glearn.priors.BetaPrime.pdf_jacobian

BetaPrime.pdf_jacobian(x)

Jacobian of the probability density function of the prior distribution.

Parameters:

xfloat or array_like[float]

Input hyperparameter or an array of hyperparameters.

Returns:

jacfloat or array_like[float]

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

See also

glearn.priors.Prior.log_pdf_jacobian()

glearn.priors.BetaPrime.pdf()

glearn.priors.BetaPrime.pdf_hessian()

Notes

The first derivative of the probability density function is

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

where \(B\) is the Beta function, \(a = \alpha-1\), and \(b = -(\alpha + \beta)\).

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

Examples

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

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

>>> # Evaluate the Jacobian of the PDF
>>> t = [0, 0.5, 1]
>>> prior.pdf_jacobian(t)
array([       nan, -1.7558299, -0.625    ])