I'm trying to compute the exact second derivatives of log marginal likelihood of Gaussian Process for learning hyperparameters.

The log marginal likelihood and its partial derivative are given in 5 chapter of GPML(http://www.gaussianprocess.org/gpml/chapters/RW5.pdf) as

$$ \log p(\mathbf{y} | \mathbf{X}, \theta) = -\dfrac{1}{2}\mathbf{y}^TK^{-1}_y \mathbf{y} -\dfrac{1}{2}\log|K_y| - \dfrac{n}{2}\log{2\pi} \tag{1} $$

$$ \dfrac{\partial}{\partial \theta_j} \log p(\mathbf{y} | \mathbf{X}, \theta) = \dfrac{1}{2} \mathbf{y}^T K^{-1} \dfrac{\partial K}{\partial \theta_j} K^{-1} \mathbf{y} - \dfrac{1}{2} tr(K^{-1} \dfrac{\partial K}{\partial \theta_j}) \tag{2} $$ where

$K_y(x^p, x^q) = \sigma^2_f \exp(-\dfrac{1}{2l^2}(x_p - x_q)^T (x_p - x_q)) + \sigma^2_n\delta_{pq}$,

$\theta = (l, \sigma_f, \sigma_n)$

refferring to matrix cookbook, I'm tackling with calculating the hessian matrix of $\log p(\mathbf{y} | \mathbf{X}, \mathbf{\theta})$, but it's not easy for me to do.

Can you compute hessian matrix of $\log p(\mathbf{y} | \mathbf{X}, \mathbf{\theta})$ by hand?


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