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Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process. (English) Zbl 1097.65019
Nonparametric Bayesian approach to Gaussian regression is considered. The training data $$(x_i,t_i)_{i=1}^N$$ consist of output $$t_i\in\mathbb{R}$$ and input vectors $$x_i\in \mathbb{R}^L$$. Their distribution is described by $\mathbf{P}(t| \;x,\Theta)\propto \exp\left( -{1\over 2}t^TC^{-1}(\vartheta)t \right),$ where $$C$$ is a matrix with entries
$c(x_i,x_j,\Theta)= \alpha\exp \left( -{1\over 2}\sum_{l=1}^L (x_i^{(l)}-x_j^{(l)})^2d_l \right) +\nu\delta_{ij},$
$$\Theta=(\alpha,d_1,\dots,d_L,\nu)$$ being the hyperparameter. It is proposed to estimate $$\Theta$$ using maximum likelihood, i.e. to minimize the negative log-likelihood $$L(\Theta)={1\over 2}\log\det C(\Theta)+{1\over 2}t^TC^{-1}(\Theta)t$$. The authors describe the form of the Hessian matrix for $$L$$ and propose a second-order trust-region algorithm for the minimization of $$L$$. Results of numerical simulations are presented.

##### MSC:
 65C60 Computational problems in statistics (MSC2010) 62G08 Nonparametric regression and quantile regression
##### Software:
Optimization Toolbox; GQTPAR
Full Text:
##### References:
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