import numba as nb
import numpy as np
from pytensor_distributions import binomial as ptd_binomial
from preliz.distributions.distributions import Discrete
from preliz.internal.distribution_helper import all_not_none, eps, pytensor_jit, pytensor_rng_jit
from preliz.internal.optimization import optimize_mean_sigma
from preliz.internal.special import mean_and_std
[docs]
class Binomial(Discrete):
R"""
Binomial distribution.
The discrete probability distribution of the number of successes
in a sequence of n independent yes/no experiments, each of which
yields success with probability p.
The pmf of this distribution is
.. math:: f(x \mid n, p) = \binom{n}{x} p^x (1-p)^{n-x}
.. plot::
:context: close-figs
from preliz import Binomial, style
style.use('preliz-doc')
ns = [5, 10, 10]
ps = [0.5, 0.5, 0.7]
for n, p in zip(ns, ps):
Binomial(n, p).plot_pdf()
======== ==========================================
Support :math:`x \in \{0, 1, \ldots, n\}`
Mean :math:`n p`
Variance :math:`n p (1 - p)`
======== ==========================================
Parameters
----------
n : int
Number of Bernoulli trials (n >= 0).
p : float
Probability of success in each trial (0 < p < 1).
"""
def __init__(self, n=None, p=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(n, p)
def _parametrization(self, n=None, p=None):
self.n = n
self.p = p
self.params = (self.n, self.p)
self.param_names = ("n", "p")
self.params_support = ((eps, np.inf), (eps, 1 - eps))
if all_not_none(n, p):
self._update(n, p)
def _update(self, n, p):
self.n = np.int64(n)
self.p = np.float64(p)
self._q = 1 - self.p
self.params = (self.n, self.p)
self.support = (0, self.n)
self.is_frozen = True
def _fit_moments(self, mean, sigma):
# crude approximation for n and p
n = mean + sigma * 2
p = mean / n
params = n, p
optimize_mean_sigma(self, mean, sigma, params)
def _fit_mle(self, sample):
self._update(*nb_fit_mle(sample))
[docs]
def pdf(self, x):
return ptd_pdf(x, self.n, self.p)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.n, self.p)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.n, self.p)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.n, self.p)
[docs]
def entropy(self):
return ptd_entropy(self.n, self.p)
[docs]
def mean(self):
return ptd_mean(self.n, self.p)
[docs]
def mode(self):
return ptd_mode(self.n, self.p)
[docs]
def var(self):
return ptd_var(self.n, self.p)
[docs]
def std(self):
return ptd_std(self.n, self.p)
[docs]
def skewness(self):
return ptd_skewness(self.n, self.p)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.n, self.p)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.n, self.p)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.n, self.p)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.n, self.p)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.n, self.p)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.n, self.p, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
# crude approximation for n and p
n = mean + sigma * 2
p = mean / n
params = n, p
return optimize_mean_sigma(self, mean, sigma, params)
def _fit_mle(self, sample):
self._update(*nb_fit_mle(sample))
@pytensor_jit
def ptd_pdf(x, n, p):
return ptd_binomial.pdf(x, n, p)
@pytensor_jit
def ptd_cdf(x, n, p):
return ptd_binomial.cdf(x, n, p)
@pytensor_jit
def ptd_ppf(q, n, p):
return ptd_binomial.ppf(q, n, p)
@pytensor_jit
def ptd_logpdf(x, n, p):
return ptd_binomial.logpdf(x, n, p)
@pytensor_jit
def ptd_entropy(n, p):
return ptd_binomial.entropy(n, p)
@pytensor_jit
def ptd_mean(n, p):
return ptd_binomial.mean(n, p)
@pytensor_jit
def ptd_mode(n, p):
return ptd_binomial.mode(n, p)
@pytensor_jit
def ptd_median(n, p):
return ptd_binomial.median(n, p)
@pytensor_jit
def ptd_var(n, p):
return ptd_binomial.var(n, p)
@pytensor_jit
def ptd_std(n, p):
return ptd_binomial.std(n, p)
@pytensor_jit
def ptd_skewness(n, p):
return ptd_binomial.skewness(n, p)
@pytensor_jit
def ptd_kurtosis(n, p):
return ptd_binomial.kurtosis(n, p)
@pytensor_jit
def ptd_lmoment1(n, p):
return ptd_binomial.lmoment1(n, p)
@pytensor_jit
def ptd_lmoment2(n, p):
return ptd_binomial.lmoment2(n, p)
@pytensor_jit
def ptd_lmoment3(n, p):
return ptd_binomial.lmoment3(n, p)
@pytensor_jit
def ptd_lmoment4(n, p):
return ptd_binomial.lmoment4(n, p)
@pytensor_rng_jit
def ptd_rvs(n, p, size, rng):
return ptd_binomial.rvs(n, p, size=size, random_state=rng)
@nb.njit(cache=True)
def nb_fit_mle(sample):
# see https://doi.org/10.1016/j.jspi.2004.02.019 for details
x_bar, x_std = mean_and_std(sample)
x_max = np.max(sample)
n = np.ceil(x_max ** (1.5) * x_std / (x_bar**0.5 * (x_max - x_bar) ** 0.5))
p = x_bar / n
return n, p
