test_distributions.py

import itertools
import sys

from .helpers import SeededTest, select_by_precision
from ..vartypes import continuous_types
from ..model import Model, Point, Deterministic
from ..blocking import DictToVarBijection
from ..distributions import (
    DensityDist, Categorical, Multinomial, VonMises, Dirichlet,
    MvStudentT, MvNormal, MatrixNormal, ZeroInflatedPoisson,
    ZeroInflatedNegativeBinomial, Constant, Poisson, Bernoulli, Beta,
    BetaBinomial, HalfStudentT, StudentT, Weibull, Pareto,
    InverseGamma, Gamma, Cauchy, HalfCauchy, Lognormal, Laplace,
    NegativeBinomial, Geometric, Exponential, ExGaussian, Normal, TruncatedNormal,
    Flat, LKJCorr, Wald, ChiSquared, HalfNormal, DiscreteUniform,
    Bound, Uniform, Triangular, Binomial, SkewNormal, DiscreteWeibull,
    Gumbel, Logistic, OrderedLogistic, LogitNormal, Interpolated,
    ZeroInflatedBinomial, HalfFlat, AR1, KroneckerNormal, Rice,
    Kumaraswamy
)

from ..distributions import continuous
from pymc3.theanof import floatX
from numpy import array, inf, log, exp
from numpy.testing import assert_almost_equal, assert_allclose, assert_equal
import numpy.random as nr
import numpy as np
import pytest

from scipy import integrate
import scipy.stats.distributions as sp
import scipy.stats
from scipy.special import logit
import theano
import theano.tensor as tt
from ..math import kronecker

def get_lkj_cases():
    """
    Log probabilities calculated using the formulas in:
    http://www.sciencedirect.com/science/article/pii/S0047259X09000876
    """
    tri = np.array([0.7, 0.0, -0.7])
    return [
        (tri, 1, 3, 1.5963125911388549),
        (tri, 3, 3, -7.7963493376312742),
        (tri, 0, 3, -np.inf),
        (np.array([1.1, 0.0, -0.7]), 1, 3, -np.inf),
        (np.array([0.7, 0.0, -1.1]), 1, 3, -np.inf)
    ]


LKJ_CASES = get_lkj_cases()


class Domain:
    def __init__(self, vals, dtype=None, edges=None, shape=None):
        avals = array(vals, dtype=dtype)
        if dtype is None and not str(avals.dtype).startswith('int'):
            avals = avals.astype(theano.config.floatX)
        vals = [array(v, dtype=avals.dtype) for v in vals]

        if edges is None:
            edges = array(vals[0]), array(vals[-1])
            vals = vals[1:-1]
        if shape is None:
            shape = avals[0].shape

        self.vals = vals
        self.shape = shape

        self.lower, self.upper = edges
        self.dtype = avals.dtype

    def __add__(self, other):
        return Domain(
            [v + other for v in self.vals],
            self.dtype,
            (self.lower + other, self.upper + other),
            self.shape)

    def __mul__(self, other):
        try:
            return Domain(
                [v * other for v in self.vals],
                self.dtype,
                (self.lower * other, self.upper * other),
                self.shape)
        except TypeError:
            return Domain(
                [v * other for v in self.vals],
                self.dtype,
                (self.lower, self.upper),
                self.shape)

    def __neg__(self):
        return Domain(
            [-v for v in self.vals],
            self.dtype,
            (-self.lower, -self.upper),
            self.shape)


def product(domains, n_samples=-1):
    """Get an iterator over a product of domains.

    Args:
        domains: a dictionary of (name, object) pairs, where the objects
                 must be "domain-like", as in, have a `.vals` property
        n_samples: int, maximum samples to return.  -1 to return whole product

    Returns:
        list of the cartesian product of the domains
    """
    try:
        names, domains = zip(*domains.items())
    except ValueError:  # domains.items() is empty
        return []
    all_vals = [zip(names, val) for val in itertools.product(*[d.vals for d in domains])]
    if n_samples > 0 and len(all_vals) > n_samples:
            return (all_vals[j] for j in nr.choice(len(all_vals), n_samples, replace=False))
    return all_vals


R = Domain([-inf, -2.1, -1, -.01, .0, .01, 1, 2.1, inf])
Rplus = Domain([0, .01, .1, .9, .99, 1, 1.5, 2, 100, inf])
Rplusbig = Domain([0, .5, .9, .99, 1, 1.5, 2, 20, inf])
Rminusbig = Domain([-inf, -2, -1.5, -1, -.99, -.9, -.5, -0.01, 0])
Unit = Domain([0, .001, .1, .5, .75, .99, 1])

Circ = Domain([-np.pi, -2.1, -1, -.01, .0, .01, 1, 2.1, np.pi])

Runif = Domain([-1, -.4, 0, .4, 1])
Rdunif = Domain([-10, 0, 10.])
Rplusunif = Domain([0, .5, inf])
Rplusdunif = Domain([2, 10, 100], 'int64')

I = Domain([-1000, -3, -2, -1, 0, 1, 2, 3, 1000], 'int64')

NatSmall = Domain([0, 3, 4, 5, 1000], 'int64')
Nat = Domain([0, 1, 2, 3, 2000], 'int64')
NatBig = Domain([0, 1, 2, 3, 5000, 50000], 'int64')
PosNat = Domain([1, 2, 3, 2000], 'int64')

Bool = Domain([0, 0, 1, 1], 'int64')


def build_model(distfam, valuedomain, vardomains, extra_args=None):
    if extra_args is None:
        extra_args = {}
    with Model() as m:
        vals = {}
        for v, dom in vardomains.items():
            vals[v] = Flat(v, dtype=dom.dtype, shape=dom.shape,
                           testval=dom.vals[0])
        vals.update(extra_args)
        distfam('value', shape=valuedomain.shape, transform=None, **vals)
    return m


def integrate_nd(f, domain, shape, dtype):
    if shape == () or shape == (1,):
        if dtype in continuous_types:
            return integrate.quad(f, domain.lower, domain.upper, epsabs=1e-8)[0]
        else:
            return sum(f(j) for j in range(domain.lower, domain.upper + 1))
    elif shape == (2,):
        def f2(a, b):
            return f([a, b])

        return integrate.dblquad(f2, domain.lower[0], domain.upper[0],
                                 lambda _: domain.lower[1],
                                 lambda _: domain.upper[1])[0]
    elif shape == (3,):
        def f3(a, b, c):
            return f([a, b, c])

        return integrate.tplquad(f3, domain.lower[0], domain.upper[0],
                                 lambda _: domain.lower[1],
                                 lambda _: domain.upper[1],
                                 lambda _, __: domain.lower[2],
                                 lambda _, __: domain.upper[2])[0]
    else:
        raise ValueError("Dont know how to integrate shape: " + str(shape))


def multinomial_logpdf(value, n, p):
    if value.sum() == n and (0 <= value).all() and (value <= n).all():
        logpdf = scipy.special.gammaln(n + 1)
        logpdf -= scipy.special.gammaln(value + 1).sum()
        logpdf += logpow(p, value).sum()
        return logpdf
    else:
        return -inf


def beta_mu_sigma(value, mu, sigma):
    kappa = mu * (1 - mu) / sigma**2 - 1
    if kappa > 0:
        return sp.beta.logpdf(value, mu * kappa, (1 - mu) * kappa)
    else:
        return -inf


class ProductDomain:
    def __init__(self, domains):
        self.vals = list(itertools.product(*[d.vals for d in domains]))
        self.shape = (len(domains),) + domains[0].shape
        self.lower = [d.lower for d in domains]
        self.upper = [d.upper for d in domains]
        self.dtype = domains[0].dtype


def Vector(D, n):
    return ProductDomain([D] * n)


def SortedVector(n):
    vals = []
    np.random.seed(42)
    for _ in range(10):
        vals.append(np.sort(np.random.randn(n)))
    return Domain(vals, edges=(None, None))


def UnitSortedVector(n):
    vals = []
    np.random.seed(42)
    for _ in range(10):
        vals.append(np.sort(np.random.rand(n)))
    return Domain(vals, edges=(None, None))


def RealMatrix(n, m):
    vals = []
    np.random.seed(42)
    for _ in range(10):
        vals.append(np.random.randn(n, m))
    return Domain(vals, edges=(None, None))


def simplex_values(n):
    if n == 1:
        yield array([1.0])
    else:
        for v in Unit.vals:
            for vals in simplex_values(n - 1):
                yield np.concatenate([[v], (1 - v) * vals])


def normal_logpdf_tau(value, mu, tau):
    return normal_logpdf_cov(value, mu, np.linalg.inv(tau)).sum()


def normal_logpdf_cov(value, mu, cov):
    return scipy.stats.multivariate_normal.logpdf(value, mu, cov).sum()


def normal_logpdf_chol(value, mu, chol):
    return normal_logpdf_cov(value, mu, np.dot(chol, chol.T)).sum()


def normal_logpdf_chol_upper(value, mu, chol):
    return normal_logpdf_cov(value, mu, np.dot(chol.T, chol)).sum()


def matrix_normal_logpdf_cov(value, mu, rowcov, colcov):
    return scipy.stats.matrix_normal.logpdf(value, mu, rowcov, colcov)


def matrix_normal_logpdf_chol(value, mu, rowchol, colchol):
    return matrix_normal_logpdf_cov(value, mu, np.dot(rowchol, rowchol.T),
                                    np.dot(colchol, colchol.T))


def kron_normal_logpdf_cov(value, mu, covs, sigma):
    cov = kronecker(*covs).eval()
    if sigma is not None:
        cov += sigma**2 * np.eye(*cov.shape)
    return scipy.stats.multivariate_normal.logpdf(value, mu, cov).sum()


def kron_normal_logpdf_chol(value, mu, chols, sigma):
    covs = [np.dot(chol, chol.T) for chol in chols]
    return kron_normal_logpdf_cov(value, mu, covs, sigma=sigma)


def kron_normal_logpdf_evd(value, mu, evds, sigma):
    covs = []
    for eigs, Q in evds:
        try:
            eigs = eigs.eval()
        except AttributeError:
            pass
        try:
            Q = Q.eval()
        except AttributeError:
            pass
        covs.append(np.dot(Q, np.dot(np.diag(eigs), Q.T)))
    return kron_normal_logpdf_cov(value, mu, covs, sigma)


def betafn(a):
    return floatX(scipy.special.gammaln(a).sum(-1) - scipy.special.gammaln(a.sum(-1)))


def logpow(v, p):
    return np.choose(v == 0, [p * np.log(v), 0])


def discrete_weibull_logpmf(value, q, beta):
    return floatX(np.log(np.power(floatX(q),
                                  np.power(floatX(value), floatX(beta)))
                  - np.power(floatX(q), np.power(floatX(value + 1),
                                                 floatX(beta)))))


def dirichlet_logpdf(value, a):
    return floatX((-betafn(a) + logpow(value, a - 1).sum(-1)).sum())


def categorical_logpdf(value, p):
    if value >= 0 and value <= len(p):
        return floatX(np.log(np.moveaxis(p, -1, 0)[value]))
    else:
        return -inf

def mvt_logpdf(value, nu, Sigma, mu=0):
    d = len(Sigma)
    dist = np.atleast_2d(value) - mu
    chol = np.linalg.cholesky(Sigma)
    trafo = np.linalg.solve(chol, dist.T).T
    logdet = np.log(np.diag(chol)).sum()

    lgamma = scipy.special.gammaln
    norm = lgamma((nu + d) / 2.)  - 0.5 * d * np.log(nu * np.pi) - lgamma(nu / 2.)
    logp = norm - logdet - (nu + d) / 2. * np.log1p((trafo * trafo).sum(-1) / nu)
    return logp.sum()

def AR1_logpdf(value, k, tau_e):
    return (sp.norm(loc=0, scale=1/np.sqrt(tau_e)).logpdf(value[0]) +
            sp.norm(loc=k*value[:-1], scale=1/np.sqrt(tau_e)).logpdf(value[1:]).sum())

def invlogit(x, eps=sys.float_info.epsilon):
    return (1. - 2. * eps) / (1. + np.exp(-x)) + eps

def orderedlogistic_logpdf(value, eta, cutpoints):
    c = np.concatenate(([-np.inf], cutpoints, [np.inf]))
    ps = np.array([invlogit(eta - cc) - invlogit(eta - cc1)
                   for cc, cc1 in zip(c[:-1], c[1:])])
    p = ps[value]
    return np.where(np.all(ps >= 0), np.log(p), -np.inf)

class Simplex:
    def __init__(self, n):
        self.vals = list(simplex_values(n))
        self.shape = (n,)
        self.dtype = Unit.dtype


class MultiSimplex:
    def __init__(self, n_dependent, n_independent):
        self.vals = []
        for simplex_value in itertools.product(simplex_values(n_dependent), repeat=n_independent):
            self.vals.append(np.vstack(simplex_value))
        self.shape = (n_independent, n_dependent)
        self.dtype = Unit.dtype


def PdMatrix(n):
    if n == 1:
        return PdMatrix1
    elif n == 2:
        return PdMatrix2
    elif n == 3:
        return PdMatrix3
    else:
        raise ValueError("n out of bounds")

PdMatrix1 = Domain([np.eye(1), [[.5]]], edges=(None, None))

PdMatrix2 = Domain([np.eye(2), [[.5, .05], [.05, 4.5]]], edges=(None, None))

PdMatrix3 = Domain(
    [np.eye(3), [[.5, .1, 0], [.1, 1, 0], [0, 0, 2.5]]], edges=(None, None))


PdMatrixChol1 = Domain([np.eye(1), [[0.001]]], edges=(None, None))
PdMatrixChol2 = Domain([np.eye(2), [[0.1, 0], [10, 1]]], edges=(None, None))
PdMatrixChol3 = Domain([np.eye(3), [[0.1, 0, 0], [10, 100, 0], [0, 1, 10]]],
                       edges=(None, None))


def PdMatrixChol(n):
    if n == 1:
        return PdMatrixChol1
    elif n == 2:
        return PdMatrixChol2
    elif n == 3:
        return PdMatrixChol3
    else:
        raise ValueError("n out of bounds")


PdMatrixCholUpper1 = Domain([np.eye(1), [[0.001]]], edges=(None, None))
PdMatrixCholUpper2 = Domain([np.eye(2), [[0.1, 10], [0, 1]]], edges=(None, None))
PdMatrixCholUpper3 = Domain([np.eye(3), [[0.1, 10, 0], [0, 100, 1], [0, 0, 10]]],
                            edges=(None, None))


def PdMatrixCholUpper(n):
    if n == 1:
        return PdMatrixCholUpper1
    elif n == 2:
        return PdMatrixCholUpper2
    elif n == 3:
        return PdMatrixCholUpper3
    else:
        raise ValueError("n out of bounds")


def RandomPdMatrix(n):
    A = np.random.rand(n, n)
    return np.dot(A, A.T) + n * np.identity(n)


class TestMatchesScipy(SeededTest):
    def pymc3_matches_scipy(self, pymc3_dist, domain, paramdomains, scipy_dist,
                            decimal=None, extra_args=None, scipy_args=None):
        if extra_args is None:
            extra_args = {}
        if scipy_args is None:
            scipy_args = {}
        model = build_model(pymc3_dist, domain, paramdomains, extra_args)
        value = model.named_vars['value']

        def logp(args):
            args.update(scipy_args)
            return scipy_dist(**args)
        self.check_logp(model, value, domain, paramdomains, logp, decimal=decimal)

    def check_logp(self, model, value, domain, paramdomains, logp_reference, decimal=None):
        domains = paramdomains.copy()
        domains['value'] = domain
        logp = model.fastlogp
        for pt in product(domains, n_samples=100):
            pt = Point(pt, model=model)
            if decimal is None:
                decimal = select_by_precision(float64=6, float32=3)
            assert_almost_equal(logp(pt), logp_reference(pt), decimal=decimal, err_msg=str(pt))

    def check_logcdf(self, pymc3_dist, domain, paramdomains, scipy_logcdf, decimal=None):
        domains = paramdomains.copy()
        domains['value'] = domain
        if decimal is None:
            decimal = select_by_precision(float64=6, float32=3)
        for pt in product(domains, n_samples=100):
            params = dict(pt)
            scipy_cdf = scipy_logcdf(**params)
            value = params.pop('value')
            dist = pymc3_dist.dist(**params)
            assert_almost_equal(dist.logcdf(value).tag.test_value, scipy_cdf,
                                decimal=decimal, err_msg=str(pt))

    def check_int_to_1(self, model, value, domain, paramdomains):
        pdf = model.fastfn(exp(model.logpt))
        for pt in product(paramdomains, n_samples=10):
            pt = Point(pt, value=value.tag.test_value, model=model)
            bij = DictToVarBijection(value, (), pt)
            pdfx = bij.mapf(pdf)
            area = integrate_nd(pdfx, domain, value.dshape, value.dtype)
            assert_almost_equal(area, 1, err_msg=str(pt))

    def checkd(self, distfam, valuedomain, vardomains, checks=None, extra_args=None):
        if checks is None:
            checks = (self.check_int_to_1, )

        if extra_args is None:
            extra_args = {}
        m = build_model(distfam, valuedomain, vardomains, extra_args=extra_args)
        for check in checks:
            check(m, m.named_vars['value'], valuedomain, vardomains)

    def test_uniform(self):
        self.pymc3_matches_scipy(
            Uniform, Runif, {'lower': -Rplusunif, 'upper': Rplusunif},
            lambda value, lower, upper: sp.uniform.logpdf(value, lower, upper - lower))
        self.check_logcdf(Uniform, Runif, {'lower': -Rplusunif, 'upper': Rplusunif},
                          lambda value, lower, upper: sp.uniform.logcdf(value, lower, upper - lower))

    def test_triangular(self):
        self.pymc3_matches_scipy(
            Triangular, Runif, {'lower': -Rplusunif, 'c': Runif, 'upper': Rplusunif},
            lambda value, c, lower, upper: sp.triang.logpdf(value, c-lower, lower, upper-lower))
        self.check_logcdf(Triangular, Runif, {'lower': -Rplusunif, 'c': Runif, 'upper': Rplusunif},
                          lambda value, c, lower, upper: sp.triang.logcdf(value, c-lower, lower, upper-lower))

    def test_bound_normal(self):
        PositiveNormal = Bound(Normal, lower=0.)
        self.pymc3_matches_scipy(PositiveNormal, Rplus, {'mu': Rplus, 'sigma': Rplus},
                                 lambda value, mu, sigma: sp.norm.logpdf(value, mu, sigma),
                                 decimal=select_by_precision(float64=6, float32=-1))
        with Model(): x = PositiveNormal('x', mu=0, sigma=1, transform=None)
        assert np.isinf(x.logp({'x':-1}))

    def test_discrete_unif(self):
        self.pymc3_matches_scipy(
            DiscreteUniform, Rdunif, {'lower': -Rplusdunif, 'upper': Rplusdunif},
            lambda value, lower, upper: sp.randint.logpmf(value, lower, upper + 1))

    def test_flat(self):
        self.pymc3_matches_scipy(Flat, Runif, {}, lambda value: 0)
        with Model():
            x = Flat('a')
            assert_allclose(x.tag.test_value, 0)
        self.check_logcdf(Flat, Runif, {}, lambda value: np.log(0.5))
        # Check infinite cases individually.
        assert 0. == Flat.dist().logcdf(np.inf).tag.test_value
        assert -np.inf == Flat.dist().logcdf(-np.inf).tag.test_value

    def test_half_flat(self):
        self.pymc3_matches_scipy(HalfFlat, Rplus, {}, lambda value: 0)
        with Model():
            x = HalfFlat('a', shape=2)
            assert_allclose(x.tag.test_value, 1)
            assert x.tag.test_value.shape == (2,)
        self.check_logcdf(HalfFlat, Runif, {}, lambda value: -np.inf)
        # Check infinite cases individually.
        assert 0. == HalfFlat.dist().logcdf(np.inf).tag.test_value
        assert -np.inf == HalfFlat.dist().logcdf(-np.inf).tag.test_value

    def test_normal(self):
        self.pymc3_matches_scipy(Normal, R, {'mu': R, 'sigma': Rplus},
                                 lambda value, mu, sigma: sp.norm.logpdf(value, mu, sigma),
                                 decimal=select_by_precision(float64=6, float32=1)
                                 )
        self.check_logcdf(Normal, R, {'mu': R, 'sigma': Rplus},
                          lambda value, mu, sigma: sp.norm.logcdf(value, mu, sigma))

    def test_truncated_normal(self):
        def scipy_logp(value, mu, sigma, lower, upper):
            return sp.truncnorm.logpdf(
                value, (lower-mu)/sigma, (upper-mu)/sigma, loc=mu, scale=sigma)

        self.pymc3_matches_scipy(
            TruncatedNormal, R,
            {'mu': R, 'sigma': Rplusbig, 'lower': -Rplusbig, 'upper': Rplusbig},
            scipy_logp,
            decimal=select_by_precision(float64=6, float32=1)
        )

    def test_half_normal(self):
        self.pymc3_matches_scipy(HalfNormal, Rplus, {'sigma': Rplus},
                                 lambda value, sigma: sp.halfnorm.logpdf(value, scale=sigma),
                                 decimal=select_by_precision(float64=6, float32=-1)
                                 )
        self.check_logcdf(HalfNormal, Rplus, {'sigma': Rplus},
                          lambda value, sigma: sp.halfnorm.logcdf(value, scale=sigma))

    def test_chi_squared(self):
        self.pymc3_matches_scipy(ChiSquared, Rplus, {'nu': Rplusdunif},
                                 lambda value, nu: sp.chi2.logpdf(value, df=nu))

    @pytest.mark.xfail(reason="Poor CDF in SciPy. See scipy/scipy#869 for details.")
    def test_wald_scipy(self):
        self.pymc3_matches_scipy(Wald, Rplus, {'mu': Rplus, 'alpha': Rplus},
                                 lambda value, mu, alpha: sp.invgauss.logpdf(value, mu=mu, loc=alpha),
                                 decimal=select_by_precision(float64=6, float32=1)
                                 )
        self.check_logcdf(Wald, Rplus, {'mu': Rplus, 'alpha': Rplus},
                          lambda value, mu, alpha: sp.invgauss.logcdf(value, mu=mu, loc=alpha))

    @pytest.mark.parametrize('value,mu,lam,phi,alpha,logp', [
        (.5, .001, .5, None, 0., -124500.7257914),
        (1., .5, .001, None, 0., -4.3733162),
        (2., 1., None, None, 0., -2.2086593),
        (5., 2., 2.5, None, 0., -3.4374500),
        (7.5, 5., None, 1., 0., -3.2199074),
        (15., 10., None, .75, 0., -4.0360623),
        (50., 15., None, .66666, 0., -6.1801249),
        (.5, .001, 0.5, None, 0., -124500.7257914),
        (1., .5, .001, None, .5, -3.3330954),
        (2., 1., None, None, 1., -0.9189385),
        (5., 2., 2.5, None, 2., -2.2128783),
        (7.5, 5., None, 1., 2.5, -2.5283764),
        (15., 10., None, .75, 5., -3.3653647),
        (50., 15., None, .666666, 10., -5.6481874)
    ])
    def test_wald(self, value, mu, lam, phi, alpha, logp):
        # Log probabilities calculated using the dIG function from the R package gamlss.
        # See e.g., doi: 10.1111/j.1467-9876.2005.00510.x, or
        # http://www.gamlss.org/.
        with Model() as model:
            Wald('wald', mu=mu, lam=lam, phi=phi, alpha=alpha, transform=None)
        pt = {'wald': value}
        decimals = select_by_precision(float64=6, float32=1)
        assert_almost_equal(model.fastlogp(pt), logp, decimal=decimals, err_msg=str(pt))

    def test_beta(self):
        self.pymc3_matches_scipy(Beta, Unit, {'alpha': Rplus, 'beta': Rplus},
                                 lambda value, alpha, beta: sp.beta.logpdf(value, alpha, beta))
        self.pymc3_matches_scipy(Beta, Unit, {'mu': Unit, 'sigma': Rplus}, beta_mu_sigma)
        self.check_logcdf(Beta, Unit, {'alpha': Rplus, 'beta': Rplus},
                                lambda value, alpha, beta: sp.beta.logcdf(value, alpha, beta))

    def test_kumaraswamy(self):
        # Scipy does not have a built-in Kumaraswamy pdf
        def scipy_log_pdf(value, a, b):
            return np.log(a) + np.log(b) + (a - 1) * np.log(value) + (b - 1) * np.log(1 - value ** a)
        self.pymc3_matches_scipy(Kumaraswamy, Unit, {'a': Rplus, 'b': Rplus}, scipy_log_pdf)

    def test_exponential(self):
        self.pymc3_matches_scipy(Exponential, Rplus, {'lam': Rplus},
                                 lambda value, lam: sp.expon.logpdf(value, 0, 1 / lam))
        self.check_logcdf(Exponential, Rplus, {'lam': Rplus},
                          lambda value, lam: sp.expon.logcdf(value, 0, 1 / lam))

    def test_geometric(self):
        self.pymc3_matches_scipy(Geometric, Nat, {'p': Unit},
                                 lambda value, p: np.log(sp.geom.pmf(value, p)))

    def test_negative_binomial(self):
        def test_fun(value, mu, alpha):
            return sp.nbinom.logpmf(value, alpha, 1 - mu / (mu + alpha))
        self.pymc3_matches_scipy(NegativeBinomial, Nat, {
                            'mu': Rplus, 'alpha': Rplus}, test_fun)

    def test_laplace(self):
        self.pymc3_matches_scipy(Laplace, R, {'mu': R, 'b': Rplus},
                                 lambda value, mu, b: sp.laplace.logpdf(value, mu, b))
        self.check_logcdf(Laplace, R, {'mu': R, 'b': Rplus},
                          lambda value, mu, b: sp.laplace.logcdf(value, mu, b))

    def test_lognormal(self):
        self.pymc3_matches_scipy(
            Lognormal, Rplus, {'mu': R, 'tau': Rplusbig},
            lambda value, mu, tau: floatX(sp.lognorm.logpdf(value, tau**-.5, 0, np.exp(mu))))
        self.check_logcdf(Lognormal, Rplus, {'mu': R, 'tau': Rplusbig},
                          lambda value, mu, tau: sp.lognorm.logcdf(value, tau**-.5, 0, np.exp(mu)))

    def test_t(self):
        self.pymc3_matches_scipy(StudentT, R, {'nu': Rplus, 'mu': R, 'lam': Rplus},
                                 lambda value, nu, mu, lam: sp.t.logpdf(value, nu, mu, lam**-0.5))
        self.check_logcdf(StudentT, R, {'nu': Rplus, 'mu': R, 'lam': Rplus},
                          lambda value, nu, mu, lam: sp.t.logcdf(value, nu, mu, lam**-0.5))

    def test_cauchy(self):
        self.pymc3_matches_scipy(Cauchy, R, {'alpha': R, 'beta': Rplusbig},
                                 lambda value, alpha, beta: sp.cauchy.logpdf(value, alpha, beta))
        self.check_logcdf(Cauchy, R, {'alpha': R, 'beta': Rplusbig},
                          lambda value, alpha, beta: sp.cauchy.logcdf(value, alpha, beta))

    def test_half_cauchy(self):
        self.pymc3_matches_scipy(HalfCauchy, Rplus, {'beta': Rplusbig},
                                 lambda value, beta: sp.halfcauchy.logpdf(value, scale=beta))
        self.check_logcdf(HalfCauchy, Rplus, {'beta': Rplusbig},
                          lambda value, beta: sp.halfcauchy.logcdf(value, scale=beta))

    def test_gamma(self):
        self.pymc3_matches_scipy(
            Gamma, Rplus, {'alpha': Rplusbig, 'beta': Rplusbig},
            lambda value, alpha, beta: sp.gamma.logpdf(value, alpha, scale=1.0 / beta))

        def test_fun(value, mu, sigma):
            return sp.gamma.logpdf(value, mu**2 / sigma**2, scale=1.0 / (mu / sigma**2))
        self.pymc3_matches_scipy(
            Gamma, Rplus, {'mu': Rplusbig, 'sigma': Rplusbig}, test_fun)

        self.check_logcdf(
            Gamma, Rplus, {'alpha': Rplusbig, 'beta': Rplusbig},
            lambda value, alpha, beta: sp.gamma.logcdf(value, alpha, scale=1.0/beta))

    def test_inverse_gamma(self):
        self.pymc3_matches_scipy(
            InverseGamma, Rplus, {'alpha': Rplus, 'beta': Rplus},
            lambda value, alpha, beta: sp.invgamma.logpdf(value, alpha, scale=beta))

    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"),
                           reason="Fails on float32 due to scaling issues")
    def test_inverse_gamma_alt_params(self):
        def test_fun(value, mu, sigma):
            alpha, beta = InverseGamma._get_alpha_beta(None, None, mu, sigma)
            return sp.invgamma.logpdf(value, alpha, scale=beta)
        self.pymc3_matches_scipy(
            InverseGamma, Rplus, {'mu': Rplus, 'sigma': Rplus}, test_fun)

    def test_pareto(self):
        self.pymc3_matches_scipy(Pareto, Rplus, {'alpha': Rplusbig, 'm': Rplusbig},
                                 lambda value, alpha, m: sp.pareto.logpdf(value, alpha, scale=m))
        self.check_logcdf(Pareto, Rplus, {'alpha': Rplusbig, 'm': Rplusbig},
                          lambda value, alpha, m: sp.pareto.logcdf(value, alpha, scale=m))

    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32 due to inf issues")
    def test_weibull(self):
        self.pymc3_matches_scipy(Weibull, Rplus, {'alpha': Rplusbig, 'beta': Rplusbig},
                                 lambda value, alpha, beta: sp.exponweib.logpdf(value, 1, alpha, scale=beta),
                                 )
        self.check_logcdf(Weibull, Rplus, {'alpha': Rplusbig, 'beta': Rplusbig},
                          lambda value, alpha, beta:
                          sp.exponweib.logcdf(value, 1, alpha, scale=beta),)

    def test_half_studentt(self):
        # this is only testing for nu=1 (halfcauchy)
        self.pymc3_matches_scipy(HalfStudentT, Rplus, {'sigma': Rplus},
                                 lambda value, sigma: sp.halfcauchy.logpdf(value, 0, sigma))

    def test_skew_normal(self):
        self.pymc3_matches_scipy(SkewNormal, R, {'mu': R, 'sigma': Rplusbig, 'alpha': R},
                                 lambda value, alpha, mu, sigma: sp.skewnorm.logpdf(value, alpha, mu, sigma))

    def test_binomial(self):
        self.pymc3_matches_scipy(Binomial, Nat, {'n': NatSmall, 'p': Unit},
                                 lambda value, n, p: sp.binom.logpmf(value, n, p))

    # Too lazy to propagate decimal parameter through the whole chain of deps
    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_beta_binomial(self):
        self.checkd(BetaBinomial, Nat, {'alpha': Rplus, 'beta': Rplus, 'n': NatSmall})

    def test_bernoulli(self):
        self.pymc3_matches_scipy(
            Bernoulli, Bool, {'logit_p': R},
            lambda value, logit_p: sp.bernoulli.logpmf(value, scipy.special.expit(logit_p)))
        self.pymc3_matches_scipy(Bernoulli, Bool, {'p': Unit},
                                 lambda value, p: sp.bernoulli.logpmf(value, p))


    def test_discrete_weibull(self):
        self.pymc3_matches_scipy(DiscreteWeibull, Nat,
                {'q': Unit, 'beta': Rplusdunif}, discrete_weibull_logpmf)

    def test_poisson(self):
        self.pymc3_matches_scipy(Poisson, Nat, {'mu': Rplus},
                                 lambda value, mu: sp.poisson.logpmf(value, mu))

    def test_bound_poisson(self):
        NonZeroPoisson = Bound(Poisson, lower=1.)
        self.pymc3_matches_scipy(NonZeroPoisson, PosNat, {'mu': Rplus},
                                lambda value, mu: sp.poisson.logpmf(value, mu))

        with Model(): x = NonZeroPoisson('x', mu=4)
        assert np.isinf(x.logp({'x':0}))

    def test_constantdist(self):
        self.pymc3_matches_scipy(Constant, I, {'c': I},
                                 lambda value, c: np.log(c == value))

    # Too lazy to propagate decimal parameter through the whole chain of deps
    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_zeroinflatedpoisson(self):
        self.checkd(ZeroInflatedPoisson, Nat, {'theta': Rplus, 'psi': Unit})

    # Too lazy to propagate decimal parameter through the whole chain of deps
    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_zeroinflatednegativebinomial(self):
        self.checkd(ZeroInflatedNegativeBinomial, Nat,
                    {'mu': Rplusbig, 'alpha': Rplusbig, 'psi': Unit})

    # Too lazy to propagate decimal parameter through the whole chain of deps
    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_zeroinflatedbinomial(self):
        self.checkd(ZeroInflatedBinomial, Nat,
                    {'n': NatSmall, 'p': Unit, 'psi': Unit})

    @pytest.mark.parametrize('n', [1, 2, 3])
    def test_mvnormal(self, n):
        self.pymc3_matches_scipy(MvNormal, RealMatrix(5, n),
                                 {'mu': Vector(R, n), 'tau': PdMatrix(n)},
                                 normal_logpdf_tau)
        self.pymc3_matches_scipy(MvNormal, Vector(R, n),
                                 {'mu': Vector(R, n), 'tau': PdMatrix(n)},
                                 normal_logpdf_tau)
        self.pymc3_matches_scipy(MvNormal, RealMatrix(5, n),
                                 {'mu': Vector(R, n), 'cov': PdMatrix(n)},
                                 normal_logpdf_cov)
        self.pymc3_matches_scipy(MvNormal, Vector(R, n),
                                 {'mu': Vector(R, n), 'cov': PdMatrix(n)},
                                 normal_logpdf_cov)
        self.pymc3_matches_scipy(MvNormal, RealMatrix(5, n),
                                 {'mu': Vector(R, n), 'chol': PdMatrixChol(n)},
                                 normal_logpdf_chol,
                                 decimal=select_by_precision(float64=6, float32=-1))
        self.pymc3_matches_scipy(MvNormal, Vector(R, n),
                                 {'mu': Vector(R, n), 'chol': PdMatrixChol(n)},
                                 normal_logpdf_chol,
                                 decimal=select_by_precision(float64=6, float32=0))

        def MvNormalUpper(*args, **kwargs):
            return MvNormal(lower=False, *args, **kwargs)

        self.pymc3_matches_scipy(MvNormalUpper, Vector(R, n),
                                 {'mu': Vector(R, n), 'chol': PdMatrixCholUpper(n)},
                                 normal_logpdf_chol_upper,
                                 decimal=select_by_precision(float64=6, float32=0))

    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32 due to inf issues")
    def test_mvnormal_indef(self):
        cov_val = np.array([[1, 0.5], [0.5, -2]])
        cov = tt.matrix('cov')
        cov.tag.test_value = np.eye(2)
        mu = floatX(np.zeros(2))
        x = tt.vector('x')
        x.tag.test_value = np.zeros(2)
        logp = MvNormal.dist(mu=mu, cov=cov).logp(x)
        f_logp = theano.function([cov, x], logp)
        assert f_logp(cov_val, np.ones(2)) == -np.inf
        dlogp = tt.grad(logp, cov)
        f_dlogp = theano.function([cov, x], dlogp)
        assert not np.all(np.isfinite(f_dlogp(cov_val, np.ones(2))))

        logp = MvNormal.dist(mu=mu, tau=cov).logp(x)
        f_logp = theano.function([cov, x], logp)
        assert f_logp(cov_val, np.ones(2)) == -np.inf
        dlogp = tt.grad(logp, cov)
        f_dlogp = theano.function([cov, x], dlogp)
        assert not np.all(np.isfinite(f_dlogp(cov_val, np.ones(2))))

    def test_mvnormal_init_fail(self):
        with Model():
            with pytest.raises(ValueError):
                x = MvNormal('x', mu=np.zeros(3), shape=3)
            with pytest.raises(ValueError):
                x = MvNormal('x', mu=np.zeros(3), cov=np.eye(3), tau=np.eye(3), shape=3)

    @pytest.mark.parametrize('n', [1, 2, 3])
    def test_matrixnormal(self, n):
        mat_scale = 1e3  # To reduce logp magnitude
        mean_scale = .1
        self.pymc3_matches_scipy(MatrixNormal, RealMatrix(n, n),
                                 {'mu': RealMatrix(n, n)*mean_scale,
                                  'rowcov': PdMatrix(n)*mat_scale,
                                  'colcov': PdMatrix(n)*mat_scale},
                                 matrix_normal_logpdf_cov)
        self.pymc3_matches_scipy(MatrixNormal, RealMatrix(2, n),
                                 {'mu': RealMatrix(2, n)*mean_scale,
                                  'rowcov': PdMatrix(2)*mat_scale,
                                  'colcov': PdMatrix(n)*mat_scale},
                                 matrix_normal_logpdf_cov)
        self.pymc3_matches_scipy(MatrixNormal, RealMatrix(3, n),
                                 {'mu': RealMatrix(3, n)*mean_scale,
                                  'rowchol': PdMatrixChol(3)*mat_scale,
                                  'colchol': PdMatrixChol(n)*mat_scale},
                                 matrix_normal_logpdf_chol,
                                 decimal=select_by_precision(float64=6, float32=-1))
        self.pymc3_matches_scipy(MatrixNormal, RealMatrix(n, 3),
                                 {'mu': RealMatrix(n, 3)*mean_scale,
                                  'rowchol': PdMatrixChol(n)*mat_scale,
                                  'colchol': PdMatrixChol(3)*mat_scale},
                                 matrix_normal_logpdf_chol,
                                 decimal=select_by_precision(float64=6, float32=0))

    @pytest.mark.parametrize('n', [2, 3])
    @pytest.mark.parametrize('m', [3])
    @pytest.mark.parametrize('sigma', [None, 1.0])
    def test_kroneckernormal(self, n, m, sigma):
        np.random.seed(5)
        N = n*m
        covs = [RandomPdMatrix(n), RandomPdMatrix(m)]
        chols = list(map(np.linalg.cholesky, covs))
        evds = list(map(np.linalg.eigh, covs))
        dom = Domain([np.random.randn(N)*0.1], edges=(None, None), shape=N)
        mu = Domain([np.random.randn(N)*0.1], edges=(None, None), shape=N)

        std_args = {'mu': mu}
        cov_args = {'covs': covs}
        chol_args = {'chols': chols}
        evd_args = {'evds': evds}
        if sigma is not None and sigma != 0:
            std_args['sigma'] = Domain([sigma], edges=(None, None))
        else:
            for args in [cov_args, chol_args, evd_args]:
                args['sigma'] = sigma

        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_cov,
             extra_args=cov_args, scipy_args=cov_args)
        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_chol,
             extra_args=chol_args, scipy_args=chol_args)
        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_evd,
             extra_args=evd_args, scipy_args=evd_args)

        dom = Domain([np.random.randn(2, N)*0.1], edges=(None, None), shape=(2, N))

        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_cov,
             extra_args=cov_args, scipy_args=cov_args)
        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_chol,
             extra_args=chol_args, scipy_args=chol_args)
        self.pymc3_matches_scipy(
             KroneckerNormal, dom, std_args, kron_normal_logpdf_evd,
             extra_args=evd_args, scipy_args=evd_args)

    @pytest.mark.parametrize('n', [1, 2])
    def test_mvt(self, n):
        self.pymc3_matches_scipy(MvStudentT, Vector(R, n),
                                 {'nu': Rplus, 'Sigma': PdMatrix(n), 'mu': Vector(R, n)},
                                 mvt_logpdf)
        self.pymc3_matches_scipy(MvStudentT, RealMatrix(2, n),
                                 {'nu': Rplus, 'Sigma': PdMatrix(n), 'mu': Vector(R, n)},
                                 mvt_logpdf)

    @pytest.mark.parametrize('n', [2, 3, 4])
    def test_AR1(self, n):
        self.pymc3_matches_scipy(AR1, Vector(R, n), {'k': Unit, 'tau_e': Rplus}, AR1_logpdf)


    @pytest.mark.parametrize('n', [2, 3])
    def test_wishart(self, n):
        # This check compares the autodiff gradient to the numdiff gradient.
        # However, due to the strict constraints of the wishart,
        # it is impossible to numerically determine the gradient as a small
        # pertubation breaks the symmetry. Thus disabling. Also, numdifftools was
        # removed in June 2019, so an alternative would be needed.
        #
        # self.checkd(Wishart, PdMatrix(n), {'n': Domain([2, 3, 4, 2000]), 'V': PdMatrix(n)},
        #             checks=[self.check_dlogp])
        pass

    @pytest.mark.parametrize('x,eta,n,lp', LKJ_CASES)
    def test_lkj(self, x, eta, n, lp):
        with Model() as model:
            LKJCorr('lkj', eta=eta, n=n, transform=None)

        pt = {'lkj': x}
        decimals = select_by_precision(float64=6, float32=4)
        assert_almost_equal(model.fastlogp(pt), lp, decimal=decimals, err_msg=str(pt))

    @pytest.mark.parametrize('n', [2, 3])
    def test_dirichlet(self, n):
        self.pymc3_matches_scipy(Dirichlet, Simplex(
            n), {'a': Vector(Rplus, n)}, dirichlet_logpdf)

    def test_dirichlet_2D(self):
        self.pymc3_matches_scipy(Dirichlet, MultiSimplex(2, 2),
                                 {'a': Vector(Vector(Rplus, 2), 2)}, dirichlet_logpdf)

    @pytest.mark.parametrize('n', [2, 3])
    def test_multinomial(self, n):
        self.pymc3_matches_scipy(Multinomial, Vector(Nat, n), {'p': Simplex(n), 'n': Nat},
                                 multinomial_logpdf)

    @pytest.mark.parametrize('p,n', [
        [[.25, .25, .25, .25], 1],
        [[.3, .6, .05, .05], 2],
        [[.3, .6, .05, .05], 10],
    ])
    def test_multinomial_mode(self, p, n):
        _p = np.array(p)
        with Model() as model:
            m = Multinomial('m', n, _p, _p.shape)
        assert_allclose(m.distribution.mode.eval().sum(), n)
        _p = np.array([p, p])
        with Model() as model:
            m = Multinomial('m', n, _p, _p.shape)
        assert_allclose(m.distribution.mode.eval().sum(axis=-1), n)

    @pytest.mark.parametrize('p, shape, n', [
        [[.25, .25, .25, .25], 4, 2],
        [[.25, .25, .25, .25], (1, 4), 3],
        # 3: expect to fail
        # [[.25, .25, .25, .25], (10, 4)],
        [[.25, .25, .25, .25], (10, 1, 4), 5],
        # 5: expect to fail
        # [[[.25, .25, .25, .25]], (2, 4), [7, 11]],
        [[[.25, .25, .25, .25],
         [.25, .25, .25, .25]], (2, 4), 13],
        [[[.25, .25, .25, .25],
         [.25, .25, .25, .25]], (1, 2, 4), [23, 29]],
        [[[.25, .25, .25, .25],
         [.25, .25, .25, .25]], (10, 2, 4), [31, 37]],
        [[[.25, .25, .25, .25],
         [.25, .25, .25, .25]], (2, 4), [17, 19]],
    ])
    def test_multinomial_random(self, p, shape, n):
        p = np.asarray(p)
        with Model() as model:
            m = Multinomial('m', n=n, p=p, shape=shape)
        m.random()

    def test_multinomial_mode_with_shape(self):
        n = [1, 10]
        p = np.asarray([[.25, .25, .25, .25], [.26, .26, .26, .22]])
        with Model() as model:
            m = Multinomial('m', n=n, p=p, shape=(2, 4))
        assert_allclose(m.distribution.mode.eval().sum(axis=-1), n)

    def test_multinomial_vec(self):
        vals = np.array([[2, 4, 4], [3, 3, 4]])
        p = np.array([0.2, 0.3, 0.5])
        n = 10

        with Model() as model_single:
            Multinomial('m', n=n, p=p, shape=len(p))

        with Model() as model_many:
            Multinomial('m', n=n, p=p, shape=vals.shape)

        assert_almost_equal(scipy.stats.multinomial.logpmf(vals, n, p),
                            np.asarray([model_single.fastlogp({'m': val}) for val in vals]),
                            decimal=4)

        assert_almost_equal(scipy.stats.multinomial.logpmf(vals, n, p),
                            model_many.free_RVs[0].logp_elemwise({'m': vals}).squeeze(),
                            decimal=4)

        assert_almost_equal(sum([model_single.fastlogp({'m': val}) for val in vals]),
                            model_many.fastlogp({'m': vals}),
                            decimal=4)

    def test_multinomial_vec_1d_n(self):
        vals = np.array([[2, 4, 4], [4, 3, 4]])
        p = np.array([0.2, 0.3, 0.5])
        ns = np.array([10, 11])

        with Model() as model:
            Multinomial('m', n=ns, p=p, shape=vals.shape)

        assert_almost_equal(sum([multinomial_logpdf(val, n, p) for val, n in zip(vals, ns)]),
                            model.fastlogp({'m': vals}),
                            decimal=4)

    def test_multinomial_vec_1d_n_2d_p(self):
        vals = np.array([[2, 4, 4], [4, 3, 4]])
        ps = np.array([[0.2, 0.3, 0.5],
                       [0.9, 0.09, 0.01]])
        ns = np.array([10, 11])

        with Model() as model:
            Multinomial('m', n=ns, p=ps, shape=vals.shape)

        assert_almost_equal(sum([multinomial_logpdf(val, n, p) for val, n, p in zip(vals, ns, ps)]),
                            model.fastlogp({'m': vals}),
                            decimal=4)

    def test_multinomial_vec_2d_p(self):
        vals = np.array([[2, 4, 4], [3, 3, 4]])
        ps = np.array([[0.2, 0.3, 0.5],
                       [0.3, 0.3, 0.4]])
        n = 10

        with Model() as model:
            Multinomial('m', n=n, p=ps, shape=vals.shape)

        assert_almost_equal(sum([multinomial_logpdf(val, n, p) for val, p in zip(vals, ps)]),
                            model.fastlogp({'m': vals}),
                            decimal=4)

    def test_categorical_bounds(self):
        with Model():
            x = Categorical('x', p=np.array([0.2, 0.3, 0.5]))
            assert np.isinf(x.logp({'x': -1}))
            assert np.isinf(x.logp({'x': 3}))

    def test_categorical_valid_p(self):
        with Model():
            x = Categorical('x', p=np.array([-0.2, 0.3, 0.5]))
            assert np.isinf(x.logp({'x': 0}))
            assert np.isinf(x.logp({'x': 1}))
            assert np.isinf(x.logp({'x': 2}))
        with Model():
            # A model where p sums to 1 but contains negative values
            x = Categorical('x', p=np.array([-0.2, 0.7, 0.5]))
            assert np.isinf(x.logp({'x': 0}))
            assert np.isinf(x.logp({'x': 1}))
            assert np.isinf(x.logp({'x': 2}))
        with Model():
            # Hard edge case from #2082
            # Early automatic normalization of p's sum would hide the negative
            # entries if there is a single or pair number of negative values
            # and the rest are zero
            x = Categorical('x', p=np.array([-1, -1, 0, 0]))
            assert np.isinf(x.logp({'x': 0}))
            assert np.isinf(x.logp({'x': 1}))
            assert np.isinf(x.logp({'x': 2}))
            assert np.isinf(x.logp({'x': 3}))

    @pytest.mark.parametrize('n', [2, 3, 4])
    def test_categorical(self, n):
        self.pymc3_matches_scipy(Categorical, Domain(range(n), 'int64'), {'p': Simplex(n)},
                                 lambda value, p: categorical_logpdf(value, p))

    @pytest.mark.parametrize('n', [2, 3, 4])
    def test_orderedlogistic(self, n):
        self.pymc3_matches_scipy(OrderedLogistic, Domain(range(n), 'int64'),
                                 {'eta': R, 'cutpoints': Vector(R, n-1)},
                                 lambda value, eta, cutpoints: orderedlogistic_logpdf(value, eta, cutpoints))

    def test_densitydist(self):
        def logp(x):
            return -log(2 * .5) - abs(x - .5) / .5
        self.checkd(DensityDist, R, {}, extra_args={'logp': logp})

    def test_get_tau_sigma(self):
        sigma = np.array([2])
        assert_almost_equal(continuous.get_tau_sigma(sigma=sigma), [1. / sigma**2, sigma])

    @pytest.mark.parametrize('value,mu,sigma,nu,logp', [
        (0.5, -50.000, 0.500, 0.500, -99.8068528),
        (1.0, -1.000, 0.001, 0.001, -1992.5922447),
        (2.0, 0.001, 1.000, 1.000, -1.6720416),
        (5.0, 0.500, 2.500, 2.500, -2.4543644),
        (7.5, 2.000, 5.000, 5.000, -2.8259429),
        (15.0, 5.000, 7.500, 7.500, -3.3093854),
        (50.0, 50.000, 10.000, 10.000, -3.6436067),
        (1000.0, 500.000, 10.000, 20.000, -27.8707323)
    ])
    def test_ex_gaussian(self, value, mu, sigma, nu, logp):
        """Log probabilities calculated using the dexGAUS function from the R package gamlss.
        See e.g., doi: 10.1111/j.1467-9876.2005.00510.x, or http://www.gamlss.org/."""
        with Model() as model:
            ExGaussian('eg', mu=mu, sigma=sigma, nu=nu)
        pt = {'eg': value}
        assert_almost_equal(model.fastlogp(pt), logp, decimal=select_by_precision(float64=6, float32=2), err_msg=str(pt))

    @pytest.mark.parametrize('value,mu,sigma,nu,logcdf', [
        (0.5, -50.000, 0.500, 0.500, 0.0000000),
        (1.0, -1.000, 0.001, 0.001, 0.0000000),
        (2.0, 0.001, 1.000, 1.000, -0.2365674),
        (5.0, 0.500, 2.500, 2.500, -0.2886489),
        (7.5, 2.000, 5.000, 5.000, -0.5655104),
        (15.0, 5.000, 7.500, 7.500, -0.4545255),
        (50.0, 50.000, 10.000, 10.000, -1.433714),
        (1000.0, 500.000, 10.000, 20.000, -1.573708e-11),
    ])
    def test_ex_gaussian_cdf(self, value, mu, sigma, nu, logcdf):
        """Log probabilities calculated using the pexGAUS function from the R package gamlss.
        See e.g., doi: 10.1111/j.1467-9876.2005.00510.x, or http://www.gamlss.org/."""
        assert_almost_equal(
            ExGaussian.dist(mu=mu, sigma=sigma, nu=nu).logcdf(value).tag.test_value,
            logcdf,
            decimal=select_by_precision(float64=6, float32=2),
            err_msg=str((value, mu, sigma, nu, logcdf)))

    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_vonmises(self):
        self.pymc3_matches_scipy(
            VonMises, R, {'mu': Circ, 'kappa': Rplus},
            lambda value, mu, kappa: floatX(sp.vonmises.logpdf(value, kappa, loc=mu)))

    def test_gumbel(self):
        def gumbel(value, mu, beta):
            return floatX(sp.gumbel_r.logpdf(value, loc=mu, scale=beta))
        self.pymc3_matches_scipy(Gumbel, R, {'mu': R, 'beta': Rplusbig}, gumbel)

        def gumbellcdf(value, mu, beta):
            return floatX(sp.gumbel_r.logcdf(value, loc=mu, scale=beta))
        self.check_logcdf(Gumbel, R, {'mu': R, 'beta': Rplusbig}, gumbellcdf)

    def test_logistic(self):
        self.pymc3_matches_scipy(Logistic, R, {'mu': R, 's': Rplus},
                                 lambda value, mu, s: sp.logistic.logpdf(value, mu, s),
                                 decimal=select_by_precision(float64=6, float32=1))
        self.check_logcdf(Logistic, R, {'mu': R, 's': Rplus},
                          lambda value, mu, s: sp.logistic.logcdf(value, mu, s),
                          decimal=select_by_precision(float64=6, float32=1))

    def test_logitnormal(self):
        self.pymc3_matches_scipy(LogitNormal, Unit, {'mu': R, 'sigma': Rplus},
                                 lambda value, mu, sigma: (sp.norm.logpdf(logit(value), mu, sigma)
                                                        - (np.log(value) + np.log1p(-value))),
                                 decimal=select_by_precision(float64=6, float32=1))

    def test_multidimensional_beta_construction(self):
        with Model():
            Beta('beta', alpha=1., beta=1., shape=(10, 20))

    def test_rice(self):
        self.pymc3_matches_scipy(Rice, Rplus, {'nu': Rplus, 'sigma': Rplusbig},
                                 lambda value, nu, sigma: sp.rice.logpdf(value, b=nu / sigma, loc=0, scale=sigma))
        self.pymc3_matches_scipy(Rice, Rplus, {'b': Rplus, 'sigma': Rplusbig},
                                 lambda value, b, sigma: sp.rice.logpdf(value, b=b, loc=0, scale=sigma))

    @pytest.mark.xfail(condition=(theano.config.floatX == "float32"), reason="Fails on float32")
    def test_interpolated(self):
        for mu in R.vals:
            for sigma in Rplus.vals:
                #pylint: disable=cell-var-from-loop
                xmin = mu - 5 * sigma
                xmax = mu + 5 * sigma

                class TestedInterpolated (Interpolated):
                    def __init__(self, **kwargs):
                        x_points = np.linspace(xmin, xmax, 100000)
                        pdf_points = sp.norm.pdf(x_points, loc=mu, scale=sigma)
                        super().__init__(x_points=x_points, pdf_points=pdf_points, **kwargs)

                def ref_pdf(value):
                    return np.where(
                        np.logical_and(value >= xmin, value <= xmax),
                        sp.norm.logpdf(value, mu, sigma),
                        -np.inf * np.ones(value.shape)
                    )

                self.pymc3_matches_scipy(TestedInterpolated, R, {}, ref_pdf)


def test_bound():
    np.random.seed(42)
    UnboundNormal = Bound(Normal)
    dist = UnboundNormal.dist(mu=0, sigma=1)
    assert dist.transform is None
    assert dist.default() == 0.
    assert isinstance(dist.random(), np.ndarray)

    LowerNormal = Bound(Normal, lower=1)
    dist = LowerNormal.dist(mu=0, sigma=1)
    assert dist.logp(0).eval() == -np.inf
    assert dist.default() > 1
    assert dist.transform is not None
    assert np.all(dist.random() > 1)

    UpperNormal = Bound(Normal, upper=-1)
    dist = UpperNormal.dist(mu=0, sigma=1)
    assert dist.logp(-0.5).eval() == -np.inf
    assert dist.default() < -1
    assert dist.transform is not None
    assert np.all(dist.random() < -1)

    ArrayNormal = Bound(Normal, lower=[1, 2], upper=[2, 3])
    dist = ArrayNormal.dist(mu=0, sigma=1, shape=2)
    assert_equal(dist.logp([0.5, 3.5]).eval(), -np.array([np.inf, np.inf]))
    assert_equal(dist.default(), np.array([1.5, 2.5]))
    assert dist.transform is not None
    with pytest.raises(ValueError) as err:
        dist.random()
    err.match('Drawing samples from distributions with array-valued')

    with Model():
        a = ArrayNormal('c', shape=2)
        assert_equal(a.tag.test_value, np.array([1.5, 2.5]))

    lower = tt.vector('lower')
    lower.tag.test_value = np.array([1, 2]).astype(theano.config.floatX)
    upper = 3
    ArrayNormal = Bound(Normal, lower=lower, upper=upper)
    dist = ArrayNormal.dist(mu=0, sigma=1, shape=2)
    logp = dist.logp([0.5, 3.5]).eval({lower: lower.tag.test_value})
    assert_equal(logp, -np.array([np.inf, np.inf]))
    assert_equal(dist.default(), np.array([2, 2.5]))
    assert dist.transform is not None

    with Model():
        a = ArrayNormal('c', shape=2)
        assert_equal(a.tag.test_value, np.array([2, 2.5]))

    rand = Bound(Binomial, lower=10).dist(n=20, p=0.3).random()
    assert rand.dtype in [np.int16, np.int32, np.int64]
    assert rand >= 10

    rand = Bound(Binomial, upper=10).dist(n=20, p=0.8).random()
    assert rand.dtype in [np.int16, np.int32, np.int64]
    assert rand <= 10

    rand = Bound(Binomial, lower=5, upper=8).dist(n=10, p=0.6).random()
    assert rand.dtype in [np.int16, np.int32, np.int64]
    assert rand >= 5 and rand <= 8

    with Model():
        BoundPoisson = Bound(Poisson, upper=6)
        BoundPoisson(name="y", mu=1)

    with Model():
        BoundNormalNamedArgs = Bound(Normal, upper=6)("y", mu=2., sd=1.)
        BoundNormalPositionalArgs = Bound(Normal, upper=6)("x", 2., 1.)


    with Model():
        BoundPoissonNamedArgs = Bound(Poisson, upper=6)("y", mu=2.)
        BoundPoissonPositionalArgs = Bound(Poisson, upper=6)("x", 2.)


class TestLatex:

    def setup_class(self):
        # True parameter values
        alpha, sigma = 1, 1
        beta = [1, 2.5]

        # Size of dataset
        size = 100

        # Predictor variable
        X = np.random.normal(size=(size, 2)).dot(np.array([[1, 0], [0, 0.2]]))

        # Simulate outcome variable
        Y = alpha + X.dot(beta) + np.random.randn(size)*sigma
        with Model() as self.model:
            # Priors for unknown model parameters
            alpha = Normal('alpha', mu=0, sigma=10)
            b = Normal('beta', mu=0, sigma=10, shape=(2,), observed=beta)
            sigma = HalfNormal('sigma', sigma=1)

            #Test Cholesky parameterization
            Z = MvNormal('Z', mu=np.zeros(2), chol=np.eye(2), shape=(2,))

            # Expected value of outcome
            mu = Deterministic('mu', floatX(alpha + tt.dot(X, b)))

            # Likelihood (sampling distribution) of observations
            Y_obs = Normal('Y_obs', mu=mu, sigma=sigma, observed=Y)
        self.distributions = [alpha, sigma, mu, b, Z, Y_obs]
        self.expected = (
            r'$\text{alpha} \sim \text{Normal}(\mathit{mu}=0.0,~\mathit{sigma}=10.0)$',
            r'$\text{sigma} \sim \text{HalfNormal}(\mathit{sigma}=1.0)$',
            r'$\text{mu} \sim \text{Deterministic}(\text{alpha},~\text{Constant},~\text{beta})$',
            r'$\text{beta} \sim \text{Normal}(\mathit{mu}=0.0,~\mathit{sigma}=10.0)$',
            r'$Z \sim \text{MvNormal}(\mathit{mu}=array, \mathit{chol}=array)$',
            r'$\text{Y_obs} \sim \text{Normal}(\mathit{mu}=\text{mu},~\mathit{sigma}=f(\text{sigma}))$'
        )

    def test__repr_latex_(self):
        for distribution, tex in zip(self.distributions, self.expected):
            assert distribution._repr_latex_() == tex

        model_tex = self.model._repr_latex_()

        for tex in self.expected:  # make sure each variable is in the model
            for segment in tex.strip('$').split(r'\sim'):
                assert segment in model_tex

    def test___latex__(self):
        for distribution, tex in zip(self.distributions, self.expected):
            assert distribution._repr_latex_() == distribution.__latex__()
        assert self.model._repr_latex_() == self.model.__latex__()


def test_discrete_trafo():
    with pytest.raises(ValueError) as err:
        Binomial.dist(n=5, p=0.5, transform='log')
    err.match('Transformations for discrete distributions')
    with Model():
        with pytest.raises(ValueError) as err:
            Binomial('a', n=5, p=0.5, transform='log')
        err.match('Transformations for discrete distributions')