (\phi(x_i))_{i=1}^m representing concave, piecewise linear function \phi,
{\bold{\eta}}: vector representing successive slopes of \phi.

Usage

quadDeriv(dx, w, eta)

Arguments

dx	Vector (0, x_i-x_{i-1})_{i=2}^m.
w	Vector of weights as in activeSetLogCon.
eta	Vector {\bold{\eta}}.

Value

m \times 2 matrix. First column contains gradient and second column diagonal of Hesse matrix.

Note

This function is not intended to be invoked by the end user.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

See Also

quadDeriv is used by the function icmaLogCon.

logcondens documentation built on Aug. 22, 2023, 5:06 p.m.