Source code for pymc.distributions.continuous

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# Contains code from AePPL, Copyright (c) 2021-2022, Aesara Developers.

# coding: utf-8
"""
A collection of common probability distributions for stochastic
nodes in PyMC.
"""

import warnings

import numpy as np
import pytensor
import pytensor.tensor as pt

from pytensor.graph.basic import Apply, Variable
from pytensor.graph.op import Op
from pytensor.raise_op import Assert
from pytensor.tensor import gamma as gammafn
from pytensor.tensor import gammaln
from pytensor.tensor.extra_ops import broadcast_shape
from pytensor.tensor.math import betaincinv, gammaincinv, tanh
from pytensor.tensor.random.basic import (
    BetaRV,
    _gamma,
    cauchy,
    exponential,
    gumbel,
    halfcauchy,
    halfnormal,
    invgamma,
    laplace,
    logistic,
    lognormal,
    normal,
    pareto,
    t,
    triangular,
    uniform,
    vonmises,
)
from pytensor.tensor.random.op import RandomVariable
from pytensor.tensor.random.utils import normalize_size_param
from pytensor.tensor.variable import TensorConstant

from pymc.logprob.abstract import _logprob_helper
from pymc.logprob.basic import icdf
from pymc.pytensorf import normalize_rng_param

try:
    from polyagamma import polyagamma_cdf, polyagamma_pdf, random_polyagamma
except ImportError:  # pragma: no cover

    def random_polyagamma(*args, **kwargs):
        raise RuntimeError("polyagamma package is not installed!")

    def polyagamma_pdf(*args, **kwargs):
        raise RuntimeError("polyagamma package is not installed!")

    def polyagamma_cdf(*args, **kwargs):
        raise RuntimeError("polyagamma package is not installed!")


from scipy import stats
from scipy.interpolate import InterpolatedUnivariateSpline

from pymc.distributions import transforms
from pymc.distributions.dist_math import (
    SplineWrapper,
    check_icdf_parameters,
    check_icdf_value,
    check_parameters,
    clipped_beta_rvs,
    i0e,
    log_diff_normal_cdf,
    log_normal,
    logpow,
    normal_lccdf,
    normal_lcdf,
    zvalue,
)
from pymc.distributions.distribution import DIST_PARAMETER_TYPES, Continuous, SymbolicRandomVariable
from pymc.distributions.shape_utils import implicit_size_from_params, rv_size_is_none
from pymc.distributions.transforms import _default_transform
from pymc.math import invlogit, logdiffexp, logit

__all__ = [
    "Uniform",
    "Flat",
    "HalfFlat",
    "Normal",
    "TruncatedNormal",
    "Beta",
    "Kumaraswamy",
    "Exponential",
    "Laplace",
    "StudentT",
    "Cauchy",
    "HalfCauchy",
    "Gamma",
    "Weibull",
    "HalfStudentT",
    "LogNormal",
    "ChiSquared",
    "HalfNormal",
    "Wald",
    "Pareto",
    "InverseGamma",
    "ExGaussian",
    "VonMises",
    "SkewNormal",
    "Triangular",
    "Gumbel",
    "Logistic",
    "LogitNormal",
    "Interpolated",
    "Rice",
    "Moyal",
    "AsymmetricLaplace",
    "PolyaGamma",
    "SkewStudentT",
]


class PositiveContinuous(Continuous):
    """Base class for positive continuous distributions"""


class UnitContinuous(Continuous):
    """Base class for continuous distributions on [0,1]"""


class CircularContinuous(Continuous):
    """Base class for circular continuous distributions"""


class BoundedContinuous(Continuous):
    """Base class for bounded continuous distributions"""

    # Indices of the arguments that define the lower and upper bounds of the distribution
    bound_args_indices: list[int] | None = None


@_default_transform.register(PositiveContinuous)
def pos_cont_transform(op, rv):
    return transforms.log


@_default_transform.register(UnitContinuous)
def unit_cont_transform(op, rv):
    return transforms.logodds


@_default_transform.register(CircularContinuous)
def circ_cont_transform(op, rv):
    return transforms.circular


@_default_transform.register(BoundedContinuous)
def bounded_cont_transform(op, rv, bound_args_indices=None):
    if bound_args_indices is None:
        raise ValueError(f"Must specify bound_args_indices for {op} bounded distribution")

    def transform_params(*args):
        lower, upper = None, None
        if bound_args_indices[0] is not None:
            lower = args[bound_args_indices[0]]
        if bound_args_indices[1] is not None:
            upper = args[bound_args_indices[1]]

        if lower is not None:
            if isinstance(lower, TensorConstant) and np.all(lower.value == -np.inf):
                lower = None
            else:
                lower = pt.as_tensor_variable(lower)

        if upper is not None:
            if isinstance(upper, TensorConstant) and np.all(upper.value == np.inf):
                upper = None
            else:
                upper = pt.as_tensor_variable(upper)

        return lower, upper

    return transforms.Interval(bounds_fn=transform_params)


def assert_negative_support(var, label, distname, value=-1e-6):
    warnings.warn(
        "The assert_negative_support function will be deprecated in future versions!"
        " See https://github.com/pymc-devs/pymc/issues/5162",
        DeprecationWarning,
    )
    msg = f"The variable specified for {label} has negative support for {distname}, "
    msg += "likely making it unsuitable for this parameter."
    return Assert(msg)(var, pt.all(pt.ge(var, 0.0)))


def get_tau_sigma(tau=None, sigma=None):
    r"""
    Find precision and standard deviation. The link between the two
    parameterizations is given by the inverse relationship:

    .. math::
        \tau = \frac{1}{\sigma^2}

    Parameters
    ----------
    tau: array-like, optional
    sigma: array-like, optional

    Results
    -------
    Returns tuple (tau, sigma)

    Notes
    -----
    If neither tau nor sigma is provided, returns (1., 1.)
    """
    if tau is not None and sigma is not None:
        raise ValueError("Can't pass both tau and sigma")
    if tau is None and sigma is None:
        sigma = pt.as_tensor_variable(1.0)
        tau = pt.as_tensor_variable(1.0)
    elif tau is None:
        sigma = pt.as_tensor_variable(sigma)
        # Keep tau negative, if sigma was negative, so that it will
        # fail when used
        tau = (sigma**-2.0) * pt.sign(sigma)
    else:
        tau = pt.as_tensor_variable(tau)
        # Keep tau negative, if sigma was negative, so that it will
        # fail when used
        sigma = pt.abs(tau) ** -0.5 * pt.sign(tau)

    return tau, sigma


[docs] class Uniform(BoundedContinuous): r""" Continuous uniform log-likelihood. The pdf of this distribution is .. math:: f(x \mid lower, upper) = \frac{1}{upper-lower} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-3, 3, 500) ls = [0., -2] us = [2., 1] for l, u in zip(ls, us): y = np.zeros(500) y[(x<u) & (x>l)] = 1.0/(u-l) plt.plot(x, y, label='lower = {}, upper = {}'.format(l, u)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.ylim(0, 1) plt.legend(loc=1) plt.show() ======== ===================================== Support :math:`x \in [lower, upper]` Mean :math:`\dfrac{lower + upper}{2}` Variance :math:`\dfrac{(upper - lower)^2}{12}` ======== ===================================== Parameters ---------- lower : tensor_like of float, default 0 Lower limit. upper : tensor_like of float, default 1 Upper limit. """ rv_op = uniform bound_args_indices = (3, 4) # Lower, Upper
[docs] @classmethod def dist(cls, lower=0, upper=1, **kwargs): lower = pt.as_tensor_variable(lower) upper = pt.as_tensor_variable(upper) return super().dist([lower, upper], **kwargs)
def support_point(rv, size, lower, upper): lower, upper = pt.broadcast_arrays(lower, upper) support_point = (lower + upper) / 2 if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point def logp(value, lower, upper): res = pt.switch( pt.bitwise_and(pt.ge(value, lower), pt.le(value, upper)), pt.fill(value, -pt.log(upper - lower)), -np.inf, ) return check_parameters( res, lower <= upper, msg="lower <= upper", ) def logcdf(value, lower, upper): res = pt.switch( pt.lt(value, lower), -np.inf, pt.switch( pt.lt(value, upper), pt.log(value - lower) - pt.log(upper - lower), 0, ), ) return check_parameters( res, lower <= upper, msg="lower <= upper", ) def icdf(value, lower, upper): res = lower + (upper - lower) * value res = check_icdf_value(res, value) return check_icdf_parameters(res, lower < upper)
@_default_transform.register(Uniform) def uniform_default_transform(op, rv): return bounded_cont_transform(op, rv, Uniform.bound_args_indices) class FlatRV(RandomVariable): name = "flat" ndim_supp = 0 ndims_params = [] dtype = "floatX" _print_name = ("Flat", "\\operatorname{Flat}") @classmethod def rng_fn(cls, rng, size): raise NotImplementedError("Cannot sample from flat variable") flat = FlatRV()
[docs] class Flat(Continuous): """ Uninformative log-likelihood that returns 0 regardless of the passed value. """ rv_op = flat def __new__(cls, *args, **kwargs): kwargs.setdefault("initval", "support_point") return super().__new__(cls, *args, **kwargs)
[docs] @classmethod def dist(cls, **kwargs): res = super().dist([], **kwargs) return res
def support_point(rv, size): return pt.zeros(size) def logp(value): return pt.zeros_like(value) def logcdf(value): return pt.switch( pt.eq(value, -np.inf), -np.inf, pt.switch(pt.eq(value, np.inf), 0, pt.log(0.5)) )
class HalfFlatRV(RandomVariable): name = "half_flat" ndim_supp = 0 ndims_params = [] dtype = "floatX" _print_name = ("HalfFlat", "\\operatorname{HalfFlat}") @classmethod def rng_fn(cls, rng, size): raise NotImplementedError("Cannot sample from half_flat variable") halfflat = HalfFlatRV()
[docs] class HalfFlat(PositiveContinuous): """Improper flat prior over the positive reals.""" rv_op = halfflat def __new__(cls, *args, **kwargs): kwargs.setdefault("initval", "support_point") return super().__new__(cls, *args, **kwargs)
[docs] @classmethod def dist(cls, **kwargs): res = super().dist([], **kwargs) return res
def support_point(rv, size): return pt.ones(size) def logp(value): return pt.switch(pt.lt(value, 0), -np.inf, pt.zeros_like(value)) def logcdf(value): return pt.switch(pt.lt(value, np.inf), -np.inf, pt.switch(pt.eq(value, np.inf), 0, -np.inf))
[docs] class Normal(Continuous): r""" Univariate normal log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, \tau) = \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (x-\mu)^2 \right\} Normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by .. math:: \tau = \dfrac{1}{\sigma^2} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-5, 5, 1000) mus = [0., 0., 0., -2.] sigmas = [0.4, 1., 2., 0.4] for mu, sigma in zip(mus, sigmas): pdf = st.norm.pdf(x, mu, sigma) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}'.format(mu, sigma)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu` Variance :math:`\dfrac{1}{\tau}` or :math:`\sigma^2` ======== ========================================== Parameters ---------- mu : tensor_like of float, default 0 Mean. sigma : tensor_like of float, optional Standard deviation (sigma > 0) (only required if tau is not specified). Defaults to 1 if neither sigma nor tau is specified. tau : tensor_like of float, optional Precision (tau > 0) (only required if sigma is not specified). Examples -------- .. code-block:: python with pm.Model(): x = pm.Normal('x', mu=0, sigma=10) with pm.Model(): x = pm.Normal('x', mu=0, tau=1/23) """ rv_op = normal
[docs] @classmethod def dist(cls, mu=0, sigma=None, tau=None, **kwargs): tau, sigma = get_tau_sigma(tau=tau, sigma=sigma) sigma = pt.as_tensor_variable(sigma) # tau = pt.as_tensor_variable(tau) # mean = median = mode = mu = pt.as_tensor_variable(floatX(mu)) # variance = 1.0 / self.tau return super().dist([mu, sigma], **kwargs)
def support_point(rv, size, mu, sigma): mu, _ = pt.broadcast_arrays(mu, sigma) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, mu, sigma): res = -0.5 * pt.pow((value - mu) / sigma, 2) - pt.log(pt.sqrt(2.0 * np.pi)) - pt.log(sigma) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def logcdf(value, mu, sigma): return check_parameters( normal_lcdf(mu, sigma, value), sigma > 0, msg="sigma > 0", ) def icdf(value, mu, sigma): res = mu + sigma * -np.sqrt(2.0) * pt.erfcinv(2 * value) res = check_icdf_value(res, value) return check_icdf_parameters( res, sigma > 0, msg="sigma > 0", )
class TruncatedNormalRV(RandomVariable): name = "truncated_normal" ndim_supp = 0 ndims_params = [0, 0, 0, 0] dtype = "floatX" _print_name = ("TruncatedNormal", "\\operatorname{TruncatedNormal}") @classmethod def rng_fn( cls, rng: np.random.RandomState, mu: np.ndarray | float, sigma: np.ndarray | float, lower: np.ndarray | float, upper: np.ndarray | float, size: list[int] | int | None, ) -> np.ndarray: # Upcast to float64. (Caller will downcast to desired dtype if needed) # (Work-around for https://github.com/scipy/scipy/issues/15928) return stats.truncnorm.rvs( a=((lower - mu) / sigma).astype("float64"), b=((upper - mu) / sigma).astype("float64"), loc=(mu).astype("float64"), scale=(sigma).astype("float64"), size=size, random_state=rng, ) truncated_normal = TruncatedNormalRV()
[docs] class TruncatedNormal(BoundedContinuous): r""" Univariate truncated normal log-likelihood. The pdf of this distribution is .. math:: f(x;\mu ,\sigma ,a,b)={\frac {\phi ({\frac {x-\mu }{\sigma }})}{ \sigma \left(\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})\right)}} Truncated normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by .. math:: \tau = \dfrac{1}{\sigma^2} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-10, 10, 1000) mus = [0., 0., 0.] sigmas = [3.,5.,7.] a1 = [-3, -5, -5] b1 = [7, 5, 4] for mu, sigma, a, b in zip(mus, sigmas,a1,b1): an, bn = (a - mu) / sigma, (b - mu) / sigma pdf = st.truncnorm.pdf(x, an,bn, loc=mu, scale=sigma) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}, a={}, b={}'.format(mu, sigma, a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in [a, b]` Mean :math:`\mu +{\frac {\phi (\alpha )-\phi (\beta )}{Z}}\sigma` Variance :math:`\sigma ^{2}\left[1+{\frac {\alpha \phi (\alpha )-\beta \phi (\beta )}{Z}}-\left({\frac {\phi (\alpha )-\phi (\beta )}{Z}}\right)^{2}\right]` ======== ========================================== Parameters ---------- mu : tensor_like of float, default 0 Mean. sigma : tensor_like of float, optional Standard deviation (sigma > 0) (only required if tau is not specified). Defaults to 1 if neither sigma nor tau is specified. tau : tensor_like of float, optional Precision (tau > 0) (only required if sigma is not specified). lower : tensor_like of float, default - numpy.inf Left bound. upper : tensor_like of float, default numpy.inf Right bound. Examples -------- .. code-block:: python with pm.Model(): x = pm.TruncatedNormal('x', mu=0, sigma=10, lower=0) with pm.Model(): x = pm.TruncatedNormal('x', mu=0, sigma=10, upper=1) with pm.Model(): x = pm.TruncatedNormal('x', mu=0, sigma=10, lower=0, upper=1) """ rv_op = truncated_normal bound_args_indices = (5, 6) # indexes for lower and upper args
[docs] @classmethod def dist( cls, mu: DIST_PARAMETER_TYPES | None = 0, sigma: DIST_PARAMETER_TYPES | None = None, *, tau: DIST_PARAMETER_TYPES | None = None, lower: DIST_PARAMETER_TYPES | None = None, upper: DIST_PARAMETER_TYPES | None = None, **kwargs, ) -> RandomVariable: tau, sigma = get_tau_sigma(tau=tau, sigma=sigma) sigma = pt.as_tensor_variable(sigma) mu = pt.as_tensor_variable(mu) lower = pt.as_tensor_variable(lower) if lower is not None else pt.constant(-np.inf) upper = pt.as_tensor_variable(upper) if upper is not None else pt.constant(np.inf) return super().dist([mu, sigma, lower, upper], **kwargs)
def support_point(rv, size, mu, sigma, lower, upper): mu, _, lower, upper = pt.broadcast_arrays(mu, sigma, lower, upper) support_point = pt.switch( pt.eq(lower, -np.inf), pt.switch( pt.eq(upper, np.inf), # lower = -inf, upper = inf mu, # lower = -inf, upper = x upper - 1, ), pt.switch( pt.eq(upper, np.inf), # lower = x, upper = inf lower + 1, # lower = x, upper = x (lower + upper) / 2, ), ) if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point def logp(value, mu, sigma, lower, upper): is_lower_bounded = not ( isinstance(lower, TensorConstant) and np.all(np.isneginf(lower.value)) ) is_upper_bounded = not (isinstance(upper, TensorConstant) and np.all(np.isinf(upper.value))) if is_lower_bounded and is_upper_bounded: norm = log_diff_normal_cdf(mu, sigma, upper, lower) elif is_lower_bounded: norm = normal_lccdf(mu, sigma, lower) elif is_upper_bounded: norm = normal_lcdf(mu, sigma, upper) else: norm = 0.0 logp = _logprob_helper(Normal.dist(mu, sigma), value) - norm if is_lower_bounded: logp = pt.switch(value < lower, -np.inf, logp) if is_upper_bounded: logp = pt.switch(value > upper, -np.inf, logp) if is_lower_bounded and is_upper_bounded: logp = check_parameters( logp, pt.le(lower, upper), msg="lower_bound <= upper_bound", ) return logp def logcdf(value, mu, sigma, lower, upper): logcdf = log_diff_normal_cdf(mu, sigma, value, lower) - log_diff_normal_cdf( mu, sigma, upper, lower ) is_lower_bounded = not ( isinstance(lower, TensorConstant) and np.all(np.isneginf(lower.value)) ) is_upper_bounded = not (isinstance(upper, TensorConstant) and np.all(np.isinf(upper.value))) if is_lower_bounded: logcdf = pt.switch(value < lower, -np.inf, logcdf) if is_upper_bounded: logcdf = pt.switch(value <= upper, logcdf, 0.0) if is_lower_bounded and is_upper_bounded: logcdf = check_parameters( logcdf, pt.le(lower, upper), msg="lower_bound <= upper_bound", ) return logcdf
@_default_transform.register(TruncatedNormal) def truncated_normal_default_transform(op, rv): return bounded_cont_transform(op, rv, TruncatedNormal.bound_args_indices)
[docs] class HalfNormal(PositiveContinuous): r""" Half-normal log-likelihood. The pdf of this distribution is .. math:: f(x \mid \tau) = \sqrt{\frac{2\tau}{\pi}} \exp\left(\frac{-x^2 \tau}{2}\right) f(x \mid \sigma) = \sqrt{\frac{2}{\pi\sigma^2}} \exp\left(\frac{-x^2}{2\sigma^2}\right) .. note:: The parameters ``sigma``/``tau`` (:math:`\sigma`/:math:`\tau`) refer to the standard deviation/precision of the unfolded normal distribution, for the standard deviation of the half-normal distribution, see below. For the half-normal, they are just two parameterisation :math:`\sigma^2 \equiv \frac{1}{\tau}` of a scale parameter. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 5, 200) for sigma in [0.4, 1., 2.]: pdf = st.halfnorm.pdf(x, scale=sigma) plt.plot(x, pdf, label=r'$\sigma$ = {}'.format(sigma)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in [0, \infty)` Mean :math:`\sqrt{\dfrac{2}{\tau \pi}}` or :math:`\dfrac{\sigma \sqrt{2}}{\sqrt{\pi}}` Variance :math:`\dfrac{1}{\tau}\left(1 - \dfrac{2}{\pi}\right)` or :math:`\sigma^2\left(1 - \dfrac{2}{\pi}\right)` ======== ========================================== Parameters ---------- sigma : tensor_like of float, optional Scale parameter :math:`\sigma` (``sigma`` > 0) (only required if ``tau`` is not specified). Defaults to 1. tau : tensor_like of float, optional Precision :math:`\tau` (tau > 0) (only required if sigma is not specified). Defaults to 1. Examples -------- .. code-block:: python with pm.Model(): x = pm.HalfNormal('x', sigma=10) with pm.Model(): x = pm.HalfNormal('x', tau=1/15) """ rv_op = halfnormal
[docs] @classmethod def dist( cls, sigma: DIST_PARAMETER_TYPES | None = None, tau: DIST_PARAMETER_TYPES | None = None, *args, **kwargs, ): tau, sigma = get_tau_sigma(tau=tau, sigma=sigma) return super().dist([0.0, sigma], **kwargs)
def support_point(rv, size, loc, sigma): support_point = loc + sigma if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point def logp(value, loc, sigma): res = -0.5 * pt.pow((value - loc) / sigma, 2) + pt.log(pt.sqrt(2.0 / np.pi)) - pt.log(sigma) res = pt.switch(pt.ge(value, loc), res, -np.inf) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def logcdf(value, loc, sigma): z = zvalue(value, mu=loc, sigma=sigma) logcdf = pt.switch( pt.lt(value, loc), -np.inf, pt.log1p(-pt.erfc(z / pt.sqrt(2.0))), ) return check_parameters( logcdf, sigma > 0, msg="sigma > 0", ) def icdf(value, loc, sigma): res = icdf(Normal.dist(loc, sigma), (value + 1.0) / 2.0) res = check_icdf_value(res, value) return res
class WaldRV(RandomVariable): name = "wald" ndim_supp = 0 ndims_params = [0, 0, 0] dtype = "floatX" _print_name = ("Wald", "\\operatorname{Wald}") @classmethod def rng_fn(cls, rng, mu, lam, alpha, size) -> np.ndarray: return np.asarray(rng.wald(mu, lam, size=size) + alpha) wald = WaldRV()
[docs] class Wald(PositiveContinuous): r""" Wald log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, \lambda) = \left(\frac{\lambda}{2\pi}\right)^{1/2} x^{-3/2} \exp\left\{ -\frac{\lambda}{2x}\left(\frac{x-\mu}{\mu}\right)^2 \right\} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 3, 500) mus = [1., 1., 1., 3.] lams = [1., .2, 3., 1.] for mu, lam in zip(mus, lams): pdf = st.invgauss.pdf(x, mu/lam, scale=lam) plt.plot(x, pdf, label=r'$\mu$ = {}, $\lambda$ = {}'.format(mu, lam)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================= Support :math:`x \in (0, \infty)` Mean :math:`\mu` Variance :math:`\dfrac{\mu^3}{\lambda}` ======== ============================= Wald distribution can be parameterized either in terms of lam or phi. The link between the two parametrizations is given by .. math:: \phi = \dfrac{\lambda}{\mu} Parameters ---------- mu : tensor_like of float, optional Mean of the distribution (mu > 0). lam : tensor_like of float, optional Relative precision (lam > 0). phi : tensor_like of float, optional Alternative shape parameter (phi > 0). alpha : tensor_like of float, default 0 Shift/location parameter (alpha >= 0). Notes ----- To instantiate the distribution specify any of the following - only mu (in this case lam will be 1) - mu and lam - mu and phi - lam and phi References ---------- .. [Tweedie1957] Tweedie, M. C. K. (1957). Statistical Properties of Inverse Gaussian Distributions I. The Annals of Mathematical Statistics, Vol. 28, No. 2, pp. 362-377 .. [Michael1976] Michael, J. R., Schucany, W. R. and Hass, R. W. (1976). Generating Random Variates Using Transformations with Multiple Roots. The American Statistician, Vol. 30, No. 2, pp. 88-90 .. [Giner2016] Göknur Giner, Gordon K. Smyth (2016) statmod: Probability Calculations for the Inverse Gaussian Distribution """ rv_op = wald
[docs] @classmethod def dist( cls, mu: DIST_PARAMETER_TYPES | None = None, lam: DIST_PARAMETER_TYPES | None = None, phi: DIST_PARAMETER_TYPES | None = None, alpha: DIST_PARAMETER_TYPES | None = 0.0, **kwargs, ): mu, lam, phi = cls.get_mu_lam_phi(mu, lam, phi) alpha = pt.as_tensor_variable(alpha) mu = pt.as_tensor_variable(mu) lam = pt.as_tensor_variable(lam) return super().dist([mu, lam, alpha], **kwargs)
def support_point(rv, size, mu, lam, alpha): mu, _, _ = pt.broadcast_arrays(mu, lam, alpha) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu @staticmethod def get_mu_lam_phi(mu, lam, phi): if mu is None: if lam is not None and phi is not None: return lam / phi, lam, phi else: if lam is None: if phi is None: return mu, 1.0, 1.0 / mu else: return mu, mu * phi, phi else: if phi is None: return mu, lam, lam / mu raise ValueError( "Wald distribution must specify either mu only, " "mu and lam, mu and phi, or lam and phi." ) def logp(value, mu, lam, alpha): centered_value = value - alpha logp = pt.switch( pt.le(centered_value, 0), -np.inf, ( logpow(lam / (2.0 * np.pi), 0.5) - logpow(centered_value, 1.5) - (0.5 * lam / centered_value * ((centered_value - mu) / mu) ** 2) ), ) return check_parameters( logp, mu > 0, lam > 0, alpha >= 0, msg="mu > 0, lam > 0, alpha >= 0", ) def logcdf(value, mu, lam, alpha): value -= alpha q = value / mu ell = lam * mu r = pt.sqrt(value * lam) a = normal_lcdf(0, 1, (q - 1.0) / r) b = 2.0 / ell + normal_lcdf(0, 1, -(q + 1.0) / r) logcdf = pt.switch( pt.le(value, 0), -np.inf, pt.switch( pt.lt(value, np.inf), a + pt.log1pexp(b - a), 0, ), ) return check_parameters( logcdf, mu > 0, lam > 0, alpha >= 0, msg="mu > 0, lam > 0, alpha >= 0", )
class BetaClippedRV(BetaRV): @classmethod def rng_fn(cls, rng, alpha, beta, size) -> np.ndarray: return np.asarray(clipped_beta_rvs(alpha, beta, size=size, random_state=rng)) beta = BetaClippedRV()
[docs] class Beta(UnitContinuous): r""" Beta log-likelihood. The pdf of this distribution is .. math:: f(x \mid \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} where :math:`B` is the Beta function. For more information, see https://en.wikipedia.org/wiki/Beta_distribution. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 1, 200) alphas = [.5, 5., 1., 2., 2.] betas = [.5, 1., 3., 2., 5.] for a, b in zip(alphas, betas): pdf = st.beta.pdf(x, a, b) plt.plot(x, pdf, label=r'$\alpha$ = {}, $\beta$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.ylim(0, 4.5) plt.legend(loc=9) plt.show() ======== ============================================================== 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 can be parameterized either in terms of alpha and beta, mean and standard deviation or mean and sample size. The link between the three parametrizations is given by .. math:: \alpha &= \mu \kappa \\ \beta &= (1 - \mu) \kappa \text{where } \kappa = \frac{\mu(1-\mu)}{\sigma^2} - 1 \alpha = \mu * \nu \beta = (1 - \mu) * \nu Parameters ---------- alpha : tensor_like of float, optional ``alpha`` > 0. If not specified, then calculated using ``mu`` and ``sigma``. beta : tensor_like of float, optional ``beta`` > 0. If not specified, then calculated using ``mu`` and ``sigma``. mu : tensor_like of float, optional Alternative mean (0 < ``mu`` < 1). sigma : tensor_like of float, optional Alternative standard deviation (0 < ``sigma`` < sqrt(``mu`` * (1 - ``mu``))). nu : tensor_like of float, optional Alternative "sample size" of a Beta distribution (``nu`` > 0). Notes ----- Beta distribution is a conjugate prior for the parameter :math:`p` of the binomial distribution. """ rv_op = pytensor.tensor.random.beta
[docs] @classmethod def dist( cls, alpha: DIST_PARAMETER_TYPES | None = None, beta: DIST_PARAMETER_TYPES | None = None, mu: DIST_PARAMETER_TYPES | None = None, sigma: DIST_PARAMETER_TYPES | None = None, nu: DIST_PARAMETER_TYPES | None = None, *args, **kwargs, ): alpha, beta = cls.get_alpha_beta(alpha, beta, mu, sigma, nu) alpha = pt.as_tensor_variable(alpha) beta = pt.as_tensor_variable(beta) return super().dist([alpha, beta], **kwargs)
def support_point(rv, size, alpha, beta): mean = alpha / (alpha + beta) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean @classmethod def get_alpha_beta(self, alpha=None, beta=None, mu=None, sigma=None, nu=None): if (alpha is not None) and (beta is not None): pass elif (mu is not None) and (sigma is not None): kappa = mu * (1 - mu) / sigma**2 - 1 alpha = mu * kappa beta = (1 - mu) * kappa elif (mu is not None) and (nu is not None): alpha = mu * nu beta = (1 - mu) * nu else: raise ValueError( "Incompatible parameterization. Either use alpha " "and beta, mu and sigma or mu and nu to specify " "distribution." ) return alpha, beta def logp(value, alpha, beta): res = ( pt.switch(pt.eq(alpha, 1.0), 0.0, (alpha - 1.0) * pt.log(value)) + pt.switch(pt.eq(beta, 1.0), 0.0, (beta - 1.0) * pt.log1p(-value)) - (pt.gammaln(alpha) + pt.gammaln(beta) - pt.gammaln(alpha + beta)) ) res = pt.switch(pt.bitwise_and(pt.ge(value, 0.0), pt.le(value, 1.0)), res, -np.inf) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def logcdf(value, alpha, beta): logcdf = pt.switch( pt.lt(value, 0), -np.inf, pt.switch( pt.lt(value, 1), pt.log(pt.betainc(alpha, beta, value)), 0, ), ) return check_parameters( logcdf, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def icdf(value, alpha, beta): res = betaincinv(alpha, beta, value) res = check_icdf_value(res, value) return check_icdf_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", )
class KumaraswamyRV(SymbolicRandomVariable): name = "kumaraswamy" signature = "[rng],[size],(),()->[rng],()" _print_name = ("Kumaraswamy", "\\operatorname{Kumaraswamy}") @classmethod def rv_op(cls, a, b, *, size=None, rng=None): a = pt.as_tensor(a) b = pt.as_tensor(b) rng = normalize_rng_param(rng) size = normalize_size_param(size) if rv_size_is_none(size): size = implicit_size_from_params(a, b, ndims_params=cls.ndims_params) next_rng, u = uniform(size=size, rng=rng).owner.outputs draws = (1 - (1 - u) ** (1 / b)) ** (1 / a) return cls( inputs=[rng, size, a, b], outputs=[next_rng, draws], )(rng, size, a, b)
[docs] class Kumaraswamy(UnitContinuous): r""" Kumaraswamy log-likelihood. The pdf of this distribution is .. math:: f(x \mid a, b) = abx^{a-1}(1-x^a)^{b-1} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 1, 200) a_s = [.5, 5., 1., 2., 2.] b_s = [.5, 1., 3., 2., 5.] for a, b in zip(a_s, b_s): pdf = a * b * x ** (a - 1) * (1 - x ** a) ** (b - 1) plt.plot(x, pdf, label=r'$a$ = {}, $b$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.ylim(0, 3.) plt.legend(loc=9) plt.show() ======== ============================================================== Support :math:`x \in (0, 1)` Mean :math:`b B(1 + \tfrac{1}{a}, b)` Variance :math:`b B(1 + \tfrac{2}{a}, b) - (b B(1 + \tfrac{1}{a}, b))^2` ======== ============================================================== Parameters ---------- a : tensor_like of float a > 0. b : tensor_like of float b > 0. """ rv_type = KumaraswamyRV rv_op = KumaraswamyRV.rv_op
[docs] @classmethod def dist(cls, a: DIST_PARAMETER_TYPES, b: DIST_PARAMETER_TYPES, *args, **kwargs): return super().dist([a, b], *args, **kwargs)
def support_point(rv, size, a, b): mean = pt.exp(pt.log(b) + pt.gammaln(1 + 1 / a) + pt.gammaln(b) - pt.gammaln(1 + 1 / a + b)) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, a, b): res = pt.log(a) + pt.log(b) + (a - 1) * pt.log(value) + (b - 1) * pt.log(1 - value**a) res = pt.switch( pt.or_(pt.lt(value, 0), pt.gt(value, 1)), -np.inf, res, ) return check_parameters( res, a > 0, b > 0, msg="a > 0, b > 0", ) def logcdf(value, a, b): res = pt.switch( pt.lt(value, 0), -np.inf, pt.switch( pt.lt(value, 1), pt.log1mexp(b * pt.log1p(-(value**a))), 0, ), ) return check_parameters( res, a > 0, b > 0, msg="a > 0, b > 0", )
[docs] class Exponential(PositiveContinuous): r""" Exponential log-likelihood. The pdf of this distribution is .. math:: f(x \mid \lambda) = \lambda \exp\left\{ -\lambda x \right\} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 3, 100) for lam in [0.5, 1., 2.]: pdf = st.expon.pdf(x, scale=1.0/lam) plt.plot(x, pdf, label=r'$\lambda$ = {}'.format(lam)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================ Support :math:`x \in [0, \infty)` Mean :math:`\dfrac{1}{\lambda}` Variance :math:`\dfrac{1}{\lambda^2}` ======== ============================ Parameters ---------- lam : tensor_like of float Rate or inverse scale (``lam`` > 0). scale: tensor_like of float Alternative parameter (scale = 1/lam). """ rv_op = exponential
[docs] @classmethod def dist(cls, lam=None, scale=None, *args, **kwargs): if lam is not None and scale is not None: raise ValueError("Incompatible parametrization. Can't specify both lam and scale.") elif lam is None and scale is None: raise ValueError("Incompatible parametrization. Must specify either lam or scale.") if scale is None: scale = pt.reciprocal(lam) scale = pt.as_tensor_variable(scale) # PyTensor exponential op is parametrized in terms of mu (1/lam) return super().dist([scale], **kwargs)
def support_point(rv, size, mu): if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, mu): res = -pt.log(mu) - value / mu res = pt.switch(pt.ge(value, 0.0), res, -np.inf) return check_parameters( res, mu >= 0, msg="mu >= 0", ) def logcdf(value, mu): lam = pt.reciprocal(mu) res = pt.switch( pt.lt(value, 0), -np.inf, pt.log1mexp(-lam * value), ) return check_parameters( res, lam >= 0, msg="lam >= 0", ) def icdf(value, mu): res = -mu * pt.log(1 - value) res = check_icdf_value(res, value) return check_icdf_parameters( res, mu >= 0, msg="mu >= 0", )
[docs] class Laplace(Continuous): r""" Laplace log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, b) = \frac{1}{2b} \exp \left\{ - \frac{|x - \mu|}{b} \right\} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-10, 10, 1000) mus = [0., 0., 0., -5.] bs = [1., 2., 4., 4.] for mu, b in zip(mus, bs): pdf = st.laplace.pdf(x, loc=mu, scale=b) plt.plot(x, pdf, label=r'$\mu$ = {}, $b$ = {}'.format(mu, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu` Variance :math:`2 b^2` ======== ======================== Parameters ---------- mu : tensor_like of float Location parameter. b : tensor_like of float Scale parameter (b > 0). """ rv_op = laplace
[docs] @classmethod def dist(cls, mu, b, *args, **kwargs): b = pt.as_tensor_variable(b) mu = pt.as_tensor_variable(mu) return super().dist([mu, b], *args, **kwargs)
def support_point(rv, size, mu, b): mu, _ = pt.broadcast_arrays(mu, b) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, mu, b): res = -pt.log(2 * b) - pt.abs(value - mu) / b return check_parameters( res, b > 0, msg="b > 0", ) def logcdf(value, mu, b): y = (value - mu) / b res = pt.switch( pt.le(value, mu), pt.log(0.5) + y, pt.switch( pt.gt(y, 1), pt.log1p(-0.5 * pt.exp(-y)), pt.log(1 - 0.5 * pt.exp(-y)), ), ) return check_parameters( res, b > 0, msg="b > 0", ) def icdf(value, mu, b): res = pt.switch( pt.le(value, 0.5), mu + b * np.log(2 * value), mu - b * np.log(2 - 2 * value) ) res = check_icdf_value(res, value) return check_icdf_parameters(res, b > 0, msg="b > 0")
class AsymmetricLaplaceRV(SymbolicRandomVariable): name = "asymmetriclaplace" signature = "[rng],[size],(),(),()->[rng],()" _print_name = ("AsymmetricLaplace", "\\operatorname{AsymmetricLaplace}") @classmethod def rv_op(cls, b, kappa, mu, *, size=None, rng=None): b = pt.as_tensor(b) kappa = pt.as_tensor(kappa) mu = pt.as_tensor(mu) rng = normalize_rng_param(rng) size = normalize_size_param(size) if rv_size_is_none(size): size = implicit_size_from_params(b, kappa, mu, ndims_params=cls.ndims_params) next_rng, u = uniform(size=size, rng=rng).owner.outputs switch = kappa**2 / (1 + kappa**2) non_positive_x = mu + kappa * pt.log(u * (1 / switch)) / b positive_x = mu - pt.log((1 - u) * (1 + kappa**2)) / (kappa * b) draws = non_positive_x * (u <= switch) + positive_x * (u > switch) return cls( inputs=[rng, size, b, kappa, mu], outputs=[next_rng, draws], )(rng, size, b, kappa, mu)
[docs] class AsymmetricLaplace(Continuous): r""" Asymmetric-Laplace log-likelihood. The pdf of this distribution is .. math:: {f(x|\\b,\kappa,\mu) = \left({\frac{\\b}{\kappa + 1/\kappa}}\right)\,e^{-(x-\mu)\\b\,s\kappa ^{s}}} where .. math:: s = sgn(x-\mu) ======== ======================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu-\frac{\\\kappa-1/\kappa}b` Variance :math:`\frac{1+\kappa^{4}}{b^2\kappa^2 }` ======== ======================== AsymmetricLaplace distribution can be parameterized either in terms of kappa or q. The link between the two parametrizations is given by .. math:: \kappa = \sqrt(\frac{q}{1-q}) Parameters ---------- kappa : tensor_like of float Symmetry parameter (kappa > 0). mu : tensor_like of float Location parameter. b : tensor_like of float Scale parameter (b > 0). q : tensor_like of float Symmetry parameter (0 < q < 1). Notes ----- The parametrization in terms of q is useful for quantile regression with q being the quantile of interest. """ rv_type = AsymmetricLaplaceRV rv_op = AsymmetricLaplaceRV.rv_op
[docs] @classmethod def dist(cls, kappa=None, mu=None, b=None, q=None, *args, **kwargs): kappa = cls.get_kappa(kappa, q) return super().dist([b, kappa, mu], *args, **kwargs)
@classmethod def get_kappa(cls, kappa=None, q=None): if kappa is not None and q is not None: raise ValueError( "Incompatible parameterization. Either use " "kappa or q to specify the distribution." ) elif q is not None: if isinstance(q, Variable): q = check_parameters(q, q > 0, q < 1, msg="0 < q < 1") else: assert np.all((np.asarray(q) > 0) | (np.asarray(q) < 1)) kappa = (q / (1 - q)) ** 0.5 return kappa def support_point(rv, size, b, kappa, mu): mean = mu - (kappa - 1 / kappa) / b if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, b, kappa, mu): value = value - mu res = pt.log(b / (kappa + (kappa**-1))) + ( -value * b * pt.sgn(value) * (kappa ** pt.sgn(value)) ) return check_parameters( res, b > 0, kappa > 0, msg="b > 0, kappa > 0", )
[docs] class LogNormal(PositiveContinuous): r""" Log-normal log-likelihood. Distribution of any random variable whose logarithm is normally distributed. A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many small independent factors. Note: Class name Lognormal is deprecated, use LogNormal now! The pdf of this distribution is .. math:: f(x \mid \mu, \tau) = \frac{1}{x} \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (\ln(x)-\mu)^2 \right\} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 3, 100) mus = [0., 0., 0.] sigmas = [.25, .5, 1.] for mu, sigma in zip(mus, sigmas): pdf = st.lognorm.pdf(x, sigma, scale=np.exp(mu)) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}'.format(mu, sigma)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================================================= Support :math:`x \in [0, \infty)` Mean :math:`\exp\{\mu + \frac{1}{2\tau}\}` Variance :math:`(\exp\{\frac{1}{\tau}\} - 1) \times \exp\{2\mu + \frac{1}{\tau}\}` ======== ========================================================================= Parameters ---------- mu : tensor_like of float, default 0 Location parameter. sigma : tensor_like of float, optional Standard deviation. (sigma > 0). (only required if tau is not specified). Defaults to 1. tau : tensor_like of float, optional Scale parameter (tau > 0). (only required if sigma is not specified). Defaults to 1. Examples -------- .. code-block:: python # Example to show that we pass in only ``sigma`` or ``tau`` but not both. with pm.Model(): x = pm.LogNormal('x', mu=2, sigma=30) with pm.Model(): x = pm.LogNormal('x', mu=2, tau=1/100) """ rv_op = lognormal
[docs] @classmethod def dist(cls, mu=0, sigma=None, tau=None, *args, **kwargs): tau, sigma = get_tau_sigma(tau=tau, sigma=sigma) mu = pt.as_tensor_variable(mu) sigma = pt.as_tensor_variable(sigma) return super().dist([mu, sigma], *args, **kwargs)
def support_point(rv, size, mu, sigma): mean = pt.exp(mu + 0.5 * sigma**2) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, mu, sigma): res = ( -0.5 * pt.pow((pt.log(value) - mu) / sigma, 2) - 0.5 * pt.log(2.0 * np.pi) - pt.log(sigma) - pt.log(value) ) res = pt.switch(pt.gt(value, 0.0), res, -np.inf) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def logcdf(value, mu, sigma): res = pt.switch( pt.le(value, 0), -np.inf, normal_lcdf(mu, sigma, pt.log(value)), ) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def icdf(value, mu, sigma): res = pt.exp(icdf(Normal.dist(mu, sigma), value)) return res
Lognormal = LogNormal
[docs] class StudentT(Continuous): r""" Student's T log-likelihood. Describes a normal variable whose precision is gamma distributed. If only nu parameter is passed, this specifies a standard (central) Student's T. The pdf of this distribution is .. math:: f(x|\mu,\lambda,\nu) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left[1+\frac{\lambda(x-\mu)^2}{\nu}\right]^{-\frac{\nu+1}{2}} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-8, 8, 200) mus = [0., 0., -2., -2.] sigmas = [1., 1., 1., 2.] dfs = [1., 5., 5., 5.] for mu, sigma, df in zip(mus, sigmas, dfs): pdf = st.t.pdf(x, df, loc=mu, scale=sigma) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}, $\nu$ = {}'.format(mu, sigma, df)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in \mathbb{R}` ======== ======================== Parameters ---------- nu : tensor_like of float Degrees of freedom, also known as normality parameter (nu > 0). mu : tensor_like of float, default 0 Location parameter. sigma : tensor_like of float, optional Scale parameter (sigma > 0). Converges to the standard deviation as nu increases (only required if lam is not specified). Defaults to 1. lam : tensor_like of float, optional Scale parameter (lam > 0). Converges to the precision as nu increases (only required if sigma is not specified). Defaults to 1. Examples -------- .. code-block:: python with pm.Model(): x = pm.StudentT('x', nu=15, mu=0, sigma=10) with pm.Model(): x = pm.StudentT('x', nu=15, mu=0, lam=1/23) """ rv_op = t
[docs] @classmethod def dist(cls, nu, mu=0, *, sigma=None, lam=None, **kwargs): nu = pt.as_tensor_variable(nu) lam, sigma = get_tau_sigma(tau=lam, sigma=sigma) sigma = pt.as_tensor_variable(sigma) return super().dist([nu, mu, sigma], **kwargs)
def support_point(rv, size, nu, mu, sigma): mu, _, _ = pt.broadcast_arrays(mu, nu, sigma) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, nu, mu, sigma): lam, _ = get_tau_sigma(sigma=sigma) res = ( gammaln((nu + 1.0) / 2.0) + 0.5 * pt.log(lam / (nu * np.pi)) - gammaln(nu / 2.0) - (nu + 1.0) / 2.0 * pt.log1p(lam * (value - mu) ** 2 / nu) ) return check_parameters( res, lam > 0, nu > 0, msg="lam > 0, nu > 0", ) def logcdf(value, nu, mu, sigma): _, sigma = get_tau_sigma(sigma=sigma) t = (value - mu) / sigma sqrt_t2_nu = pt.sqrt(t**2 + nu) x = (t + sqrt_t2_nu) / (2.0 * sqrt_t2_nu) res = pt.log(pt.betainc(nu / 2.0, nu / 2.0, x)) return check_parameters( res, nu > 0, sigma > 0, msg="nu > 0, sigma > 0", ) def icdf(value, nu, mu, sigma): res = pt.switch( pt.lt(value, 0.5), -pt.sqrt(nu) * pt.sqrt((1.0 / betaincinv(nu * 0.5, 0.5, 2.0 * value)) - 1.0), pt.sqrt(nu) * pt.sqrt((1.0 / betaincinv(nu * 0.5, 0.5, 2.0 * (1 - value))) - 1.0), ) res = mu + res * sigma res = check_icdf_value(res, value) return check_icdf_parameters( res, nu > 0, sigma > 0, msg="nu > 0, sigma > 0", )
class SkewStudentTRV(RandomVariable): name = "skewstudentt" ndim_supp = 0 ndims_params = [0, 0, 0, 0] dtype = "floatX" _print_name = ("SkewStudentT", "\\operatorname{SkewStudentT}") @classmethod def rng_fn(cls, rng, a, b, mu, sigma, size=None) -> np.ndarray: return np.asarray( stats.jf_skew_t.rvs(a=a, b=b, loc=mu, scale=sigma, size=size, random_state=rng) ) skewstudentt = SkewStudentTRV()
[docs] class SkewStudentT(Continuous): r""" Skewed Student's T distribution log-likelihood. This follows Jones and Faddy (2003) The pdf of this distribution is .. math:: f(t)=f(t ; a, b)=C_{a, b}^{-1}\left\{1+\frac{t}{\left(a+b+t^2\right)^{1 / 2}}\right\}^{a+1 / 2}\left\{1-\frac{t}{\left(a+b+t^2\right)^{1 / 2}}\right\}^{b+1 / 2} where .. math:: C_{a, b}=2^{a+b-1} B(a, b)(a+b)^{1 / 2} ======== ============================================================= Support :math:`x \in [\infty, \infty)` Mean :math:`E(T)=\frac{(a-b) \sqrt{(a+b)}}{2} \frac{\Gamma\left(a-\frac{1}{2}\right) \Gamma\left(b-\frac{1}{2}\right)}{\Gamma(a) \Gamma(b)}` ======== ============================================================= Parameters ---------- a : tensor_like of float First kurtosis parameter (a > 0). b : tensor_like of float Second kurtosis parameter (b > 0). mu : tensor_like of float Location parameter. sigma : tensor_like of float Scale parameter (sigma > 0). Converges to the standard deviation as a and b become close (only required if lam is not specified). Defaults to 1. lam : tensor_like of float, optional Scale parameter (lam > 0). Converges to the precision as a and b become close (only required if sigma is not specified). Defaults to 1. """ rv_op = skewstudentt
[docs] @classmethod def dist(cls, a, b, *, mu=0, sigma=None, lam=None, **kwargs): a = pt.as_tensor_variable(a) b = pt.as_tensor_variable(b) lam, sigma = get_tau_sigma(tau=lam, sigma=sigma) sigma = pt.as_tensor_variable(sigma) return super().dist([a, b, mu, sigma], **kwargs)
def support_point(rv, size, a, b, mu, sigma): a, b, mu, _ = pt.broadcast_arrays(a, b, mu, sigma) Et = mu + (a - b) * pt.sqrt(a + b) * gammafn(a - 0.5) * gammafn(b - 0.5) / ( 2 * gammafn(a) * gammafn(b) ) if not rv_size_is_none(size): Et = pt.full(size, Et) return Et def logp(value, a, b, mu, sigma): _, sigma = get_tau_sigma(sigma=sigma) x = (value - mu) / sigma a_ = (a + 0.5) * pt.log(1 + x / pt.sqrt(a + b + x**2)) b_ = (b + 0.5) * pt.log(1 - x / pt.sqrt(a + b + x**2)) c = (a + b - 1) * pt.log(2) + pt.special.betaln(a, b) + 0.5 * pt.log(a + b) res = a_ + b_ - c - pt.log(sigma) return check_parameters( res, a > 0, b > 0, sigma > 0, msg="a > 0, b > 0, sigma > 0", ) def logcdf(value, a, b, mu, sigma): _, sigma = get_tau_sigma(sigma=sigma) x = (value - mu) / sigma y = (1 + x / pt.sqrt(a + b + x**2)) * 0.5 res = pt.log(pt.betainc(a, b, y)) return check_parameters( res, a > 0, b > 0, sigma > 0, msg="a > 0, b > 0, sigma > 0", ) def icdf(value, a, b, mu, sigma): _, sigma = get_tau_sigma(sigma=sigma) bval = betaincinv(a, b, value) num = (2 * bval - 1) * pt.sqrt(a + b) denom = 2 * pt.sqrt(bval * (1 - bval)) res = num / denom res = mu + res * sigma res = check_icdf_value(res, value) return check_icdf_parameters( res, a > 0, b > 0, sigma > 0, msg="a > 0, b > 0, sigma > 0", )
[docs] class Pareto(BoundedContinuous): r""" Pareto log-likelihood. Often used to characterize wealth distribution, or other examples of the 80/20 rule. The pdf of this distribution is .. math:: f(x \mid \alpha, m) = \frac{\alpha m^{\alpha}}{x^{\alpha+1}} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 4, 1000) alphas = [1., 2., 5., 5.] ms = [1., 1., 1., 2.] for alpha, m in zip(alphas, ms): pdf = st.pareto.pdf(x, alpha, scale=m) plt.plot(x, pdf, label=r'$\alpha$ = {}, m = {}'.format(alpha, m)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================================================= Support :math:`x \in [m, \infty)` Mean :math:`\dfrac{\alpha m}{\alpha - 1}` for :math:`\alpha \ge 1` Variance :math:`\dfrac{m \alpha}{(\alpha - 1)^2 (\alpha - 2)}` for :math:`\alpha > 2` ======== ============================================================= Parameters ---------- alpha : tensor_like of float Shape parameter (alpha > 0). m : tensor_like of float Scale parameter (m > 0). """ rv_op = pareto bound_args_indices = (4, None) # lower-bounded by `m`
[docs] @classmethod def dist(cls, alpha, m, **kwargs): alpha = pt.as_tensor_variable(alpha) m = pt.as_tensor_variable(m) return super().dist([alpha, m], **kwargs)
def support_point(rv, size, alpha, m): median = m * 2 ** (1 / alpha) if not rv_size_is_none(size): median = pt.full(size, median) return median def logp(value, alpha, m): res = pt.log(alpha) + logpow(m, alpha) - logpow(value, alpha + 1.0) res = pt.switch(pt.ge(value, m), res, -np.inf) return check_parameters( res, alpha > 0, m > 0, msg="alpha > 0, m > 0", ) def logcdf(value, alpha, m): arg = (m / value) ** alpha res = pt.switch( pt.lt(value, m), -np.inf, pt.switch( pt.le(arg, 1e-5), pt.log1p(-arg), pt.log(1 - arg), ), ) return check_parameters( res, alpha > 0, m > 0, msg="alpha > 0, m > 0", ) def icdf(value, alpha, m): res = m * pt.pow(1 - value, -1 / alpha) res = check_icdf_value(res, value) return check_icdf_parameters( res, alpha > 0, m > 0, msg="alpha > 0, m > 0", )
@_default_transform.register(Pareto) def pareto_default_transform(op, rv): return bounded_cont_transform(op, rv, Pareto.bound_args_indices)
[docs] class Cauchy(Continuous): r""" Cauchy log-likelihood. Also known as the Lorentz or the Breit-Wigner distribution. The pdf of this distribution is .. math:: f(x \mid \alpha, \beta) = \frac{1}{\pi \beta [1 + (\frac{x-\alpha}{\beta})^2]} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-5, 5, 500) alphas = [0., 0., 0., -2.] betas = [.5, 1., 2., 1.] for a, b in zip(alphas, betas): pdf = st.cauchy.pdf(x, loc=a, scale=b) plt.plot(x, pdf, label=r'$\alpha$ = {}, $\beta$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in \mathbb{R}` Mode :math:`\alpha` Mean undefined Variance undefined ======== ======================== Parameters ---------- alpha : tensor_like of float Location parameter. beta : tensor_like of float Scale parameter > 0. """ rv_op = cauchy
[docs] @classmethod def dist(cls, alpha, beta, *args, **kwargs): alpha = pt.as_tensor_variable(alpha) beta = pt.as_tensor_variable(beta) return super().dist([alpha, beta], **kwargs)
def support_point(rv, size, alpha, beta): alpha, _ = pt.broadcast_arrays(alpha, beta) if not rv_size_is_none(size): alpha = pt.full(size, alpha) return alpha def logp(value, alpha, beta): res = -pt.log(np.pi) - pt.log(beta) - pt.log1p(pt.pow((value - alpha) / beta, 2)) return check_parameters( res, beta > 0, msg="beta > 0", ) def logcdf(value, alpha, beta): res = pt.log(0.5 + pt.arctan((value - alpha) / beta) / np.pi) return check_parameters( res, beta > 0, msg="beta > 0", ) def icdf(value, alpha, beta): res = alpha + beta * pt.tan(np.pi * (value - 0.5)) res = check_icdf_value(res, value) return check_icdf_parameters( res, beta > 0, msg="beta > 0", )
[docs] class HalfCauchy(PositiveContinuous): r""" Half-Cauchy log-likelihood. The pdf of this distribution is .. math:: f(x \mid \beta) = \frac{2}{\pi \beta [1 + (\frac{x}{\beta})^2]} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 5, 200) for b in [0.5, 1.0, 2.0]: pdf = st.cauchy.pdf(x, scale=b) plt.plot(x, pdf, label=r'$\beta$ = {}'.format(b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in [0, \infty)` Mode 0 Mean undefined Variance undefined ======== ======================== Parameters ---------- beta : tensor_like of float Scale parameter (beta > 0). """ rv_op = halfcauchy
[docs] @classmethod def dist(cls, beta, *args, **kwargs): beta = pt.as_tensor_variable(beta) return super().dist([0.0, beta], **kwargs)
def support_point(rv, size, loc, beta): if not rv_size_is_none(size): beta = pt.full(size, beta) return beta def logp(value, loc, beta): res = pt.log(2) + _logprob_helper(Cauchy.dist(loc, beta), value) res = pt.switch(pt.ge(value, loc), res, -np.inf) return check_parameters( res, beta > 0, msg="beta > 0", ) def logcdf(value, loc, beta): res = pt.switch( pt.lt(value, loc), -np.inf, pt.log(2 * pt.arctan((value - loc) / beta) / np.pi), ) return check_parameters( res, beta > 0, msg="beta > 0", ) def icdf(value, loc, beta): res = loc + beta * pt.tan(np.pi * (value) / 2.0) res = check_icdf_value(res, value) return check_icdf_parameters( res, beta > 0, msg="beta > 0", )
[docs] class Gamma(PositiveContinuous): r""" Gamma log-likelihood. Represents the sum of alpha exponentially distributed random variables, each of which has rate beta. The pdf of this distribution is .. math:: f(x \mid \alpha, \beta) = \frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 20, 200) alphas = [1., 2., 3., 7.5] betas = [.5, .5, 1., 1.] for a, b in zip(alphas, betas): pdf = st.gamma.pdf(x, a, scale=1.0/b) plt.plot(x, pdf, label=r'$\alpha$ = {}, $\beta$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== =============================== Support :math:`x \in (0, \infty)` Mean :math:`\dfrac{\alpha}{\beta}` Variance :math:`\dfrac{\alpha}{\beta^2}` ======== =============================== Gamma distribution can be parameterized either in terms of alpha and beta or mean and standard deviation. The link between the two parametrizations is given by .. math:: \alpha &= \frac{\mu^2}{\sigma^2} \\ \beta &= \frac{\mu}{\sigma^2} Parameters ---------- alpha : tensor_like of float, optional Shape parameter (alpha > 0). beta : tensor_like of float, optional Rate parameter (beta > 0). mu : tensor_like of float, optional Alternative shape parameter (mu > 0). sigma : tensor_like of float, optional Alternative scale parameter (sigma > 0). """ # gamma is temporarily a deprecation wrapper in PyTensor rv_op = _gamma
[docs] @classmethod def dist(cls, alpha=None, beta=None, mu=None, sigma=None, **kwargs): alpha, beta = cls.get_alpha_beta(alpha, beta, mu, sigma) alpha = pt.as_tensor_variable(alpha) beta = pt.as_tensor_variable(beta) # PyTensor gamma op is parametrized in terms of scale (1/beta) scale = pt.reciprocal(beta) return super().dist([alpha, scale], **kwargs)
@classmethod def get_alpha_beta(cls, alpha=None, beta=None, mu=None, sigma=None): if (alpha is not None) and (beta is not None): pass elif (mu is not None) and (sigma is not None): if isinstance(sigma, Variable): sigma = check_parameters(sigma, sigma > 0, msg="sigma > 0") else: assert np.all(np.asarray(sigma) > 0) alpha = mu**2 / sigma**2 beta = mu / sigma**2 else: raise ValueError( "Incompatible parameterization. Either use " "alpha and beta, or mu and sigma to specify " "distribution." ) return alpha, beta def support_point(rv, size, alpha, scale): mean = alpha * scale if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, alpha, scale): beta = pt.reciprocal(scale) res = -pt.gammaln(alpha) + logpow(beta, alpha) - beta * value + logpow(value, alpha - 1) res = pt.switch(pt.ge(value, 0.0), res, -np.inf) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def logcdf(value, alpha, scale): beta = pt.reciprocal(scale) res = pt.switch( pt.lt(value, 0), -np.inf, pt.log(pt.gammainc(alpha, beta * value)), ) return check_parameters(res, 0 < alpha, 0 < beta, msg="alpha > 0, beta > 0") def icdf(value, alpha, scale): res = scale * gammaincinv(alpha, value) res = check_icdf_value(res, value) return check_icdf_parameters( res, alpha > 0, scale > 0, msg="alpha > 0, beta > 0", )
[docs] class InverseGamma(PositiveContinuous): r""" Inverse gamma log-likelihood, 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 import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 3, 500) alphas = [1., 2., 3., 3.] betas = [1., 1., 1., .5] for a, b in zip(alphas, betas): pdf = st.invgamma.pdf(x, a, scale=b) plt.plot(x, pdf, label=r'$\alpha$ = {}, $\beta$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ====================================================== 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` ======== ====================================================== Parameters ---------- alpha : tensor_like of float, optional Shape parameter (alpha > 0). beta : tensor_like of float, optional Scale parameter (beta > 0). mu : tensor_like of float, optional Alternative shape parameter (mu > 0). sigma : tensor_like of float, optional Alternative scale parameter (sigma > 0). """ rv_op = invgamma
[docs] @classmethod def dist(cls, alpha=None, beta=None, mu=None, sigma=None, *args, **kwargs): alpha, beta = cls._get_alpha_beta(alpha, beta, mu, sigma) alpha = pt.as_tensor_variable(alpha) beta = pt.as_tensor_variable(beta) return super().dist([alpha, beta], **kwargs)
def support_point(rv, size, alpha, beta): mean = beta / (alpha - 1.0) mode = beta / (alpha + 1.0) support_point = pt.switch(alpha > 1, mean, mode) if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point @classmethod def _get_alpha_beta(cls, alpha, beta, mu, sigma): if alpha is not None: if beta is not None: pass else: beta = 1 elif (mu is not None) and (sigma is not None): if isinstance(sigma, Variable): sigma = check_parameters(sigma, sigma > 0, msg="sigma > 0") else: assert np.all(np.asarray(sigma) > 0) alpha = (2 * sigma**2 + mu**2) / sigma**2 beta = mu * (mu**2 + sigma**2) / sigma**2 else: raise ValueError( "Incompatible parameterization. Either use " "alpha and (optionally) beta, or mu and sigma to specify " "distribution." ) return alpha, beta def logp(value, alpha, beta): res = -pt.gammaln(alpha) + logpow(beta, alpha) - beta / value + logpow(value, -alpha - 1) res = pt.switch(pt.ge(value, 0.0), res, -np.inf) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def logcdf(value, alpha, beta): res = pt.switch( pt.lt(value, 0), -np.inf, pt.log(pt.gammaincc(alpha, beta / value)), ) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", )
[docs] class ChiSquared: r""" :math:`\chi^2` log-likelihood. This is the distribution from the sum of the squares of :math:`\nu` independent standard normal random variables or a special case of the gamma distribution with :math:`\alpha = \nu/2` and :math:`\beta = 1/2`. The pdf of this distribution is .. math:: f(x \mid \nu) = \frac{x^{(\nu-2)/2}e^{-x/2}}{2^{\nu/2}\Gamma(\nu/2)} Read more about the :math:`\chi^2` distribution at https://en.wikipedia.org/wiki/Chi-squared_distribution .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 15, 200) for df in [1, 2, 3, 6, 9]: pdf = st.chi2.pdf(x, df) plt.plot(x, pdf, label=r'$\nu$ = {}'.format(df)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.ylim(0, 0.6) plt.legend(loc=1) plt.show() ======== =============================== Support :math:`x \in [0, \infty)` Mean :math:`\nu` Variance :math:`2 \nu` ======== =============================== Parameters ---------- nu : tensor_like of float Degrees of freedom (nu > 0). """ def __new__(cls, name, nu, **kwargs): return Gamma(name, alpha=nu / 2, beta=1 / 2, **kwargs)
[docs] @classmethod def dist(cls, nu, **kwargs): return Gamma.dist(alpha=nu / 2, beta=1 / 2, **kwargs)
class WeibullBetaRV(RandomVariable): name = "weibull" ndim_supp = 0 ndims_params = [0, 0] dtype = "floatX" _print_name = ("Weibull", "\\operatorname{Weibull}") def __call__(self, alpha, beta, size=None, **kwargs): return super().__call__(alpha, beta, size=size, **kwargs) @classmethod def rng_fn(cls, rng, alpha, beta, size) -> np.ndarray: if size is None: size = np.broadcast_shapes(alpha.shape, beta.shape) return np.asarray(beta * rng.weibull(alpha, size=size)) weibull_beta = WeibullBetaRV()
[docs] class Weibull(PositiveContinuous): r""" Weibull log-likelihood. The pdf of this distribution is .. math:: f(x \mid \alpha, \beta) = \frac{\alpha x^{\alpha - 1} \exp(-(\frac{x}{\beta})^{\alpha})}{\beta^\alpha} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 3, 200) alphas = [.5, 1., 1.5, 5., 5.] betas = [1., 1., 1., 1., 2] for a, b in zip(alphas, betas): pdf = st.weibull_min.pdf(x, a, scale=b) plt.plot(x, pdf, label=r'$\alpha$ = {}, $\beta$ = {}'.format(a, b)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.ylim(0, 2.5) plt.legend(loc=1) plt.show() ======== ==================================================== Support :math:`x \in [0, \infty)` Mean :math:`\beta \Gamma(1 + \frac{1}{\alpha})` Variance :math:`\beta^2 \Gamma(1 + \frac{2}{\alpha} - \mu^2/\beta^2)` ======== ==================================================== Parameters ---------- alpha : float Shape parameter (alpha > 0). beta : float Scale parameter (beta > 0). """ rv_op = weibull_beta
[docs] @classmethod def dist(cls, alpha, beta, *args, **kwargs): alpha = pt.as_tensor_variable(alpha) beta = pt.as_tensor_variable(beta) return super().dist([alpha, beta], *args, **kwargs)
def support_point(rv, size, alpha, beta): mean = beta * pt.gamma(1 + 1 / alpha) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logcdf(value, alpha, beta): a = (value / beta) ** alpha res = pt.switch( pt.lt(value, 0), -np.inf, pt.log1mexp(-a), ) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def logp(value, alpha, beta): res = ( pt.log(alpha) - pt.log(beta) + (alpha - 1.0) * pt.log(value / beta) - pt.pow(value / beta, alpha) ) res = pt.switch(pt.ge(value, 0.0), res, -np.inf) return check_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", ) def icdf(value, alpha, beta): res = beta * (-pt.log(1 - value)) ** (1 / alpha) res = check_icdf_value(res, value) return check_icdf_parameters( res, alpha > 0, beta > 0, msg="alpha > 0, beta > 0", )
class HalfStudentTRV(SymbolicRandomVariable): name = "halfstudentt" signature = "[rng],[size],(),()->[rng],()" _print_name = ("HalfStudentT", "\\operatorname{HalfStudentT}") @classmethod def rv_op(cls, nu, sigma, *, size=None, rng=None) -> np.ndarray: nu = pt.as_tensor(nu) sigma = pt.as_tensor(sigma) rng = normalize_rng_param(rng) size = normalize_size_param(size) next_rng, t_draws = t(df=nu, scale=sigma, size=size, rng=rng).owner.outputs draws = pt.abs(t_draws) return cls(inputs=[rng, size, nu, sigma], outputs=[next_rng, draws])(rng, size, nu, sigma)
[docs] class HalfStudentT(PositiveContinuous): r""" Half Student's T log-likelihood. The pdf of this distribution is .. math:: f(x \mid \sigma,\nu) = \frac{2\;\Gamma\left(\frac{\nu+1}{2}\right)} {\Gamma\left(\frac{\nu}{2}\right)\sqrt{\nu\pi\sigma^2}} \left(1+\frac{1}{\nu}\frac{x^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 5, 200) sigmas = [1., 1., 2., 1.] nus = [.5, 1., 1., 30.] for sigma, nu in zip(sigmas, nus): pdf = st.t.pdf(x, df=nu, loc=0, scale=sigma) plt.plot(x, pdf, label=r'$\sigma$ = {}, $\nu$ = {}'.format(sigma, nu)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in [0, \infty)` ======== ======================== Parameters ---------- nu : tensor_like of float Degrees of freedom, also known as normality parameter (nu > 0). sigma : tensor_like of float, optional Scale parameter (sigma > 0). Converges to the standard deviation as nu increases (only required if lam is not specified). Defaults to 1. lam : tensor_like of float, optional Scale parameter (lam > 0). Converges to the precision as nu increases (only required if sigma is not specified). Defaults to 1. Examples -------- .. code-block:: python # Only pass in one of lam or sigma, but not both. with pm.Model(): x = pm.HalfStudentT('x', sigma=10, nu=10) with pm.Model(): x = pm.HalfStudentT('x', lam=4, nu=10) """ rv_type = HalfStudentTRV rv_op = HalfStudentTRV.rv_op
[docs] @classmethod def dist(cls, nu, sigma=None, lam=None, *args, **kwargs): lam, sigma = get_tau_sigma(lam, sigma) return super().dist([nu, sigma], *args, **kwargs)
def support_point(rv, size, nu, sigma): sigma, _ = pt.broadcast_arrays(sigma, nu) if not rv_size_is_none(size): sigma = pt.full(size, sigma) return sigma def logp(value, nu, sigma): res = ( pt.log(2) + gammaln((nu + 1.0) / 2.0) - gammaln(nu / 2.0) - 0.5 * pt.log(nu * np.pi * sigma**2) - (nu + 1.0) / 2.0 * pt.log1p(value**2 / (nu * sigma**2)) ) res = pt.switch( pt.lt(value, 0), -np.inf, res, ) return check_parameters( res, sigma > 0, nu > 0, msg="sigma > 0, nu > 0", )
class ExGaussianRV(SymbolicRandomVariable): name = "exgaussian" signature = "[rng],[size],(),(),()->[rng],()" _print_name = ("ExGaussian", "\\operatorname{ExGaussian}") @classmethod def rv_op(cls, mu, sigma, nu, *, size=None, rng=None): mu = pt.as_tensor(mu) sigma = pt.as_tensor(sigma) nu = pt.as_tensor(nu) rng = normalize_rng_param(rng) size = normalize_size_param(size) if rv_size_is_none(size): size = implicit_size_from_params(mu, sigma, nu, ndims_params=cls.ndims_params) next_rng, normal_draws = normal(loc=mu, scale=sigma, size=size, rng=rng).owner.outputs final_rng, exponential_draws = exponential(scale=nu, size=size, rng=next_rng).owner.outputs draws = normal_draws + exponential_draws return cls(inputs=[rng, size, mu, sigma, nu], outputs=[final_rng, draws])( rng, size, mu, sigma, nu )
[docs] class ExGaussian(Continuous): r""" Exponentially modified Gaussian log-likelihood. Results from the convolution of a normal distribution with an exponential distribution. The pdf of this distribution is .. math:: f(x \mid \mu, \sigma, \tau) = \frac{1}{\nu}\; \exp\left\{\frac{\mu-x}{\nu}+\frac{\sigma^2}{2\nu^2}\right\} \Phi\left(\frac{x-\mu}{\sigma}-\frac{\sigma}{\nu}\right) where :math:`\Phi` is the cumulative distribution function of the standard normal distribution. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-6, 9, 200) mus = [0., -2., 0., -3.] sigmas = [1., 1., 3., 1.] nus = [1., 1., 1., 4.] for mu, sigma, nu in zip(mus, sigmas, nus): pdf = st.exponnorm.pdf(x, nu/sigma, loc=mu, scale=sigma) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}, $\nu$ = {}'.format(mu, sigma, nu)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ======================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu + \nu` Variance :math:`\sigma^2 + \nu^2` ======== ======================== Parameters ---------- mu : tensor_like of float, default 0 Mean of the normal distribution. sigma : tensor_like of float Standard deviation of the normal distribution (sigma > 0). nu : tensor_like of float Mean of the exponential distribution (nu > 0). References ---------- .. [Rigby2005] Rigby R.A. and Stasinopoulos D.M. (2005). "Generalized additive models for location, scale and shape" Applied Statististics., 54, part 3, pp 507-554. .. [Lacouture2008] Lacouture, Y. and Couseanou, D. (2008). "How to use MATLAB to fit the ex-Gaussian and other probability functions to a distribution of response times". Tutorials in Quantitative Methods for Psychology, Vol. 4, No. 1, pp 35-45. """ rv_type = ExGaussianRV rv_op = ExGaussianRV.rv_op
[docs] @classmethod def dist(cls, mu=0.0, sigma=None, nu=None, *args, **kwargs): return super().dist([mu, sigma, nu], *args, **kwargs)
def support_point(rv, size, mu, sigma, nu): mu, nu, _ = pt.broadcast_arrays(mu, nu, sigma) support_point = mu + nu if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point def logp(value, mu, sigma, nu): # Alogithm is adapted from dexGAUS.R from gamlss res = pt.switch( pt.gt(nu, 0.05 * sigma), ( -pt.log(nu) + (mu - value) / nu + 0.5 * (sigma / nu) ** 2 + normal_lcdf(mu + (sigma**2) / nu, sigma, value) ), log_normal(value, mean=mu, sigma=sigma), ) return check_parameters( res, sigma > 0, nu > 0, msg="nu > 0, sigma > 0", ) def logcdf(value, mu, sigma, nu): # Alogithm is adapted from pexGAUS.R from gamlss res = pt.switch( pt.gt(nu, 0.05 * sigma), logdiffexp( normal_lcdf(mu, sigma, value), ( (mu - value) / nu + 0.5 * (sigma / nu) ** 2 + normal_lcdf(mu + (sigma**2) / nu, sigma, value) ), ), normal_lcdf(mu, sigma, value), ) return check_parameters( res, sigma > 0, nu > 0, msg="sigma > 0, nu > 0", )
[docs] class VonMises(CircularContinuous): r""" Univariate VonMises log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, \kappa) = \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)} where :math:`I_0` is the modified Bessel function of order 0. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-np.pi, np.pi, 200) mus = [0., 0., 0., -2.5] kappas = [.01, 0.5, 4., 2.] for mu, kappa in zip(mus, kappas): pdf = st.vonmises.pdf(x, kappa, loc=mu) plt.plot(x, pdf, label=r'$\mu$ = {}, $\kappa$ = {}'.format(mu, kappa)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in [-\pi, \pi]` Mean :math:`\mu` Variance :math:`1-\frac{I_1(\kappa)}{I_0(\kappa)}` ======== ========================================== Parameters ---------- mu : tensor_like of float, default 0.0 Mean. kappa : tensor_like of float, default 1.0 Concentration (:math:`\frac{1}{\kappa}` is analogous to :math:`\sigma^2`). """ rv_op = vonmises
[docs] @classmethod def dist(cls, mu=0.0, kappa=1.0, *args, **kwargs): mu = pt.as_tensor_variable(mu) kappa = pt.as_tensor_variable(kappa) return super().dist([mu, kappa], *args, **kwargs)
def support_point(rv, size, mu, kappa): mu, _ = pt.broadcast_arrays(mu, kappa) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, mu, kappa): res = kappa * pt.cos(mu - value) - pt.log(2 * np.pi) - pt.log(pt.i0(kappa)) res = pt.switch(pt.bitwise_and(pt.ge(value, -np.pi), pt.le(value, np.pi)), res, -np.inf) return check_parameters( res, kappa > 0, msg="kappa > 0", )
class SkewNormalRV(RandomVariable): name = "skewnormal" ndim_supp = 0 ndims_params = [0, 0, 0] dtype = "floatX" _print_name = ("SkewNormal", "\\operatorname{SkewNormal}") @classmethod def rng_fn(cls, rng, mu, sigma, alpha, size=None) -> np.ndarray: return np.asarray( stats.skewnorm.rvs(a=alpha, loc=mu, scale=sigma, size=size, random_state=rng) ) skewnormal = SkewNormalRV()
[docs] class SkewNormal(Continuous): r""" Univariate skew-normal log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, \tau, \alpha) = 2 \Phi((x-\mu)\sqrt{\tau}\alpha) \phi(x,\mu,\tau) .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-4, 4, 200) for alpha in [-6, 0, 6]: pdf = st.skewnorm.pdf(x, alpha, loc=0, scale=1) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}, $\alpha$ = {}'.format(0, 1, alpha)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu + \sigma \sqrt{\frac{2}{\pi}} \frac {\alpha }{{\sqrt {1+\alpha ^{2}}}}` Variance :math:`\sigma^2 \left( 1-\frac{2\alpha^2}{(\alpha^2+1) \pi} \right)` ======== ========================================== Skew-normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by .. math:: \tau = \dfrac{1}{\sigma^2} Parameters ---------- mu : tensor_like of float, default 0 Location parameter. sigma : tensor_like of float, optional Scale parameter (sigma > 0). Defaults to 1. tau : tensor_like of float, optional Alternative scale parameter (tau > 0). Defaults to 1. alpha : tensor_like of float, default 1 Skewness parameter. Notes ----- When alpha=0 we recover the Normal distribution and mu becomes the mean, tau the precision and sigma the standard deviation. In the limit of alpha approaching plus/minus infinite we get a half-normal distribution. """ rv_op = skewnormal
[docs] @classmethod def dist(cls, alpha=1, mu=0.0, sigma=None, tau=None, *args, **kwargs): tau, sigma = get_tau_sigma(tau=tau, sigma=sigma) alpha = pt.as_tensor_variable(alpha) mu = pt.as_tensor_variable(mu) tau = pt.as_tensor_variable(tau) sigma = pt.as_tensor_variable(sigma) return super().dist([mu, sigma, alpha], *args, **kwargs)
def support_point(rv, size, mu, sigma, alpha): mean = mu + sigma * (2 / np.pi) ** 0.5 * alpha / (1 + alpha**2) ** 0.5 if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, mu, sigma, alpha): tau, _ = get_tau_sigma(sigma=sigma) res = ( pt.log(1 + pt.erf(((value - mu) * pt.sqrt(tau) * alpha) / pt.sqrt(2))) + (-tau * (value - mu) ** 2 + pt.log(tau / np.pi / 2.0)) / 2.0 ) return check_parameters( res, tau > 0, msg="tau > 0", )
[docs] class Triangular(BoundedContinuous): r""" Continuous Triangular log-likelihood. The pdf of this distribution is .. math:: \begin{cases} 0 & \text{for } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c, \\[4pt] \frac{2}{b-a} & \text{for } x = c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b, \\[4pt] 0 & \text{for } b < x. \end{cases} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-2, 10, 500) lowers = [0., -1, 2] cs = [2., 0., 6.5] uppers = [4., 1, 8] for lower, c, upper in zip(lowers, cs, uppers): scale = upper - lower c_ = (c - lower) / scale pdf = st.triang.pdf(x, loc=lower, c=c_, scale=scale) plt.plot(x, pdf, label='lower = {}, c = {}, upper = {}'.format(lower, c, upper)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================================================================ Support :math:`x \in [lower, upper]` Mean :math:`\dfrac{lower + upper + c}{3}` Variance :math:`\dfrac{upper^2 + lower^2 +c^2 - lower*upper - lower*c - upper*c}{18}` ======== ============================================================================ Parameters ---------- lower : tensor_like of float, default 0 Lower limit. c : tensor_like of float, default 0.5 Mode. upper : tensor_like of float, default 1 Upper limit. """ rv_op = triangular bound_args_indices = (3, 5) # lower, upper
[docs] @classmethod def dist(cls, lower=0, upper=1, c=0.5, *args, **kwargs): lower = pt.as_tensor_variable(lower) upper = pt.as_tensor_variable(upper) c = pt.as_tensor_variable(c) return super().dist([lower, c, upper], *args, **kwargs)
def support_point(rv, size, lower, c, upper): mean = (lower + upper + c) / 3 if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, lower, c, upper): res = pt.switch( pt.lt(value, c), pt.log(2 * (value - lower) / ((upper - lower) * (c - lower))), pt.log(2 * (upper - value) / ((upper - lower) * (upper - c))), ) res = pt.switch(pt.bitwise_and(pt.le(lower, value), pt.le(value, upper)), res, -np.inf) return check_parameters( res, lower <= c, c <= upper, msg="lower <= c <= upper", ) def logcdf(value, lower, c, upper): res = pt.switch( pt.le(value, lower), -np.inf, pt.switch( pt.le(value, c), pt.log(((value - lower) ** 2) / ((upper - lower) * (c - lower))), pt.switch( pt.lt(value, upper), pt.log1p(-((upper - value) ** 2) / ((upper - lower) * (upper - c))), 0, ), ), ) return check_parameters( res, lower <= c, c <= upper, msg="lower <= c <= upper", ) def icdf(value, lower, c, upper): res = pt.switch( pt.lt(value, ((c - lower) / (upper - lower))), lower + np.sqrt((upper - lower) * (c - lower) * value), upper - np.sqrt((upper - lower) * (upper - c) * (1 - value)), ) res = check_icdf_value(res, value) return check_icdf_parameters( res, lower <= c, c <= upper, msg="lower <= c <= upper", )
@_default_transform.register(Triangular) def triangular_default_transform(op, rv): return bounded_cont_transform(op, rv, Triangular.bound_args_indices)
[docs] class Gumbel(Continuous): r""" Univariate right-skewed Gumbel log-likelihood. This distribution is typically used for modeling maximum (or extreme) values. Those looking to find the extreme minimum provided by the left-skewed Gumbel should invert the sign of all x and mu values. The pdf of this distribution is .. math:: f(x \mid \mu, \beta) = \frac{1}{\beta}e^{-(z + e^{-z})} where .. math:: z = \frac{x - \mu}{\beta}. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-10, 20, 200) mus = [0., 4., -1.] betas = [2., 2., 4.] for mu, beta in zip(mus, betas): pdf = st.gumbel_r.pdf(x, loc=mu, scale=beta) plt.plot(x, pdf, label=r'$\mu$ = {}, $\beta$ = {}'.format(mu, beta)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu + \beta\gamma`, where :math:`\gamma` is the Euler-Mascheroni constant Variance :math:`\frac{\pi^2}{6} \beta^2` ======== ========================================== Parameters ---------- mu : tensor_like of float Location parameter. beta : tensor_like of float Scale parameter (beta > 0). """ rv_op = gumbel
[docs] @classmethod def dist(cls, mu, beta, **kwargs): mu = pt.as_tensor_variable(mu) beta = pt.as_tensor_variable(beta) return super().dist([mu, beta], **kwargs)
def support_point(rv, size, mu, beta): mean = mu + beta * np.euler_gamma if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, mu, beta): z = (value - mu) / beta res = -z - pt.exp(-z) - pt.log(beta) return check_parameters( res, beta > 0, msg="beta > 0", ) def logcdf(value, mu, beta): res = -pt.exp(-(value - mu) / beta) return check_parameters( res, beta > 0, msg="beta > 0", ) def icdf(value, mu, beta): res = mu - beta * pt.log(-pt.log(value)) res = check_icdf_value(res, value) return check_icdf_parameters( res, beta > 0, msg="beta > 0", )
class RiceRV(RandomVariable): name = "rice" ndim_supp = 0 ndims_params = [0, 0] dtype = "floatX" _print_name = ("Rice", "\\operatorname{Rice}") @classmethod def rng_fn(cls, rng, b, sigma, size=None) -> np.ndarray: return np.asarray(stats.rice.rvs(b=b, scale=sigma, size=size, random_state=rng)) rice = RiceRV()
[docs] class Rice(PositiveContinuous): r""" Rice distribution. .. math:: f(x\mid \nu ,\sigma )= {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right), .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0, 8, 500) nus = [0., 0., 4., 4.] sigmas = [1., 2., 1., 2.] for nu, sigma in zip(nus, sigmas): pdf = st.rice.pdf(x, nu / sigma, scale=sigma) plt.plot(x, pdf, label=r'$\nu$ = {}, $\sigma$ = {}'.format(nu, sigma)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================================================== Support :math:`x \in (0, \infty)` Mean :math:`\sigma {\sqrt {\pi /2}}\,\,L_{{1/2}}(-\nu ^{2}/2\sigma ^{2})` Variance :math:`2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{{1/2}}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)` ======== ============================================================== Parameters ---------- nu : tensor_like of float, optional Noncentrality parameter (only required if b is not specified). sigma : tensor_like of float, default 1 scale parameter. b : tensor_like of float, optional Shape parameter (alternative to nu, only required if nu is not specified). Notes ----- The distribution :math:`\mathrm{Rice}\left(|\nu|,\sigma\right)` is the distribution of :math:`R=\sqrt{X^2+Y^2}` where :math:`X\sim N(\nu \cos{\theta}, \sigma^2)`, :math:`Y\sim N(\nu \sin{\theta}, \sigma^2)` are independent and for any real :math:`\theta`. The distribution is defined with either nu or b. The link between the two parametrizations is given by .. math:: b = \dfrac{\nu}{\sigma} """ rv_op = rice
[docs] @classmethod def dist(cls, nu=None, sigma=None, b=None, *args, **kwargs): nu, b, sigma = cls.get_nu_b(nu, b, sigma) b = pt.as_tensor_variable(b) sigma = pt.as_tensor_variable(sigma) return super().dist([b, sigma], *args, **kwargs)
@classmethod def get_nu_b(cls, nu, b, sigma): if sigma is None: sigma = 1.0 if nu is None and b is not None: nu = b * sigma return nu, b, sigma elif nu is not None and b is None: b = nu / sigma return nu, b, sigma raise ValueError("Rice distribution must specify either nu" " or b.") def support_point(rv, size, nu, sigma): nu_sigma_ratio = -(nu**2) / (2 * sigma**2) mean = ( sigma * np.sqrt(np.pi / 2) * pt.exp(nu_sigma_ratio / 2) * ( (1 - nu_sigma_ratio) * pt.i0(-nu_sigma_ratio / 2) - nu_sigma_ratio * pt.i1(-nu_sigma_ratio / 2) ) ) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, b, sigma): x = value / sigma res = pt.switch( pt.le(value, 0), -np.inf, pt.log(x * pt.exp((-(x - b) * (x - b)) / 2) * i0e(x * b) / sigma), ) return check_parameters( res, sigma >= 0, b >= 0, msg="sigma >= 0, b >= 0", )
[docs] class Logistic(Continuous): r""" Logistic log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, s) = \frac{\exp\left(-\frac{x - \mu}{s}\right)}{s \left(1 + \exp\left(-\frac{x - \mu}{s}\right)\right)^2} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-5, 5, 200) mus = [0., 0., 0., -2.] ss = [.4, 1., 2., .4] for mu, s in zip(mus, ss): pdf = st.logistic.pdf(x, loc=mu, scale=s) plt.plot(x, pdf, label=r'$\mu$ = {}, $s$ = {}'.format(mu, s)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in \mathbb{R}` Mean :math:`\mu` Variance :math:`\frac{s^2 \pi^2}{3}` ======== ========================================== Parameters ---------- mu : tensor_like of float, default 0 Mean. s : tensor_like of float, default 1 Scale (s > 0). """ rv_op = logistic
[docs] @classmethod def dist(cls, mu=0.0, s=1.0, *args, **kwargs): mu = pt.as_tensor_variable(mu) s = pt.as_tensor_variable(s) return super().dist([mu, s], *args, **kwargs)
def support_point(rv, size, mu, s): mu, _ = pt.broadcast_arrays(mu, s) if not rv_size_is_none(size): mu = pt.full(size, mu) return mu def logp(value, mu, s): z = (value - mu) / s res = -z - pt.log(s) - 2.0 * pt.log1p(pt.exp(-z)) return check_parameters( res, s > 0, msg="s > 0", ) def logcdf(value, mu, s): res = -pt.log1pexp(-(value - mu) / s) return check_parameters( res, s > 0, msg="s > 0", ) def icdf(value, mu, s): res = mu + s * pt.log(value / (1 - value)) res = check_icdf_value(res, value) return check_icdf_parameters( res, s > 0, msg="s > 0", )
class LogitNormalRV(SymbolicRandomVariable): name = "logit_normal" signature = "[rng],[size],(),()->[rng],()" _print_name = ("logitNormal", "\\operatorname{logitNormal}") @classmethod def rv_op(cls, mu, sigma, *, size=None, rng=None): mu = pt.as_tensor(mu) sigma = pt.as_tensor(sigma) rng = normalize_rng_param(rng) size = normalize_size_param(size) next_rng, normal_draws = normal(loc=mu, scale=sigma, size=size, rng=rng).owner.outputs draws = pt.expit(normal_draws) return cls( inputs=[rng, size, mu, sigma], outputs=[next_rng, draws], )(rng, size, mu, sigma)
[docs] class LogitNormal(UnitContinuous): r""" Logit-Normal log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu, \tau) = \frac{1}{x(1-x)} \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (logit(x)-\mu)^2 \right\} .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st from scipy.special import logit import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0.0001, 0.9999, 500) mus = [0., 0., 0., 1.] sigmas = [0.3, 1., 2., 1.] for mu, sigma in zip(mus, sigmas): pdf = st.norm.pdf(logit(x), loc=mu, scale=sigma) * 1/(x * (1-x)) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}'.format(mu, sigma)) plt.legend(loc=1) plt.show() ======== ========================================== Support :math:`x \in (0, 1)` Mean no analytical solution Variance no analytical solution ======== ========================================== Parameters ---------- mu : tensor_like of float, default 0 Location parameter. sigma : tensor_like of float, optional Scale parameter (sigma > 0). Defaults to 1. tau : tensor_like of float, optional Scale parameter (tau > 0). Defaults to 1. """ rv_type = LogitNormalRV rv_op = LogitNormalRV.rv_op
[docs] @classmethod def dist(cls, mu=0, sigma=None, tau=None, **kwargs): _, sigma = get_tau_sigma(tau=tau, sigma=sigma) return super().dist([mu, sigma], **kwargs)
def support_point(rv, size, mu, sigma): median, _ = pt.broadcast_arrays(invlogit(mu), sigma) if not rv_size_is_none(size): median = pt.full(size, median) return median def logp(value, mu, sigma): tau, _ = get_tau_sigma(sigma=sigma) res = pt.switch( pt.or_(pt.le(value, 0), pt.ge(value, 1)), -np.inf, ( -0.5 * tau * (logit(value) - mu) ** 2 + 0.5 * pt.log(tau / (2.0 * np.pi)) - pt.log(value * (1 - value)) ), ) return check_parameters( res, tau > 0, msg="tau > 0", )
def _interpolated_argcdf(p, pdf, cdf, x): index = np.searchsorted(cdf, p) - 1 slope = (pdf[index + 1] - pdf[index]) / (x[index + 1] - x[index]) # First term (constant) of the Taylor expansion around slope = 0 small_slopes = np.where( np.abs(pdf[index]) <= 1e-8, np.zeros(index.shape), (p - cdf[index]) / pdf[index] ) # This warning happens when we divide by slope = 0: we can ignore it # because the other result will be returned with warnings.catch_warnings(): warnings.filterwarnings("ignore", ".*invalid value encountered in.*", RuntimeWarning) large_slopes = ( -pdf[index] + np.sqrt(pdf[index] ** 2 + 2 * slope * (p - cdf[index])) ) / slope return x[index] + np.where(np.abs(slope) <= 1e-8, small_slopes, large_slopes) class InterpolatedRV(RandomVariable): name = "interpolated" ndim_supp = 0 ndims_params = [1, 1, 1] dtype = "floatX" _print_name = ("Interpolated", "\\operatorname{Interpolated}") @classmethod def rng_fn(cls, rng, x, pdf, cdf, size=None) -> np.ndarray: p = rng.uniform(size=size) return np.asarray(_interpolated_argcdf(p, pdf, cdf, x)) interpolated = InterpolatedRV()
[docs] class Interpolated(BoundedContinuous): r""" Univariate probability distribution defined as a linear interpolation of probability density function evaluated on some lattice of points. The lattice can be uneven, so the steps between different points can have different size and it is possible to vary the precision between regions of the support. The probability density function values don not have to be normalized, as the interpolated density is any way normalized to make the total probability equal to $1$. Both parameters ``x_points`` and values ``pdf_points`` are not variables, but plain array-like objects, so they are constant and cannot be sampled. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import pymc as pm import arviz as az from scipy.stats import gamma plt.style.use('arviz-darkgrid') rv = gamma(1.99) x = np.linspace(rv.ppf(0.01), rv.ppf(0.99), 1000) points = np.linspace(x[0], x[-1], 50) pdf = rv.pdf(points) interpolated = pm.Interpolated.dist(points, pdf) fig, ax = plt.subplots(1, 1) ax.plot(x, rv.pdf(x), 'C0', linestyle = '--', label='Original Gamma pdf', alpha=0.8, lw=2) ax.plot(points, pdf, color='black', marker='o', label='Lattice Points', alpha=0.5, linestyle='') ax.plot(x, np.exp(pm.logp(interpolated, x).eval()), 'C1', label='Interpolated pdf', alpha=0.8, lw=3) r = pm.draw(interpolated, draws=1000) ax.hist(r, density=True, alpha=0.4, align ='mid', color='grey') ax.legend(loc='best', frameon=False) plt.show() ======== =========================================== Support :math:`x \in [x\_points[0], x\_points[-1]]` ======== =========================================== Parameters ---------- x_points : array_like A monotonically growing list of values. Must be non-symbolic. pdf_points : array_like Probability density function evaluated on lattice ``x_points``. Must be non-symbolic. """ rv_op = interpolated
[docs] @classmethod def dist(cls, x_points, pdf_points, *args, **kwargs): interp = InterpolatedUnivariateSpline(x_points, pdf_points, k=1, ext="zeros") Z = interp.integral(x_points[0], x_points[-1]) cdf_points = interp.antiderivative()(x_points) / Z pdf_points = pdf_points / Z x_points = pt.constant(x_points) pdf_points = pt.constant(pdf_points) cdf_points = pt.constant(cdf_points) # lower = pt.as_tensor_variable(x_points[0]) # upper = pt.as_tensor_variable(x_points[-1]) # median = _interpolated_argcdf(0.5, pdf_points, cdf_points, x_points) return super().dist([x_points, pdf_points, cdf_points], **kwargs)
def support_point(rv, size, x_points, pdf_points, cdf_points): """ Estimates the expectation integral using the trapezoid rule; cdf_points are not used. """ x_fx = pt.mul(x_points, pdf_points) # x_i * f(x_i) for all xi's in x_points support_point = pt.sum(pt.mul(pt.diff(x_points), x_fx[1:] + x_fx[:-1])) / 2 if not rv_size_is_none(size): support_point = pt.full(size, support_point) return support_point def logp(value, x_points, pdf_points, cdf_points): # x_points and pdf_points are expected to be non-symbolic arrays wrapped # within a tensor.constant. We use the .data method to retrieve them interp = InterpolatedUnivariateSpline(x_points.data, pdf_points.data, k=1, ext="zeros") Z = interp.integral(x_points.data[0], x_points.data[-1]) # interp and Z are converted to symbolic variables here interp_op = SplineWrapper(interp) Z = pt.constant(Z) return pt.log(interp_op(value) / Z)
@_default_transform.register(Interpolated) def interpolated_default_transform(op, rv): def transform_params(*params): _, _, _, x_points, _, _ = params return x_points[0], x_points[-1] return transforms.Interval(bounds_fn=transform_params) class MoyalRV(RandomVariable): name = "moyal" ndim_supp = 0 ndims_params = [0, 0] dtype = "floatX" _print_name = ("Moyal", "\\operatorname{Moyal}") @classmethod def rng_fn(cls, rng, mu, sigma, size=None) -> np.ndarray: return np.asarray(stats.moyal.rvs(mu, sigma, size=size, random_state=rng)) moyal = MoyalRV()
[docs] class Moyal(Continuous): r""" Moyal log-likelihood. The pdf of this distribution is .. math:: f(x \mid \mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(z + e^{-z}\right)}, where .. math:: z = \frac{x-\mu}{\sigma}. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np import scipy.stats as st import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(-10, 20, 200) mus = [-1., 0., 4.] sigmas = [2., 2., 4.] for mu, sigma in zip(mus, sigmas): pdf = st.moyal.pdf(x, loc=mu, scale=sigma) plt.plot(x, pdf, label=r'$\mu$ = {}, $\sigma$ = {}'.format(mu, sigma)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================================================== Support :math:`x \in (-\infty, \infty)` Mean :math:`\mu + \sigma\left(\gamma + \log 2\right)`, where :math:`\gamma` is the Euler-Mascheroni constant Variance :math:`\frac{\pi^{2}}{2}\sigma^{2}` ======== ============================================================== Parameters ---------- mu : tensor_like of float, default 0 Location parameter. sigma : tensor_like of float, default 1 Scale parameter (sigma > 0). """ rv_op = moyal
[docs] @classmethod def dist(cls, mu=0, sigma=1.0, *args, **kwargs): mu = pt.as_tensor_variable(mu) sigma = pt.as_tensor_variable(sigma) return super().dist([mu, sigma], *args, **kwargs)
def support_point(rv, size, mu, sigma): mean = mu + sigma * (np.euler_gamma + pt.log(2)) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, mu, sigma): scaled = (value - mu) / sigma res = -(1 / 2) * (scaled + pt.exp(-scaled)) - pt.log(sigma) - (1 / 2) * pt.log(2 * np.pi) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def logcdf(value, mu, sigma): scaled = (value - mu) / sigma res = pt.log(pt.erfc(pt.exp(-scaled / 2) * (2**-0.5))) return check_parameters( res, sigma > 0, msg="sigma > 0", ) def icdf(value, mu, sigma): res = sigma * -pt.log(2.0 * pt.erfcinv(value) ** 2) + mu res = check_icdf_value(res, value) return check_icdf_parameters( res, sigma > 0, msg="sigma > 0", )
class PolyaGammaRV(RandomVariable): """Polya-Gamma random variable.""" name = "polyagamma" ndim_supp = 0 ndims_params = [0, 0] dtype = "floatX" _print_name = ("PG", "\\operatorname{PG}") def __call__(self, h=1.0, z=0.0, size=None, **kwargs): return super().__call__(h, z, size=size, **kwargs) @classmethod def rng_fn(cls, rng, h, z, size=None) -> np.ndarray: """ Generate a random sample from the distribution with the given parameters Parameters ---------- rng : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator} A seed to initialize the random number generator. If None, then fresh, unpredictable entropy will be pulled from the OS. If an ``int`` or ``array_like[ints]`` is passed, then it will be passed to `SeedSequence` to derive the initial `BitGenerator` state. One may also pass in a `SeedSequence` instance. Additionally, when passed a `BitGenerator`, it will be wrapped by `Generator`. If passed a `Generator`, it will be returned unaltered. h : scalar or sequence The shape parameter of the distribution. z : scalar or sequence The exponential tilting parameter. size : int or tuple of ints, optional The number of elements to draw from the distribution. If size is ``None`` (default) then a single value is returned. If a tuple of integers is passed, the returned array will have the same shape. If the element(s) of size is not an integer type, it will be truncated to the largest integer smaller than its value (e.g (2.1, 1) -> (2, 1)). This parameter only applies if `h` and `z` are scalars. """ # handle the kind of rng passed to the sampler bg = rng._bit_generator if isinstance(rng, np.random.RandomState) else rng return np.asarray( random_polyagamma(h, z, size=size, random_state=bg).astype(pytensor.config.floatX) ) polyagamma = PolyaGammaRV() class _PolyaGammaLogDistFunc(Op): __props__ = ("get_pdf",) def __init__(self, get_pdf=False): self.get_pdf = get_pdf def make_node(self, x, h, z): x = pt.as_tensor_variable(x) h = pt.as_tensor_variable(h) z = pt.as_tensor_variable(z) bshape = broadcast_shape(x, h, z) shape = [None] * len(bshape) return Apply(self, [x, h, z], [pt.TensorType(pytensor.config.floatX, shape)()]) def perform(self, node, ins, outs): x, h, z = ins[0], ins[1], ins[2] outs[0][0] = ( polyagamma_pdf(x, h, z, return_log=True) if self.get_pdf else polyagamma_cdf(x, h, z, return_log=True) ).astype(pytensor.config.floatX)
[docs] class PolyaGamma(PositiveContinuous): r""" The Polya-Gamma distribution. The distribution is parametrized by ``h`` (shape parameter) and ``z`` (exponential tilting parameter). The pdf of this distribution is .. math:: f(x \mid h, z) = cosh^h(\frac{z}{2})e^{-\frac{1}{2}xz^2}f(x \mid h, 0), where :math:`f(x \mid h, 0)` is the pdf of a :math:`PG(h, 0)` variable. Notice that the pdf of this distribution is expressed as an alternating-sign sum of inverse-Gaussian densities. .. math:: X = \Sigma_{k=1}^{\infty}\frac{Ga(h, 1)}{d_k}, where :math:`d_k = 2(k - 0.5)^2\pi^2 + z^2/2`, :math:`Ga(h, 1)` is a gamma random variable with shape parameter ``h`` and scale parameter ``1``. .. plot:: :context: close-figs import matplotlib.pyplot as plt import numpy as np from polyagamma import polyagamma_pdf import arviz as az plt.style.use('arviz-darkgrid') x = np.linspace(0.01, 5, 500);x.sort() hs = [1., 5., 10., 15.] zs = [0.] * 4 for h, z in zip(hs, zs): pdf = polyagamma_pdf(x, h=h, z=z) plt.plot(x, pdf, label=r'$h$ = {}, $z$ = {}'.format(h, z)) plt.xlabel('x', fontsize=12) plt.ylabel('f(x)', fontsize=12) plt.legend(loc=1) plt.show() ======== ============================= Support :math:`x \in (0, \infty)` Mean :math:`\dfrac{h}{4}` if :math:`z=0`, :math:`\dfrac{tanh(z/2)h}{2z}` otherwise. Variance :math:`0.041666688h` if :math:`z=0`, :math:`\dfrac{h(sinh(z) - z)(1 - tanh^2(z/2))}{4z^3}` otherwise. ======== ============================= Parameters ---------- h : tensor_like of float, default 1 The shape parameter of the distribution (h > 0). z : tensor_like of float, default 0 The exponential tilting parameter of the distribution. Examples -------- .. code-block:: python rng = np.random.default_rng() with pm.Model(): x = pm.PolyaGamma('x', h=1, z=5.5) with pm.Model(): x = pm.PolyaGamma('x', h=25, z=-2.3, rng=rng, size=(100, 5)) References ---------- .. [1] Polson, Nicholas G., James G. Scott, and Jesse Windle. "Bayesian inference for logistic models using Pólya–Gamma latent variables." Journal of the American statistical Association 108.504 (2013): 1339-1349. .. [2] Windle, Jesse, Nicholas G. Polson, and James G. Scott. "Sampling Polya-Gamma random variates: alternate and approximate techniques." arXiv preprint arXiv:1405.0506 (2014). .. [3] Luc Devroye. "On exact simulation algorithms for some distributions related to Jacobi theta functions." Statistics & Probability Letters, Volume 79, Issue 21, (2009): 2251-2259. .. [4] Windle, J. (2013). Forecasting high-dimensional, time-varying variance-covariance matrices with high-frequency data and sampling Pólya-Gamma random variates for posterior distributions derived from logistic likelihoods.(PhD thesis). Retrieved from http://hdl.handle.net/2152/21842 """ rv_op = polyagamma
[docs] @classmethod def dist(cls, h=1.0, z=0.0, **kwargs): h = pt.as_tensor_variable(h) z = pt.as_tensor_variable(z) msg = f"The variable {h} specified for PolyaGamma has non-positive " msg += "values, making it unsuitable for this parameter." Assert(msg)(h, pt.all(pt.gt(h, 0.0))) return super().dist([h, z], **kwargs)
def support_point(rv, size, h, z): mean = pt.switch(pt.eq(z, 0), h / 4, tanh(z / 2) * (h / (2 * z))) if not rv_size_is_none(size): mean = pt.full(size, mean) return mean def logp(value, h, z): res = pt.switch( pt.le(value, 0), -np.inf, _PolyaGammaLogDistFunc(get_pdf=True)(value, h, z), ) return check_parameters( res, h > 0, msg="h > 0", ) def logcdf(value, h, z): res = pt.switch( pt.le(value, 0), -np.inf, _PolyaGammaLogDistFunc(get_pdf=False)(value, h, z), ) return check_parameters( res, h > 0, msg="h > 0", )