glearn.priors.InverseGamma.pdf_hessian

InverseGamma.pdf_hessian(x)

Hessian of the probability density function of the prior distribution.

Parameters:

xfloat or array_like[float]

Input hyperparameter or an array of hyperparameters.

Returns:

hessfloat or array_like[float]

The Hessian of the probability density function of the input hyperparameter(s).

See also

glearn.priors.Prior.log_pdf_hessian()

glearn.priors.InverseGamma.pdf()

glearn.priors.InverseGamma.pdf_jacobian()

Notes

The second derivative of the probability density function is

\[\frac{\mathrm{d}^2}{\mathrm{d}\theta^2} p(\theta \vert \alpha, \beta) = \frac{\theta^{\alpha-1} e^{-\beta \theta} \beta^{\alpha}}{\Gamma(\alpha)} \frac{\left(a^2 + a - 2ab - 2b + b^2 \right)}{\theta^2},\]

where \(\Gamma\) is the Gamma function, \(a = \alpha+1\), and \(b = \frac{\beta}{\theta}\).

When an array of hyperparameters are given, it is assumed that prior for each hyperparameter is independent of others.

Examples

Create the inverse Gamma distribution with the shape parameter \(\alpha=4\) and rate parameter \(\beta=2\).

>>> from glearn import priors
>>> prior = priors.InverseGamma(4, 2)

>>> # Evaluate the Hessian of the PDF
>>> t = [0, 0.5, 1]
>>> prior.pdf_hessian(t)
array([[         nan,   0.        ,   0.        ],
       [  0.        , -12.50347615,   0.        ],
       [  0.        ,   0.        ,   3.60894089]])