import numba as nb
import numpy as np
from pytensor_distributions import inversegamma as ptd_inversegamma
from preliz.distributions.distributions import Continuous
from preliz.internal.distribution_helper import (
all_not_none,
any_not_none,
eps,
pytensor_jit,
pytensor_rng_jit,
)
from preliz.internal.optimization import optimize_ml
[docs]
class InverseGamma(Continuous):
r"""
Inverse gamma distribution, the reciprocal of the gamma distribution.
The pdf of this distribution is
.. math::
f(x \mid \alpha, \beta) =
\frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{-\alpha - 1}
\exp\left(\frac{-\beta}{x}\right)
.. plot::
:context: close-figs
from preliz import InverseGamma, style
style.use('preliz-doc')
alphas = [1., 2., 3.]
betas = [1., 1., .5]
for alpha, beta in zip(alphas, betas):
InverseGamma(alpha, beta).plot_pdf(support=(0, 3))
======== ===============================
Support :math:`x \in (0, \infty)`
Mean :math:`\dfrac{\beta}{\alpha-1}` for :math:`\alpha > 1`
Variance :math:`\dfrac{\beta^2}{(\alpha-1)^2(\alpha - 2)}` for :math:`\alpha > 2`
======== ===============================
Inverse gamma distribution has 2 alternative parameterization. In terms of alpha and
beta or mu (mean) and sigma (standard deviation).
The link between the 2 alternatives is given by
.. math::
\alpha &= \frac{\mu^2}{\sigma^2} + 2 \\
\beta &= \frac{\mu^3}{\sigma^2} + \mu
Parameters
----------
alpha : float
Shape parameter (alpha > 0).
beta : float
Scale parameter (beta > 0).
mu : float
Mean (mu > 0).
sigma : float
Standard deviation (sigma > 0)
"""
def __init__(self, alpha=None, beta=None, mu=None, sigma=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(alpha, beta, mu, sigma)
def _parametrization(self, alpha=None, beta=None, mu=None, sigma=None):
if any_not_none(alpha, beta) and any_not_none(mu, sigma):
raise ValueError(
"Incompatible parametrization. Either use alpha and beta or mu and sigma."
)
self.param_names = ("alpha", "beta")
self.params_support = ((eps, np.inf), (eps, np.inf))
if any_not_none(mu, sigma):
self.mu = mu
self.sigma = sigma
self.param_names = ("mu", "sigma")
if all_not_none(mu, sigma):
alpha, beta = ptd_inversegamma.from_mu_sigma(mu, sigma)
self.alpha = alpha
self.beta = beta
if all_not_none(self.alpha, self.beta):
self._update(self.alpha, self.beta)
def _update(self, alpha, beta):
self.alpha = np.float64(alpha)
self.beta = np.float64(beta)
self.mu = _to_mu(self.alpha, self.beta)
self.sigma = _to_sigma(self.alpha, self.beta)
if self.param_names[0] == "alpha":
self.params = (self.alpha, self.beta)
elif self.param_names[1] == "sigma":
self.params = (self.mu, self.sigma)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.alpha, self.beta)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.alpha, self.beta)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.alpha, self.beta)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.alpha, self.beta)
[docs]
def entropy(self):
return ptd_entropy(self.alpha, self.beta)
[docs]
def mean(self):
return ptd_mean(self.alpha, self.beta)
[docs]
def mode(self):
return ptd_mode(self.alpha, self.beta)
[docs]
def var(self):
return ptd_var(self.alpha, self.beta)
[docs]
def std(self):
return ptd_std(self.alpha, self.beta)
[docs]
def skewness(self):
return ptd_skewness(self.alpha, self.beta)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.alpha, self.beta)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.alpha, self.beta)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.alpha, self.beta)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.alpha, self.beta)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.alpha, self.beta)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.alpha, self.beta, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
alpha, beta = _from_mu_sigma(mean, sigma)
self._update(alpha, beta)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, alpha, beta):
return ptd_inversegamma.pdf(x, alpha, beta)
@pytensor_jit
def ptd_cdf(x, alpha, beta):
return ptd_inversegamma.cdf(x, alpha, beta)
@pytensor_jit
def ptd_ppf(q, alpha, beta):
return ptd_inversegamma.ppf(q, alpha, beta)
@pytensor_jit
def ptd_logpdf(x, alpha, beta):
return ptd_inversegamma.logpdf(x, alpha, beta)
@pytensor_jit
def ptd_entropy(alpha, beta):
return ptd_inversegamma.entropy(alpha, beta)
@pytensor_jit
def ptd_mean(alpha, beta):
return ptd_inversegamma.mean(alpha, beta)
@pytensor_jit
def ptd_mode(alpha, beta):
return ptd_inversegamma.mode(alpha, beta)
@pytensor_jit
def ptd_median(alpha, beta):
return ptd_inversegamma.median(alpha, beta)
@pytensor_jit
def ptd_var(alpha, beta):
return ptd_inversegamma.var(alpha, beta)
@pytensor_jit
def ptd_std(alpha, beta):
return ptd_inversegamma.std(alpha, beta)
@pytensor_jit
def ptd_skewness(alpha, beta):
return ptd_inversegamma.skewness(alpha, beta)
@pytensor_jit
def ptd_kurtosis(alpha, beta):
return ptd_inversegamma.kurtosis(alpha, beta)
@pytensor_jit
def ptd_lmoment1(alpha, beta):
return ptd_inversegamma.lmoment1(alpha, beta)
@pytensor_jit
def ptd_lmoment2(alpha, beta):
return ptd_inversegamma.lmoment2(alpha, beta)
@pytensor_jit
def ptd_lmoment3(alpha, beta):
return ptd_inversegamma.lmoment3(alpha, beta)
@pytensor_jit
def ptd_lmoment4(alpha, beta):
return ptd_inversegamma.lmoment4(alpha, beta)
@pytensor_rng_jit
def ptd_rvs(alpha, beta, size, rng):
return ptd_inversegamma.rvs(alpha, beta, size=size, random_state=rng)
def _from_mu_sigma(mu, sigma):
alpha = mu**2 / sigma**2 + 2
beta = mu**3 / sigma**2 + mu
return alpha, beta
@nb.vectorize(nopython=True, cache=True)
def _to_mu(alpha, beta):
if alpha > 1:
return beta / (alpha - 1)
else:
return np.nan
@nb.vectorize(nopython=True, cache=True)
def _to_sigma(alpha, beta):
if alpha > 2:
return beta / ((alpha - 1) * (alpha - 2) ** 0.5)
else:
return np.nan
