basic.py

import abc
from functools import partial
from typing import List, Optional, Union

import numpy as np
import scipy.stats as stats

import aesara
from aesara.tensor.basic import as_tensor_variable
from aesara.tensor.random.op import RandomVariable, default_supp_shape_from_params
from aesara.tensor.random.type import RandomGeneratorType, RandomStateType
from aesara.tensor.random.utils import broadcast_params
from aesara.tensor.random.var import (
    RandomGeneratorSharedVariable,
    RandomStateSharedVariable,
)


try:
    broadcast_shapes = np.broadcast_shapes
except AttributeError:
    from numpy.lib.stride_tricks import _broadcast_shape

    def broadcast_shapes(*shapes):
        return _broadcast_shape(*[np.empty(x, dtype=[]) for x in shapes])


def get_partial_wrapper(rv, name, *args, **kwargs):
    func = partial(rv, *args, **kwargs)
    func.__name__ = name
    func.__module__ = rv.__module__
    return func


class ScipyRandomVariable(RandomVariable):
    r"""A class for straightforward `RandomVariable`\s that use SciPy-based samplers.

    By "straightforward" we mean `RandomVariable`\s for which the output shape
    is entirely determined by broadcasting the distribution parameters
    (e.g. basic scalar distributions).

    The more sophisticated shape logic performed by `RandomVariable` is avoided
    in order to reduce the amount of unnecessary steps taken to correct SciPy's
    shape-reducing defects.

    """

    @classmethod
    @abc.abstractmethod
    def rng_fn_scipy(cls, rng, *args, **kwargs):
        r"""

        `RandomVariable`\s implementations that want to use SciPy-based samplers
        need to implement this method instead of the base
        `RandomVariable.rng_fn`; otherwise their broadcast dimensions will be
        dropped by SciPy.

        """

    @classmethod
    def rng_fn(cls, *args, **kwargs):
        size = args[-1]
        res = cls.rng_fn_scipy(*args, **kwargs)

        if np.ndim(res) == 0:
            # The sample is an `np.number`, and is not writeable, or non-NumPy
            # type, so we need to clone/create a usable NumPy result
            res = np.asarray(res)

        if size is None:
            # SciPy will sometimes drop broadcastable dimensions; we need to
            # check and, if necessary, add them back
            exp_shape = broadcast_shapes(*[np.shape(a) for a in args[1:-1]])
            if res.shape != exp_shape:
                return np.broadcast_to(res, exp_shape).copy()

        return res


class UniformRV(RandomVariable):
    r"""A uniform continuous random variable.

    The probability density function for `uniform` within the interval :math:`[l, h)` is:

    .. math::
        \begin{split}
            f(x; l, h) = \begin{cases}
                          \frac{1}{h-l}\quad \text{for $l \leq x \leq h$},\\
                           0\quad \text{otherwise}.
                       \end{cases}
        \end{split}

    """
    name = "uniform"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("U", "\\operatorname{U}")

    def __call__(self, low=0.0, high=1.0, size=None, **kwargs):
        r"""Draw samples from a uniform distribution.

        The results are undefined when `high < low`.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        low
           Lower boundary :math:`l` of the output interval; all values generated
           will be greater than or equal to `low`.
        high
           Upper boundary :math:`h` of the output interval; all values generated
           will be less than or equal to `high`.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(low, high, size=size, **kwargs)


uniform = UniformRV()


class TriangularRV(RandomVariable):
    r"""A triangular continuous random variable.

    The probability density function for `triangular` within the interval :math:`[l, r)`
    and mode :math:`m` (where the peak of the distribution occurs) is:

    .. math::

        \begin{split}
            f(x; l, m, r) = \begin{cases}
                                \frac{2(x-l)}{(r-l)(m-l)}\quad \text{for $l \leq x \leq m$},\\
                                \frac{2(r-x)}{(r-l)(r-m)}\quad \text{for $m \leq x \leq r$},\\
                                0\quad \text{otherwise}.
                            \end{cases}
        \end{split}

    """
    name = "triangular"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "floatX"
    _print_name = ("Triang", "\\operatorname{Triang}")

    def __call__(self, left, mode, right, size=None, **kwargs):
        r"""Draw samples from a triangular distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        left
           Lower boundary :math:`l` of the output interval; all values generated
           will be greater than or equal to `left`.
        mode
           Mode :math:`m` of the distribution, where the peak occurs. Must be such
           that `left <= mode <= right`.
        right
           Upper boundary :math:`r` of the output interval; all values generated
           will be less than or equal to `right`. Must be larger than `left`.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(left, mode, right, size=size, **kwargs)


triangular = TriangularRV()


class BetaRV(RandomVariable):
    r"""A beta continuous random variable.

    The probability density function for `beta` in terms of its parameters :math:`\alpha`
    and :math:`\beta` is:

    .. math::

        f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1}

    for :math:`0 \leq x \leq 1`. :math:`B` is the beta function defined as:

    .. math::

        B(\alpha, \beta) = \int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \mathrm{d}t

    """
    name = "beta"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Beta", "\\operatorname{Beta}")

    def __call__(self, alpha, beta, size=None, **kwargs):
        r"""Draw samples from a beta distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        alpha
            Alpha parameter :math:`\alpha` of the distribution. Must be positive.
        beta
            Beta parameter :math:`\beta` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(alpha, beta, size=size, **kwargs)


beta = BetaRV()


class RayleighRV(RandomVariable):
    r"""A rayleigh continuous random variable.

    The probability density function for `rayleigh` is:

    .. math::

        P(x;scale) = \\frac{x}{scale^2}e^{\\frac{-x^2}{2 \\cdotp scale^2}}


    """
    name = "rayleigh"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "floatX"
    _print_name = ("Rayleigh", "\\operatorname{Rayleigh}")

    def __call__(self, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a rayleigh distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        scale
            Scale, also equals the mode. Must be non-negative. Default is 1
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(scale, size=size, **kwargs)


rayleigh = RayleighRV()


class PowerRV(RandomVariable):
    r"""A power continuous random variable.

    The probability density function for `power` is:

    .. math::

        P(x; a) = ax^{a-1}, 0 \\le x \\le 1, a>0.

    """
    name = "power"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "floatX"
    _print_name = ("Power", "\\operatorname{Power}")

    def __call__(self, a, size=None, **kwargs):
        r"""Draw samples from a rayleigh distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        a
            Parameter of the distribution. Must be non-negative.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(a, size=size, **kwargs)


power = PowerRV()


class ZipfRV(RandomVariable):
    r"""A zipf discrete random variable.

    The probability density function for `zipf` is:

    .. math::

        p(x) = \\frac{x^{-a}}{\\zeta(a)},

    where :math:`\\zeta` is the Riemann Zeta function.

    """
    name = "zipf"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "int64"
    _print_name = ("Zipf", "\\operatorname{Zipf}")

    def __call__(self, a, size=None, **kwargs):
        r"""Draw samples from a rayleigh distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        a
            Distribution parameter. Must be greater than 1.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(a, size=size, **kwargs)


zipf = ZipfRV()


class NormalRV(RandomVariable):
    r"""A normal continuous random variable.

    The probability density function for `normal` in terms of its location parameter (mean)
    :math:`\mu` and scale parameter (standard deviation) :math:`\sigma` is:

    .. math::

        f(x; \mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

    for :math:`\sigma > 0`.

    """
    name = "normal"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("N", "\\operatorname{N}")

    def __call__(self, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a normal distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        loc
            Mean :math:`\mu` of the normal distribution.
        scale
            Standard deviation :math:`\sigma` of the normal distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)


normal = NormalRV()
standard_normal = get_partial_wrapper(normal, "standard_normal", loc=0.0, scale=1.0)


class HalfNormalRV(ScipyRandomVariable):
    r"""A half-normal continuous random variable.

    The probability density function for `halfnormal` in terms of its location parameter
    :math:`\mu` and scale parameter :math:`\sigma` is:

    .. math::

        f(x; \mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

    for :math:`x \geq 0` and :math:`\sigma > 0`.

    """
    name = "halfnormal"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("N**+", "\\operatorname{N^{+}}")

    def __call__(self, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a half-normal distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        loc
            Location parameter :math:`\mu` of the distribution.
        scale
            Scale parameter :math:`\sigma` of the distribution.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, loc, scale, size):
        r"""Draw sample from a half-normal distribution using Scipy's generator.

        Parameters
        ----------
        loc
            Location parameter :math:`\mu` of the distribution.
        scale
            Scale parameter :math:`\sigma` of the distribution.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return stats.halfnorm.rvs(loc, scale, random_state=rng, size=size)


halfnormal = HalfNormalRV()


class LogNormalRV(RandomVariable):
    r"""A lognormal continuous random variable.

    The probability density function for `lognormal` in terms of the mean
    parameter :math:`\mu` and sigma parameter :math:`\sigma` is:

    .. math::

        f(x; \mu, \sigma) = \frac{1}{x \sqrt{2 \pi \sigma^2}} e^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}}

    for :math:`x > 0` and :math:`\sigma > 0`.

    """
    name = "lognormal"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("LogN", "\\operatorname{LogN}")

    def __call__(self, mean=0.0, sigma=1.0, size=None, **kwargs):
        r"""Draw sample from a lognormal distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        mean
            Mean :math:`\mu` of the random variable's natural logarithm.
        sigma
            Standard deviation :math:`\sigma` of the random variable's natural logarithm.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(mean, sigma, size=size, **kwargs)


lognormal = LogNormalRV()


class GammaRV(ScipyRandomVariable):
    r"""A gamma continuous random variable.

    The probability density function for `gamma` in terms of the shape parameter
    :math:`\alpha` and rate parameter :math:`\beta` is:

    .. math::

        f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}

    for :math:`x \geq 0`, :math:`\alpha > 0` and :math:`\beta > 0`. :math:`\Gamma` is
    the gamma function:

    .. math::

        \Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t} \mathrm{d}t

    """
    name = "gamma"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Gamma", "\\operatorname{Gamma}")

    def __call__(self, shape, rate, size=None, **kwargs):
        r"""Draw samples from a gamma distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        shape
            The shape :math:`\alpha` of the gamma distribution. Must be positive.
        rate
            The rate :math:`\beta` of the gamma distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(shape, 1.0 / rate, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, shape, scale, size):
        return stats.gamma.rvs(shape, scale=scale, size=size, random_state=rng)


gamma = GammaRV()
standard_gamma = get_partial_wrapper(gamma, "standard_gamma", rate=1.0)


class ChiSquareRV(RandomVariable):
    r"""A chi square continuous random variable.

    The probability density function for `chisquare` in terms of the number of degrees of
    freedom :math:`k` is:

    .. math::

        f(x; k) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2-1} e^{-x/2}

    for :math:`k > 2`. :math:`\Gamma` is the gamma function:

    .. math::

        \Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t} \mathrm{d}t


    This variable is obtained by summing the squares :math:`k` independent, standard normally
    distributed random variables.

    """
    name = "chisquare"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "floatX"
    _print_name = ("ChiSquare", "\\operatorname{ChiSquare}")

    def __call__(self, df, size=None, **kwargs):
        r"""Draw samples from a chisquare distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        df
            The number :math:`k` of degrees of freedom. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(df, size=size, **kwargs)


chisquare = ChiSquareRV()


class ParetoRV(ScipyRandomVariable):
    r"""A pareto continuous random variable.

    The probability density function for `pareto` in terms of its shape parameter :math:`b` and
    scale parameter :math:`x_m` is:

    .. math::

        f(x; b, x_m) = \frac{b x_m^b}{x^{b+1}}

    and is defined for :math:`x \geq x_m`.

    """
    name = "pareto"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Pareto", "\\operatorname{Pareto}")

    def __call__(self, b, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a pareto distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        b
            The shape :math:`b` (or exponent) of the pareto distribution. Must be positive.
        scale
            The scale :math:`x_m` of the pareto distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(b, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, b, scale, size):
        return stats.pareto.rvs(b, scale=scale, size=size, random_state=rng)


pareto = ParetoRV()


class GumbelRV(ScipyRandomVariable):
    r"""A gumbel continuous random variable.

    The probability density function for `gumbel` in terms of its location parameter :math:`\mu` and
    scale parameter :math:`\beta` is:

    .. math::

        f(x; \mu, \beta) = \frac{\exp(-(x + e^{(x-\mu)/\beta})}{\beta}

    for :math:`\beta > 0`.

    """
    name = "gumbel"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Gumbel", "\\operatorname{Gumbel}")

    def __call__(
        self,
        loc: Union[np.ndarray, float],
        scale: Union[np.ndarray, float] = 1.0,
        size: Optional[Union[List[int], int]] = None,
        **kwargs,
    ) -> RandomVariable:
        r"""Draw samples from a gumbel distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        loc
            The location parameter :math:`\mu` of the distribution.
        scale
            The scale :math:`\beta` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(
        cls,
        rng: Union[np.random.Generator, np.random.RandomState],
        loc: Union[np.ndarray, float],
        scale: Union[np.ndarray, float],
        size: Optional[Union[List[int], int]],
    ) -> np.ndarray:
        return stats.gumbel_r.rvs(loc=loc, scale=scale, size=size, random_state=rng)


gumbel = GumbelRV()


class ExponentialRV(RandomVariable):
    r"""An exponential continuous random variable.

    The probability density function for `exponential` in terms of its scale parameter :math:`\beta` is:

    .. math::

        f(x; \beta) = \frac{\exp(-x / \beta)}{\beta}

    for :math:`x \geq 0` and :math:`\beta > 0`.

    """
    name = "exponential"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "floatX"
    _print_name = ("Exp", "\\operatorname{Exp}")

    def __call__(self, scale=1.0, size=None, **kwargs):
        r"""Draw samples from an exponential distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        scale
            The scale :math:`\beta` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(scale, size=size, **kwargs)


exponential = ExponentialRV()
standard_exponential = get_partial_wrapper(
    exponential, "standard_exponential", scale=1.0
)


class WeibullRV(RandomVariable):
    r"""A weibull continuous random variable.

    The probability density function for `weibull` in terms of its shape parameter :math:`k` is :

    .. math::

        f(x; k) = k x^{k-1} e^{-x^k}

    for :math:`x \geq 0` and :math:`k > 0`.

    """
    name = "weibull"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "floatX"
    _print_name = ("Weibull", "\\operatorname{Weibull}")

    def __call__(self, shape, size=None, **kwargs):
        r"""Draw samples from a weibull distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        shape
            The shape :math:`k` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(shape, size=size, **kwargs)


weibull = WeibullRV()


class LogisticRV(RandomVariable):
    r"""A logistic continuous random variable.

    The probability density function for `logistic` in terms of its location parameter :math:`\mu` and
    scale parameter :math:`s` is :

    .. math::

        f(x; \mu, s) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}

    for :math:`s > 0`.

    """
    name = "logistic"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Logistic", "\\operatorname{Logistic}")

    def __call__(self, loc=0, scale=1, size=None, **kwargs):
        r"""Draw samples from a logistic distribution.

        Signature
        ---------

        `(), () -> ()`


        Parameters
        ----------
        loc
            The location parameter :math:`\mu` of the distribution.
        scale
            The scale :math:`s` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)


logistic = LogisticRV()


class VonMisesRV(RandomVariable):
    r"""A von Misses continuous random variable.

    The probability density function for `vonmisses` in terms of its mode :math:`\mu` and
    dispersion parameter :math:`\kappa` is :

    .. math::

        f(x; \mu, \kappa) = \frac{e^{\kappa \cos(x-\mu)}}{2 \pi I_0(\kappa)}

    for :math:`x \in [-\pi, \pi]` and :math:`\kappa > 0`. :math:`I_0` is the modified Bessel
    function of order 0.

    """
    name = "vonmises"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("VonMises", "\\operatorname{VonMises}")

    def __call__(self, mu, kappa, size=None, **kwargs):
        r"""Draw samples from a von Mises distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        mu
            The mode :math:`\mu` of the distribution.
        kappa
            The dispersion parameter :math:`\kappa` of the distribution. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(mu, kappa, size=size, **kwargs)


vonmises = VonMisesRV()


def safe_multivariate_normal(mean, cov, size=None, rng=None):
    """A shape consistent multivariate normal sampler.

    What we mean by "shape consistent": SciPy will return scalars when the
    arguments are vectors with dimension of size 1.  We require that the output
    be at least 1D, so that it's consistent with the underlying random
    variable.

    """
    res = np.atleast_1d(
        stats.multivariate_normal(mean=mean, cov=cov, allow_singular=True).rvs(
            size=size, random_state=rng
        )
    )

    if size is not None:
        res = res.reshape(list(size) + [-1])

    return res


class MvNormalRV(RandomVariable):
    r"""A multivariate normal random variable.

    The probability density function for `multivariate_normal` in term of its location parameter
    :math:`\boldsymbol{\mu}` and covariance matrix :math:`\Sigma` is

    .. math::

        f(\boldsymbol{x}; \boldsymbol{\mu}, \Sigma) = \det(2 \pi \Sigma)^{-1/2}  \exp\left(-\frac{1}{2} (\boldsymbol{x} - \boldsymbol{\mu})^T \Sigma (\boldsymbol{x} - \boldsymbol{\mu})\right)

    where :math:`\Sigma` is a positive semi-definite matrix.

    """
    name = "multivariate_normal"
    ndim_supp = 1
    ndims_params = [1, 2]
    dtype = "floatX"
    _print_name = ("N", "\\operatorname{N}")

    def __call__(self, mean=None, cov=None, size=None, **kwargs):
        r""" "Draw samples from a multivariate normal distribution.

        Signature
        ---------

        `(n), (n,n) -> (n)`

        Parameters
        ----------
        mean
            Location parameter (mean) :math:`\boldsymbol{\mu}` of the distribution. Vector
            of length `N`.
        cov
            Covariance matrix :math:`\Sigma` of the distribution. Must be a symmetric
            and positive-semidefinite `NxN` matrix.
        size
            Given a size of, for example, `(m, n, k)`, `m * n * k` independent,
            identically distributed samples are generated. Because each sample
            is `N`-dimensional, the output shape is `(m, n, k, N)`. If no shape
            is specified, a single `N`-dimensional sample is returned.

        """
        dtype = aesara.config.floatX if self.dtype == "floatX" else self.dtype

        if mean is None:
            mean = np.array([0.0], dtype=dtype)
        if cov is None:
            cov = np.array([[1.0]], dtype=dtype)
        return super().__call__(mean, cov, size=size, **kwargs)

    @classmethod
    def rng_fn(cls, rng, mean, cov, size):
        if mean.ndim > 1 or cov.ndim > 2:
            # Neither SciPy nor NumPy implement parameter broadcasting for
            # multivariate normals (or any other multivariate distributions),
            # so we need to implement that here

            size = tuple(size or ())
            if size:
                mean = np.broadcast_to(mean, size + mean.shape[-1:])
                cov = np.broadcast_to(cov, size + cov.shape[-2:])
            else:
                mean, cov = broadcast_params([mean, cov], cls.ndims_params)

            res = np.empty(mean.shape)
            for idx in np.ndindex(mean.shape[:-1]):
                m = mean[idx]
                c = cov[idx]
                res[idx] = safe_multivariate_normal(m, c, rng=rng)
            return res
        else:
            return safe_multivariate_normal(mean, cov, size=size, rng=rng)


multivariate_normal = MvNormalRV()


class DirichletRV(RandomVariable):
    r"""A Dirichlet continuous random variable.

    The probability density function for `dirichlet` in terms of the vector of
    concentration parameters :math:`\boldsymbol{\alpha}` is:

    .. math::

        f(x; \boldsymbol{\alpha}) = \prod_{i=1}^k x_i^{\alpha_i-1}

    where :math:`x` is a vector, such that :math:`x_i > 0\;\forall i` and
    :math:`\sum_{i=1}^k x_i = 1`.

    """
    name = "dirichlet"
    ndim_supp = 1
    ndims_params = [1]
    dtype = "floatX"
    _print_name = ("Dir", "\\operatorname{Dir}")

    def __call__(self, alphas, size=None, **kwargs):
        r"""Draw samples from a dirichlet distribution.

        Signature
        ---------

        `(k) -> (k)`

        Parameters
        ----------
        alphas
            A sequence of concentration parameters :math:`\boldsymbol{\alpha}` of the
            distribution. A sequence of length `k` will produce samples of length `k`.
        size
            Given a size of, for example, `(r, s, t)`, `r * s * t` independent,
            identically distributed samples are generated. Because each sample
            is `k`-dimensional, the output shape is `(r, s, t, k)`. If no shape
            is specified, a single `k`-dimensional sample is returned.

        """
        return super().__call__(alphas, size=size, **kwargs)

    @classmethod
    def rng_fn(cls, rng, alphas, size):
        if alphas.ndim > 1:
            if size is None:
                size = ()

            size = tuple(np.atleast_1d(size))

            if size:
                alphas = np.broadcast_to(alphas, size + alphas.shape[-1:])

            samples_shape = alphas.shape
            samples = np.empty(samples_shape)
            for index in np.ndindex(*samples_shape[:-1]):
                samples[index] = rng.dirichlet(alphas[index])

            return samples
        else:
            return rng.dirichlet(alphas, size=size)


dirichlet = DirichletRV()


class PoissonRV(RandomVariable):
    r"""A poisson discrete random variable.

    The probability mass function for `poisson` in terms of the expected number
    of events :math:`\lambda` is:

    .. math::

        f(k; \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}

    for :math:`\lambda > 0`.

    """
    name = "poisson"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "int64"
    _print_name = ("Pois", "\\operatorname{Pois}")

    def __call__(self, lam=1.0, size=None, **kwargs):
        r"""Draw samples from a poisson distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        lam
            Expected number of events :math:`\lambda`. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(lam, size=size, **kwargs)


poisson = PoissonRV()


class GeometricRV(RandomVariable):
    r"""A geometric discrete random variable.

    The probability mass function for `geometric` for the number of successes :math:`k`
    before the first failure in terms of the probability of success :math:`p` of a single
    trial is:

    .. math::

        f(k; p) = p^{k-1}(1-p)

    for :math:`0 \geq p \geq 1`.

    """
    name = "geometric"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "int64"
    _print_name = ("Geom", "\\operatorname{Geom}")

    def __call__(self, p, size=None, **kwargs):
        r"""Draw samples from a geometric distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        p
            Probability of success :math:`p` of an individual trial.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n *
            k` independent, identically distributed samples are returned.
            Default is `None` in which case a single sample is returned.

        """
        return super().__call__(p, size=size, **kwargs)


geometric = GeometricRV()


class HyperGeometricRV(RandomVariable):
    r"""A hypergeometric discrete random variable.

    The probability mass function for `hypergeometric` for the number of
    successes :math:`k` in :math:`n` draws without replacement, from a
    finite population of size :math:`N` with :math:`K` desired items is:

    .. math::

        f(k; n, N, K) = \frac{{K \choose k} {N-K \choose n-k}}{{N \choose n}}

    """
    name = "hypergeometric"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "int64"
    _print_name = ("HyperGeom", "\\operatorname{HyperGeom}")

    def __call__(self, ngood, nbad, nsample, size=None, **kwargs):
        r"""Draw samples from a geometric distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        ngood
            Number :math:`K` of desirable items in the population. Positive integer.
        nbad
            Number :math:`N-K` of undesirable items in the population. Positive integer.
        nsample
            Number :math:`n` of items sampled. Must be less than :math:`N`,
            i.e. `ngood + nbad`.` Positive integer.
        size
           Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None` in which case a single sample is returned.

        """
        return super().__call__(ngood, nbad, nsample, size=size, **kwargs)


hypergeometric = HyperGeometricRV()


class CauchyRV(ScipyRandomVariable):
    r"""A Cauchy continuous random variable.

    The probability density function for `cauchy` in terms of its location
    parameter :math:`x_0` and scale parameter :math:`\gamma` is:

    .. math::

        f(x; x_0, \gamma) = \frac{1}{\pi \gamma \left(1 + (\frac{x-x_0}{\gamma})^2\right)}

    where :math:`\gamma > 0`.

    """
    name = "cauchy"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("C", "\\operatorname{C}")

    def __call__(self, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a Cauchy distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        loc
            Location parameter :math:`x_0` of the distribution.
        scale
            Scale parameter :math:`\gamma` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None` in which case a single sample is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, loc, scale, size):
        return stats.cauchy.rvs(loc=loc, scale=scale, random_state=rng, size=size)


cauchy = CauchyRV()
standard_cauchy = get_partial_wrapper(cauchy, "standard_cauchy", loc=0.0, scale=1.0)


class HalfCauchyRV(ScipyRandomVariable):
    r"""A half-Cauchy continuous random variable.

    The probability density function for `halfcauchy` in terms of its location
    parameter :math:`x_0` and scale parameter :math:`\gamma` is:

    .. math::

        f(x; x_0, \gamma) = \frac{1}{\pi \gamma \left(1 + (\frac{x-x_0}{\gamma})^2\right)}

    for :math:`x \geq 0` where :math:`\gamma > 0`.

    """
    name = "halfcauchy"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("C**+", "\\operatorname{C^{+}}")

    def __call__(self, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a half-Cauchy distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        loc
            Location parameter :math:`x_0` of the distribution.
        scale
            Scale parameter :math:`\gamma` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None`, in which case a single sample is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, loc, scale, size):
        return stats.halfcauchy.rvs(loc=loc, scale=scale, random_state=rng, size=size)


halfcauchy = HalfCauchyRV()


class InvGammaRV(ScipyRandomVariable):
    r"""An inverse-gamma continuous random variable.

    The probability density function for `invgamma` in terms of its shape
    parameter :math:`\alpha` and scale parameter :math:`\beta` is:

    .. math::

        f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-(\alpha+1)} \exp\left(-\frac{\beta}{x}\right)

    for :math:`x > 0`, where :math:`\alpha > 0` and :math:`\beta > 0`. :math:`Gamma` is the gamma function :

    .. math::

        \Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t} \mathrm{d}t

    """
    name = "invgamma"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("InvGamma", "\\operatorname{Gamma^{-1}}")

    def __call__(self, shape, scale, size=None, **kwargs):
        r"""Draw samples from an inverse-gamma distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        shape
            Shape parameter :math:`\alpha` of the distribution. Must be positive.
        scale
            Scale parameter :math:`\beta` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed sample are returned. Default is
           `None`, in which case a single sample is returned.

        """
        return super().__call__(shape, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, shape, scale, size):
        return stats.invgamma.rvs(shape, scale=scale, size=size, random_state=rng)


invgamma = InvGammaRV()


class WaldRV(RandomVariable):
    r"""A Wald (or inverse Gaussian) continuous random variable.

    The probability density function for `wald` in terms of its mean
    parameter :math:`\mu` and shape parameter :math:`\lambda` is:

    .. math::

        f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}} \exp\left(-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}\right)

    for :math:`x > 0`, where :math:`\mu > 0` and :math:`\lambda > 0`.

    """
    name = "wald"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name_ = ("Wald", "\\operatorname{Wald}")

    def __call__(self, mean=1.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a Wald distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        mean
            Mean parameter :math:`\mu` of the distribution. Must be positive.
        shape
            Shape parameter :math:`\lambda` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None`, in which case a single sample is returned.

        """
        return super().__call__(mean, scale, size=size, **kwargs)


wald = WaldRV()


class TruncExponentialRV(ScipyRandomVariable):
    r"""A truncated exponential continuous random variable.

    The probability density function for `truncexp` in terms of its shape
    parameter :math:`b`, location parameter :math:`\alpha` and scale
    parameter :math:`\beta` is:

    .. math::

        f(x; b, \alpha, \beta) = \frac{\exp(-(x-\alpha)/\beta)}{\beta (1-\exp(-b))}

    for :math:`0 \leq x \leq b` and :math:`\beta > 0`.

    """
    name = "truncexpon"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "floatX"
    _print_name = ("TruncExp", "\\operatorname{TruncExp}")

    def __call__(self, b, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a truncated exponential distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        b
            Shape parameter :math:`b` of the distribution. Must be positive.
        loc
            Location parameter :math:`\alpha` of the distribution.
        scale
            Scale parameter :math:`\beta` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None` in which case a single sample is returned.

        """
        return super().__call__(b, loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, b, loc, scale, size):
        return stats.truncexpon.rvs(
            b, loc=loc, scale=scale, size=size, random_state=rng
        )


truncexpon = TruncExponentialRV()


class StudentTRV(ScipyRandomVariable):
    r"""A Student's t continuous random variable.

    The probability density function for `t` in terms of its degrees of freedom
    parameter :math:`\nu`, location parameter :math:`\mu` and scale
    parameter :math:`\sigma` is:

    .. math::

        f(x; \nu, \alpha, \beta) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{1}{\pi\nu\sigma}\right)^{\frac{1}{2}} \left[1+\frac{(x-\mu)^2}{\nu\sigma}\right]^{-\frac{\nu+1}{2}}

    for :math:`\nu > 0`, :math:`\sigma > 0`.

    """
    name = "t"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "floatX"
    _print_name = ("StudentT", "\\operatorname{StudentT}")

    def __call__(self, df, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a Student's t distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        df
            Degrees of freedom parameter :math:`\nu` of the distribution. Must be
            positive.
        loc
            Location parameter :math:`\mu` of the distribution.
        scale
            Scale parameter :math:`\sigma` of the distribution. Must be
            positive.
        size
           Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
           independent, identically distributed samples are returned. Default is
           `None` in which case a single sample is returned.

        """
        return super().__call__(df, loc, scale, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, df, loc, scale, size):
        return stats.t.rvs(df, loc=loc, scale=scale, size=size, random_state=rng)


t = StudentTRV()
standard_t = get_partial_wrapper(t, "standard_t", loc=0.0, scale=1.0)


class BernoulliRV(ScipyRandomVariable):
    r"""A Bernoulli discrete random variable.

    The probability mass function for `bernoulli` in terms of the probability
    of success :math:`p` of a single trial is:


    .. math::

        \begin{split}
            f(k; p) = \begin{cases}
                                (1-p)\quad \text{if $k = 0$},\\
                                p\quad \text{if $k=1$}\\
                        \end{cases}
        \end{split}

    where :math:`0 \leq p \leq 1`.

    """
    name = "bernoulli"
    ndim_supp = 0
    ndims_params = [0]
    dtype = "int64"
    _print_name = ("Bern", "\\operatorname{Bern}")

    def __call__(self, p, size=None, **kwargs):
        r"""Draw samples from a Bernoulli distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        p
            Probability of success :math:`p` of a single trial.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are returned. Default
            is `None` in which case a single sample is returned.

        """
        return super().__call__(p, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, p, size):
        return stats.bernoulli.rvs(p, size=size, random_state=rng)


bernoulli = BernoulliRV()


class LaplaceRV(RandomVariable):
    r"""A Laplace continuous random variable.

    The probability density function for `laplace` in terms of its location
    parameter :math:`\mu` and scale parameter :math:`\lambda` is:

    .. math::

        f(x; \mu, \lambda) = \frac{1}{2 \lambda} \exp\left(-\frac{|x-\mu|}{\lambda}\right)

    with :math:`\lambda > 0`.

    """
    name = "laplace"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("Laplace", "\\operatorname{Laplace}")

    def __call__(self, loc=0.0, scale=1.0, size=None, **kwargs):
        r"""Draw samples from a Laplace distribution.

        Signature
        ---------

        `(), () -> ()`


        Parameters
        ----------
        loc
            Location parameter :math:`\mu` of the distribution.
        scale
            Scale parameter :math:`\lambda` of the distribution. Must be
            positive.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are returned. Default
            is `None` in which case a single sample is returned.

        """
        return super().__call__(loc, scale, size=size, **kwargs)


laplace = LaplaceRV()


class BinomialRV(RandomVariable):
    r"""A binomial discrete random variable.

    The probability mass function for `binomial` for the number :math:`k` of successes
    in terms of the probability of success :math:`p` of a single trial and the number
    :math:`n` of trials is:

    .. math::

            f(k; p, n) = {n \choose k} p^k (1-p)^{n-k}

    """
    name = "binomial"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "int64"
    _print_name = ("Binom", "\\operatorname{Binom}")

    def __call__(self, n, p, size=None, **kwargs):
        r"""Draw samples from a binomial distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        n
            Number of trials :math:`n`. Must be a positive integer.
        p
            Probability of success :math:`p` of a single trial.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are returned. Default
            is `None` in which case a single sample is returned.

        """
        return super().__call__(n, p, size=size, **kwargs)


binomial = BinomialRV()


class NegBinomialRV(ScipyRandomVariable):
    r"""A negative binomial discrete random variable.

    The probability mass function for `nbinom` for the number :math:`k` of draws
    before observing the :math:`n`\th success in terms of the probability of
    success :math:`p` of a single trial is:

    .. math::

            f(k; p, n) = {k+n-1 \choose n-1} p^n (1-p)^{k}

    """
    name = "nbinom"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "int64"
    _print_name = ("NB", "\\operatorname{NB}")

    def __call__(self, n, p, size=None, **kwargs):
        r"""Draw samples from a negative binomial distribution.

        Signature
        ---------

        `(), () -> ()`

        Parameters
        ----------
        n
            Number of successes :math:`n`. Must be a positive integer.
        p
            Probability of success :math:`p` of a single trial.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are returned. Default
            is `None` in which case a single sample is returned.

        """
        return super().__call__(n, p, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, n, p, size):
        return stats.nbinom.rvs(n, p, size=size, random_state=rng)


nbinom = NegBinomialRV()
negative_binomial = NegBinomialRV()


class BetaBinomialRV(ScipyRandomVariable):
    r"""A beta-binomial discrete random variable.

    The probability mass function for `betabinom` in terms of its shape
    parameters :math:`n \geq 0`, :math:`a > 0`, :math:`b > 0` and the probability
    :math:`p` is:

    .. math::

            f(k; p, n, a, b) = {n \choose k} \frac{\operatorname{B}(k+a, n-k+b)}{\operatorname{B}(a,b)}

    where :math:`\operatorname{B}` is the beta function:

    .. math::

        \operatorname{B}(a, b) = \int_0^1 t^{a-1} (1-t)^{b-1} \mathrm{d}t

    """
    name = "beta_binomial"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "int64"
    _print_name = ("BetaBinom", "\\operatorname{BetaBinom}")

    def __call__(self, n, a, b, size=None, **kwargs):
        r"""Draw samples from a beta-binomial distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        n
            Shape parameter :math:`n`. Must be a positive integer.
        a
            Shape parameter :math:`a`. Must be positive.
        b
            Shape parameter :math:`b`. Must be positive.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are returned. Default
            is `None` in which case a single sample is returned.

        """
        return super().__call__(n, a, b, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, n, a, b, size):
        return stats.betabinom.rvs(n, a, b, size=size, random_state=rng)


betabinom = BetaBinomialRV()


class GenGammaRV(ScipyRandomVariable):
    r"""A generalized gamma continuous random variable.

    The probability density function of `gengamma` in terms of its scale parameter
    :math:`\alpha` and other parameters :math:`p` and :math:`\lambda` is:

    .. math::

            f(x; \alpha, \lambda, p) = \frac{p/\lambda^\alpha}{\Gamma(\alpha/p)} x^{\alpha-1} e^{-(x/\lambda)^p}

    for :math:`x > 0`, where :math:`\alpha, \lambda, p > 0`.

    """
    name = "gengamma"
    ndim_supp = 0
    ndims_params = [0, 0, 0]
    dtype = "floatX"
    _print_name = ("GG", "\\operatorname{GG}")

    def __call__(self, alpha=1.0, p=1.0, lambd=1.0, size=None, **kwargs):
        r"""Draw samples from a generalized gamma distribution.

        Signature
        ---------

        `(), (), () -> ()`

        Parameters
        ----------
        alpha
            Parameter :math:`\alpha`. Must be positive.
        p
            Parameter :math:`p`. Must be positive.
        lambd
            Scale parameter :math:`\lambda`. Must be positive.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are
            returned. Default is `None` in which case a single sample
            is returned.

        """
        return super().__call__(alpha, p, lambd, size=size, **kwargs)

    @classmethod
    def rng_fn_scipy(cls, rng, alpha, p, lambd, size):
        return stats.gengamma.rvs(
            alpha / p, p, scale=lambd, size=size, random_state=rng
        )


gengamma = GenGammaRV()


class MultinomialRV(RandomVariable):
    r"""A multinomial discrete random variable.

    The probability mass function of `multinomial` in terms of the number
    of experiments :math:`n` and the probabilities :math:`p_1, \dots, p_k`
    of the :math:`k` different possible outcomes is:


    .. math::

        f(x_1,\dots,x_k; n, p_1, \dots, p_k) = \frac{n!}{x_1! \dots x_k!} \prod_{i=1}^k x_i^{p_i}


    where :math:`n>0` and :math:`\sum_{i=1}^k p_i = 1`.

    Notes
    -----
    The length of the support dimension is determined by the last
    dimension in the *second* parameter (i.e.  the probabilities vector).

    """
    name = "multinomial"
    ndim_supp = 1
    ndims_params = [0, 1]
    dtype = "int64"
    _print_name = ("MN", "\\operatorname{MN}")

    def __call__(self, n, p, size=None, **kwargs):
        r"""Draw samples from a discrete multinomial distribution.

        Signature
        ---------

        `(), (n) -> (n)`

        Parameters
        ----------
        n
            Number of experiments :math:`n`. Must be a positive integer.
        p
            Probabilities of each of the :math:`k` different outcomes.
        size
            Given a size of, for example, `(r, s, t)`, `r * s * t` independent,
            identically distributed samples are generated. Because each sample
            is `k`-dimensional, the output shape is `(r, s, t, k)`. If no shape
            is specified, a single `k`-dimensional sample is returned.

        """
        return super().__call__(n, p, size=size, **kwargs)

    def _supp_shape_from_params(self, dist_params, rep_param_idx=1, param_shapes=None):
        return default_supp_shape_from_params(
            self.ndim_supp, dist_params, rep_param_idx, param_shapes
        )

    @classmethod
    def rng_fn(cls, rng, n, p, size):
        if n.ndim > 0 or p.ndim > 1:
            size = tuple(size or ())

            if size:
                n = np.broadcast_to(n, size)
                p = np.broadcast_to(p, size + p.shape[-1:])
            else:
                n, p = broadcast_params([n, p], cls.ndims_params)

            res = np.empty(p.shape, dtype=cls.dtype)
            for idx in np.ndindex(p.shape[:-1]):
                res[idx] = rng.multinomial(n[idx], p[idx])
            return res
        else:
            return rng.multinomial(n, p, size=size)


multinomial = MultinomialRV()

vsearchsorted = np.vectorize(np.searchsorted, otypes=[int], signature="(n),()->()")


class CategoricalRV(RandomVariable):
    r"""A categorical discrete random variable.

    The probability mass function of `categorical` in terms of its :math:`N` event
    probabilities :math:`p_1, \dots, p_N` is:

    .. math::

        P(k=i) = p_k

    where :math:`\sum_i p_i = 1`.

    """

    name = "categorical"
    ndim_supp = 0
    ndims_params = [1]
    dtype = "int64"
    _print_name = ("Cat", "\\operatorname{Cat}")

    def __call__(self, p, size=None, **kwargs):
        r"""Draw samples from a discrete categorical distribution.

        Signature
        ---------

        `(j) -> ()`

        Parameters
        ----------
        p
            An array that contains the :math:`N` event probabilities.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed random samples are
            returned. Default is `None`, in which case a single sample
            is returned.

        """
        return super().__call__(p, size=size, **kwargs)

    @classmethod
    def rng_fn(cls, rng, p, size):
        if size is None:
            size = p.shape[:-1]
        else:
            # Check that `size` does not define a shape that would be broadcasted
            # to `p.shape[:-1]` in the call to `vsearchsorted` below.
            if len(size) < (p.ndim - 1):
                raise ValueError("`size` is incompatible with the shape of `p`")
            for s, ps in zip(reversed(size), reversed(p.shape[:-1])):
                if s == 1 and ps != 1:
                    raise ValueError("`size` is incompatible with the shape of `p`")

        unif_samples = rng.uniform(size=size)
        samples = vsearchsorted(p.cumsum(axis=-1), unif_samples)

        return samples


categorical = CategoricalRV()


class RandIntRV(RandomVariable):
    r"""A discrete uniform random variable.

    Only available for `RandomStateType`. Use `integers` with `RandomGeneratorType`\s.

    """

    name = "randint"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "int64"
    _print_name = ("randint", "\\operatorname{randint}")

    def __call__(self, low, high=None, size=None, **kwargs):
        r"""Draw samples from a discrete uniform distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        low
            Lower boundary of the output interval. All values generated will
            be greater than or equal to `low`, unless `high=None`, in which case
            all values generated are greater than or equal to `0` and
            smaller than `low` (exclusive).
        high
            Upper boundary of the output interval.  All values generated
            will be smaller than `high` (exclusive).
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are
            returned. Default is `None`, in which case a single
            sample is returned.

        """
        if high is None:
            low, high = 0, low
        return super().__call__(low, high, size=size, **kwargs)

    def make_node(self, rng, *args, **kwargs):
        if not isinstance(
            getattr(rng, "type", None), (RandomStateType, RandomStateSharedVariable)
        ):
            raise TypeError("`randint` is only available for `RandomStateType`s")
        return super().make_node(rng, *args, **kwargs)


randint = RandIntRV()


class IntegersRV(RandomVariable):
    r"""A discrete uniform random variable.

    Only available for `RandomGeneratorType`. Use `randint` with `RandomStateType`\s.

    """
    name = "integers"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "int64"
    _print_name = ("integers", "\\operatorname{integers}")

    def __call__(self, low, high=None, size=None, **kwargs):
        r"""Draw samples from a discrete uniform distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        low
            Lower boundary of the output interval.  All values generated
            will be greater than or equal to `low` (inclusive).
        high
            Upper boundary of the output interval.  All values generated
            will be smaller than `high` (exclusive).
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n * k`
            independent, identically distributed samples are
            returned. Default is `None`, in which case a single sample
            is returned.

        """
        if high is None:
            low, high = 0, low
        return super().__call__(low, high, size=size, **kwargs)

    def make_node(self, rng, *args, **kwargs):
        if not isinstance(
            getattr(rng, "type", None),
            (RandomGeneratorType, RandomGeneratorSharedVariable),
        ):
            raise TypeError("`integers` is only available for `RandomGeneratorType`s")
        return super().make_node(rng, *args, **kwargs)


integers = IntegersRV()


class ChoiceRV(RandomVariable):
    """Randomly choose an element in a sequence."""

    name = "choice"
    ndim_supp = 0
    ndims_params = [1, 1, 0]
    dtype = None
    _print_name = ("choice", "\\operatorname{choice}")

    @classmethod
    def rng_fn(cls, rng, a, p, replace, size):
        return rng.choice(a, size, replace, p)

    def _supp_shape_from_params(self, *args, **kwargs):
        raise NotImplementedError()

    def _infer_shape(self, size, dist_params, param_shapes=None):
        (a, p, _) = dist_params

        if isinstance(p.type, aesara.tensor.type_other.NoneTypeT):
            shape = super()._infer_shape(size, (a,), param_shapes)
        else:
            shape = super()._infer_shape(size, (a, p), param_shapes)

        return shape

    def __call__(self, a, size=None, replace=True, p=None, **kwargs):
        r"""Generate a random sample from an array.

        Signature
        ---------

        `(x) -> ()`

        Parameters
        ----------
        a
            The array from which to randomly sample an element. If an int,
            a sample is generated from `aesara.tensor.arange(a)`.
        size
            Sample shape. If the given size is `(m, n, k)`, then `m * n *
            k` independent samples are returned. Default is `None`, in
            which case a single sample is returned.
        replace
            When `True`, sampling is performed with replacement.
        p
            The probabilities associated with each entry in `a`. If not
            given, all elements have equal probability.
        """
        a = as_tensor_variable(a)

        if a.ndim == 0:
            a = aesara.tensor.arange(a)

        if p is None:
            p = aesara.tensor.type_other.NoneConst

        if isinstance(replace, bool):
            replace = aesara.tensor.constant(np.array(replace))

        return super().__call__(a, p, replace, size=size, dtype=a.dtype, **kwargs)


choice = ChoiceRV()


class PermutationRV(RandomVariable):
    """Randomly shuffle a sequence."""

    name = "permutation"
    ndim_supp = 1
    ndims_params = [1]
    dtype = None
    _print_name = ("permutation", "\\operatorname{permutation}")

    @classmethod
    def rng_fn(cls, rng, x, size):
        return rng.permutation(x if x.ndim > 0 else x.item())

    def _infer_shape(self, size, dist_params, param_shapes=None):
        param_shapes = param_shapes or [p.shape for p in dist_params]

        (x,) = dist_params
        (x_shape,) = param_shapes

        if x.ndim == 0:
            return (x,)
        else:
            return x_shape

    def __call__(self, x, **kwargs):
        r"""Randomly permute a sequence or a range of values.

        Signature
        ---------

        `(x) -> (x)`

        Parameters
        ----------
        x
            If `x` is an integer, randomly permute `np.arange(x)`. If `x` is a sequence,
            shuffle its elements randomly.

        """
        x = as_tensor_variable(x)
        return super().__call__(x, dtype=x.dtype, **kwargs)


permutation = PermutationRV()


random = uniform


class FRV(RandomVariable):
    r"""An F-distribution continuous random variable.

    The probability density function for `F` in terms of its degrees of freedom parameters `dfn` and `dfd` is:

    .. math::

        f(x; dfn, dfd) = \frac{\sqrt{\frac{(dfn \cdot x)^{dfn} \cdot dfd^{dfd}}{(dfn \cdot x + dfd)^{dfn + dfd}}}}{x \cdot \text{Beta}(\frac{dfn}{2}, \frac{dfd}{2})}

    for :math:`x > 0`, :math:`dfn > 0`, and :math:`dfd > 0`.

    """
    name = "f"
    ndim_supp = 0
    ndims_params = [0, 0]
    dtype = "floatX"
    _print_name = ("F", "\\operatorname{F}")

    def __call__(self, dfn=1, dfd=1, size=None, **kwargs):
        r"""Draw samples from an F-distribution.

        Signature
        ---------

        `() -> ()`

        Parameters
        ----------
        dfn
            Degrees of freedom parameter for the numerator. Must be positive.
        dfd
            Degrees of freedom parameter for the denominator. Must be positive.
        size
            Sample shape. If the given size is, e.g. `(m, n, k)` then `m * n * k`
            independent, identically distributed random variables are
            returned. Default is `None` in which case a single random variable
            is returned.

        """
        return super().__call__(dfn, dfd, size=size, **kwargs)


f = FRV()


__all__ = [
    "permutation",
    "choice",
    "integers",
    "randint",
    "categorical",
    "multinomial",
    "betabinom",
    "nbinom",
    "binomial",
    "laplace",
    "bernoulli",
    "truncexpon",
    "wald",
    "invgamma",
    "halfcauchy",
    "cauchy",
    "hypergeometric",
    "geometric",
    "poisson",
    "dirichlet",
    "multivariate_normal",
    "vonmises",
    "logistic",
    "weibull",
    "exponential",
    "gumbel",
    "pareto",
    "chisquare",
    "gamma",
    "lognormal",
    "halfnormal",
    "normal",
    "beta",
    "triangular",
    "uniform",
    "standard_cauchy",
    "standard_exponential",
    "standard_gamma",
    "standard_normal",
    "standard_t",
    "negative_binomial",
    "gengamma",
    "t",
    "random",
    "rayleigh",
    "power",
    "zipf",
    "f",
]