"""Beta distribution."""
import numpy as np
from pytensor_distributions import beta as ptd_beta
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
from preliz.internal.special import mean_and_std
[docs]
class Beta(Continuous):
r"""
Beta distribution.
The pdf of this distribution is
.. math::
f(x \mid \alpha, \beta) =
\frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}
.. plot::
:context: close-figs
from preliz import Beta, style
style.use('preliz-doc')
alphas = [.5, 5., 2.]
betas = [.5, 5., 5.]
for alpha, beta in zip(alphas, betas):
ax = Beta(alpha, beta).plot_pdf()
ax.set_ylim(0, 5)
======== ==============================================================
Support :math:`x \in (0, 1)`
Mean :math:`\dfrac{\alpha}{\alpha + \beta}`
Variance :math:`\dfrac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}`
======== ==============================================================
Beta distribution has 3 alternative parameterizations. In terms of alpha and
beta, mean and sigma (standard deviation) or mean and nu (concentration).
The link between the 3 alternatives is given by
.. math::
\alpha &= \mu \nu \\
\beta &= (1 - \mu) \nu
\text{where } \nu = \frac{\mu(1-\mu)}{\sigma^2} - 1
Parameters
----------
alpha : float
alpha > 0
beta : float
beta > 0
mu : float
mean (0 < ``mu`` < 1).
sigma : float
standard deviation (``sigma`` < sqrt(``mu`` * (1 - ``mu``))).
nu : float
concentration > 0
"""
def __init__(self, alpha=None, beta=None, mu=None, sigma=None, nu=None):
super().__init__()
self.support = (0, 1)
self._parametrization(alpha, beta, mu, sigma, nu)
def _parametrization(self, alpha=None, beta=None, mu=None, sigma=None, nu=None):
if any_not_none(alpha, beta) and any_not_none(mu, sigma, nu) or all_not_none(sigma, nu):
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")
self.params_support = ((eps, 1 - eps), (eps, 1 - eps))
if all_not_none(mu, sigma):
alpha, beta = self._from_mu_sigma(mu, sigma)
if any_not_none(mu, nu) and sigma is None:
self.mu = mu
self.nu = nu
self.param_names = ("mu", "nu")
self.params_support = ((eps, 1 - eps), (eps, np.inf))
if all_not_none(mu, nu):
alpha, beta = self._from_mu_nu(mu, nu)
self.alpha = alpha
self.beta = beta
if all_not_none(self.alpha, self.beta):
self._update(self.alpha, self.beta)
def _from_mu_sigma(self, mu, sigma):
nu = mu * (1 - mu) / sigma**2 - 1
alpha = mu * nu
beta = (1 - mu) * nu
return alpha, beta
def _from_mu_nu(self, mu, nu):
alpha = mu * nu
beta = (1 - mu) * nu
return alpha, beta
def _to_mu_sigma(self, alpha, beta):
alpha_plus_beta = alpha + beta
mu = alpha / alpha_plus_beta
sigma = (alpha * beta) ** 0.5 / alpha_plus_beta / (alpha_plus_beta + 1) ** 0.5
return mu, sigma
def _update(self, alpha, beta):
self.alpha = np.float64(alpha)
self.beta = np.float64(beta)
self.mu, self.sigma = self._to_mu_sigma(self.alpha, self.beta)
self.nu = self.mu * (1 - self.mu) / self.sigma**2 - 1
if self.param_names[0] == "alpha":
self.params = (self.alpha, self.beta)
elif self.param_names[1] == "sigma":
self.params = (self.mu, self.sigma)
elif self.param_names[1] == "nu":
self.params = (self.mu, self.nu)
self.is_frozen = True
def _fit_moments(self, mean, sigma):
alpha, beta = self._from_mu_sigma(mean, sigma)
alpha = max(0.5, alpha)
beta = max(0.5, beta)
self._update(alpha, beta)
def _fit_mle(self, sample):
mean, std = mean_and_std(sample)
self._fit_moments(mean, std)
optimize_ml(self, sample)
[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 sf(self, x):
return ptd_sf(x, self.alpha, self.beta)
[docs]
def isf(self, q):
return ptd_isf(q, self.alpha, self.beta)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.alpha, self.beta)
[docs]
def logcdf(self, x):
return ptd_logcdf(x, self.alpha, self.beta)
[docs]
def logsf(self, x):
return ptd_logsf(x, self.alpha, self.beta)
def logisf(self, q):
return ptd_logisf(q, 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)
@pytensor_jit
def ptd_pdf(x, alpha, beta):
return ptd_beta.pdf(x, alpha, beta)
@pytensor_jit
def ptd_cdf(x, alpha, beta):
return ptd_beta.cdf(x, alpha, beta)
@pytensor_jit
def ptd_ppf(q, alpha, beta):
return ptd_beta.ppf(q, alpha, beta)
@pytensor_jit
def ptd_sf(x, alpha, beta):
return ptd_beta.sf(x, alpha, beta)
@pytensor_jit
def ptd_isf(q, alpha, beta):
return ptd_beta.isf(q, alpha, beta)
@pytensor_jit
def ptd_logpdf(x, alpha, beta):
return ptd_beta.logpdf(x, alpha, beta)
@pytensor_jit
def ptd_logcdf(x, alpha, beta):
return ptd_beta.logcdf(x, alpha, beta)
@pytensor_jit
def ptd_logsf(x, alpha, beta):
return ptd_beta.logsf(x, alpha, beta)
@pytensor_jit
def ptd_logisf(q, alpha, beta):
return ptd_beta.logisf(q, alpha, beta)
@pytensor_jit
def ptd_entropy(alpha, beta):
return ptd_beta.entropy(alpha, beta)
@pytensor_jit
def ptd_mean(alpha, beta):
return ptd_beta.mean(alpha, beta)
@pytensor_jit
def ptd_mode(alpha, beta):
return ptd_beta.mode(alpha, beta)
@pytensor_jit
def ptd_median(alpha, beta):
return ptd_beta.median(alpha, beta)
@pytensor_jit
def ptd_var(alpha, beta):
return ptd_beta.var(alpha, beta)
@pytensor_jit
def ptd_std(alpha, beta):
return ptd_beta.std(alpha, beta)
@pytensor_jit
def ptd_skewness(alpha, beta):
return ptd_beta.skewness(alpha, beta)
@pytensor_jit
def ptd_kurtosis(alpha, beta):
return ptd_beta.kurtosis(alpha, beta)
@pytensor_jit
def ptd_lmoment1(alpha, beta):
return ptd_beta.lmoment1(alpha, beta)
@pytensor_jit
def ptd_lmoment2(alpha, beta):
return ptd_beta.lmoment2(alpha, beta)
@pytensor_jit
def ptd_lmoment3(alpha, beta):
return ptd_beta.lmoment3(alpha, beta)
@pytensor_jit
def ptd_lmoment4(alpha, beta):
return ptd_beta.lmoment4(alpha, beta)
@pytensor_rng_jit
def ptd_rvs(alpha, beta, size, rng):
return ptd_beta.rvs(alpha, beta, size=size, random_state=rng)
