# Continuous¶

 Uniform([lower, upper]) Continuous uniform log-likelihood. Flat(*args, **kwargs) Uninformative log-likelihood that returns 0 regardless of the passed value. HalfFlat(*args, **kwargs) Improper flat prior over the positive reals. Normal([mu, sigma, tau, sd]) Univariate normal log-likelihood. HalfNormal([sigma, tau, sd]) Half-normal log-likelihood. SkewNormal([mu, sigma, tau, alpha, sd]) Univariate skew-normal log-likelihood. Beta([alpha, beta, mu, sigma, sd]) Beta log-likelihood. Kumaraswamy(a, b, *args, **kwargs) Kumaraswamy log-likelihood. Exponential(lam, *args, **kwargs) Exponential log-likelihood. Laplace(mu, b, *args, **kwargs) Laplace log-likelihood. StudentT(nu[, mu, lam, sigma, sd]) Student’s T log-likelihood. HalfStudentT([nu, sigma, lam, sd]) Half Student’s T log-likelihood Cauchy(alpha, beta, *args, **kwargs) Cauchy log-likelihood. HalfCauchy(beta, *args, **kwargs) Half-Cauchy log-likelihood. Gamma([alpha, beta, mu, sigma, sd]) Gamma log-likelihood. Weibull(alpha, beta, *args, **kwargs) Weibull log-likelihood. Lognormal([mu, sigma, tau, sd]) Log-normal log-likelihood. ChiSquared(nu, *args, **kwargs) $$\chi^2$$ log-likelihood. Wald([mu, lam, phi, alpha]) Wald log-likelihood. Pareto(alpha, m[, transform]) Pareto log-likelihood. InverseGamma([alpha, beta, mu, sigma, sd]) Inverse gamma log-likelihood, the reciprocal of the gamma distribution. ExGaussian([mu, sigma, nu, sd]) Exponentially modified Gaussian log-likelihood. VonMises([mu, kappa, transform]) Univariate VonMises log-likelihood. Triangular([lower, upper, c]) Continuous Triangular log-likelihood Gumbel([mu, beta]) Univariate Gumbel log-likelihood Logistic([mu, s]) Logistic log-likelihood. LogitNormal([mu, sigma, tau, sd]) Logit-Normal log-likelihood. Interpolated(x_points, pdf_points, *args, …) Univariate probability distribution defined as a linear interpolation of probability density function evaluated on some lattice of points.

A collection of common probability distributions for stochastic nodes in PyMC.

class pymc3.distributions.continuous.Uniform(lower=0, upper=1, *args, **kwargs)

Continuous uniform log-likelihood.

The pdf of this distribution is

$f(x \mid lower, upper) = \frac{1}{upper-lower}$
 Support $$x \in [lower, upper]$$ Mean $$\dfrac{lower + upper}{2}$$ Variance $$\dfrac{(upper - lower)^2}{12}$$
Parameters: lower : float Lower limit. upper : float Upper limit.
logp(value)

Calculate log-probability of Uniform distribution at specified value.

Parameters: value : numeric Value for which log-probability is calculated. TensorVariable
random(point=None, size=None)

Draw random values from Uniform distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Flat(*args, **kwargs)

Uninformative log-likelihood that returns 0 regardless of the passed value.

logp(value)

Calculate log-probability of Flat distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Raises ValueError as it is not possible to sample from Flat distribution

Parameters: point : dict, optional size : int, optional ValueError
class pymc3.distributions.continuous.HalfFlat(*args, **kwargs)

Improper flat prior over the positive reals.

logp(value)

Calculate log-probability of HalfFlat distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Raises ValueError as it is not possible to sample from HalfFlat distribution

Parameters: point : dict, optional size : int, optional ValueError
class pymc3.distributions.continuous.Normal(mu=0, sigma=None, tau=None, sd=None, **kwargs)

Univariate normal log-likelihood.

The pdf of this distribution is

$f(x \mid \mu, \tau) = \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (x-\mu)^2 \right\}$

Normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

$\tau = \dfrac{1}{\sigma^2}$
 Support $$x \in \mathbb{R}$$ Mean $$\mu$$ Variance $$\dfrac{1}{\tau}$$ or $$\sigma^2$$
Parameters: mu : float Mean. sigma : float Standard deviation (sigma > 0) (only required if tau is not specified). tau : float Precision (tau > 0) (only required if sigma is not specified).

Examples

with pm.Model():
x = pm.Normal('x', mu=0, sigma=10)

with pm.Model():
x = pm.Normal('x', mu=0, tau=1/23)

logp(value)

Calculate log-probability of Normal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Normal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.TruncatedNormal(mu=0, sigma=None, tau=None, lower=None, upper=None, transform='auto', sd=None, *args, **kwargs)

Univariate truncated normal log-likelihood.

The pdf of this distribution is

$f(x;\mu ,\sigma ,a,b)={\frac {\phi ({\frac {x-\mu }{\sigma }})}{ \sigma \left(\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})\right)}}$

Truncated normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

$\tau = \dfrac{1}{\sigma^2}$
Parameters: mu : float Mean. sigma : float Standard deviation (sigma > 0). lower : float (optional) Left bound. upper : float (optional) Right bound.

Examples

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)

logp(value)

Calculate log-probability of TruncatedNormal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from TruncatedNormal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Beta(alpha=None, beta=None, mu=None, sigma=None, sd=None, *args, **kwargs)

Beta log-likelihood.

The pdf of this distribution is

$f(x \mid \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}$
 Support $$x \in (0, 1)$$ Mean $$\dfrac{\alpha}{\alpha + \beta}$$ Variance $$\dfrac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$

Beta 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

\begin{align}\begin{aligned}\begin{split}\alpha &= \mu \kappa \\ \beta &= (1 - \mu) \kappa\end{split}\\\text{where } \kappa = \frac{\mu(1-\mu)}{\sigma^2} - 1\end{aligned}\end{align}
Parameters: alpha : float alpha > 0. beta : float beta > 0. mu : float Alternative mean (0 < mu < 1). sigma : float Alternative standard deviation (0 < sigma < sqrt(mu * (1 - mu))).

Notes

Beta distribution is a conjugate prior for the parameter $$p$$ of the binomial distribution.

logp(value)

Calculate log-probability of Beta distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Beta distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Kumaraswamy(a, b, *args, **kwargs)

Kumaraswamy log-likelihood.

The pdf of this distribution is

$f(x \mid a, b) = abx^{a-1}(1-x^a)^{b-1}$
 Support $$x \in (0, 1)$$ Mean $$b B(1 + \tfrac{1}{a}, b)$$ Variance $$b B(1 + \tfrac{2}{a}, b) - (b B(1 + \tfrac{1}{a}, b))^2$$
Parameters: a : float a > 0. b : float b > 0.
logp(value)

Calculate log-probability of Kumaraswamy distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Kumaraswamy distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Exponential(lam, *args, **kwargs)

Exponential log-likelihood.

The pdf of this distribution is

$f(x \mid \lambda) = \lambda \exp\left\{ -\lambda x \right\}$
 Support $$x \in [0, \infty)$$ Mean $$\dfrac{1}{\lambda}$$ Variance $$\dfrac{1}{\lambda^2}$$
Parameters: lam : float Rate or inverse scale (lam > 0)
logcdf(value)

Compute the log CDF for the Exponential distribution

References

 [Machler20125] Martin Mächler (2012). “Accurately computing log(1-exp(-|a|)) Assessed by the Rmpfr package”
logp(value)

Calculate log-probability of Exponential distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Exponential distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Laplace(mu, b, *args, **kwargs)

Laplace log-likelihood.

The pdf of this distribution is

$f(x \mid \mu, b) = \frac{1}{2b} \exp \left\{ - \frac{|x - \mu|}{b} \right\}$
 Support $$x \in \mathbb{R}$$ Mean $$\mu$$ Variance $$2 b^2$$
Parameters: mu : float Location parameter. b : float Scale parameter (b > 0).
logp(value)

Calculate log-probability of Laplace distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Laplace distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.StudentT(nu, mu=0, lam=None, sigma=None, sd=None, *args, **kwargs)

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

$f(x|\mu,\lambda,\nu) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left[1+\frac{\lambda(x-\mu)^2}{\nu}\right]^{-\frac{\nu+1}{2}}$
 Support $$x \in \mathbb{R}$$
Parameters: nu : float Degrees of freedom, also known as normality parameter (nu > 0). mu : float Location parameter. sigma : float Scale parameter (sigma > 0). Converges to the standard deviation as nu increases. (only required if lam is not specified) lam : float Scale parameter (lam > 0). Converges to the precision as nu increases. (only required if sigma is not specified)

Examples

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)

logp(value)

Calculate log-probability of StudentT distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from StudentT distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Cauchy(alpha, beta, *args, **kwargs)

Cauchy log-likelihood.

Also known as the Lorentz or the Breit-Wigner distribution.

The pdf of this distribution is

$f(x \mid \alpha, \beta) = \frac{1}{\pi \beta [1 + (\frac{x-\alpha}{\beta})^2]}$
 Support $$x \in \mathbb{R}$$ Mode $$\alpha$$ Mean undefined Variance undefined
Parameters: alpha : float Location parameter beta : float Scale parameter > 0
logp(value)

Calculate log-probability of Cauchy distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Cauchy distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.HalfCauchy(beta, *args, **kwargs)

Half-Cauchy log-likelihood.

The pdf of this distribution is

$f(x \mid \beta) = \frac{2}{\pi \beta [1 + (\frac{x}{\beta})^2]}$
 Support $$x \in [0, \infty)$$ Mode 0 Mean undefined Variance undefined
Parameters: beta : float Scale parameter (beta > 0).
logp(value)

Calculate log-probability of HalfCauchy distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from HalfCauchy distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Gamma(alpha=None, beta=None, mu=None, sigma=None, sd=None, *args, **kwargs)

Gamma log-likelihood.

Represents the sum of alpha exponentially distributed random variables, each of which has mean beta.

The pdf of this distribution is

$f(x \mid \alpha, \beta) = \frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)}$
 Support $$x \in (0, \infty)$$ Mean $$\dfrac{\alpha}{\beta}$$ Variance $$\dfrac{\alpha}{\beta^2}$$

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

$\begin{split}\alpha &= \frac{\mu^2}{\sigma^2} \\ \beta &= \frac{\mu}{\sigma^2}\end{split}$
Parameters: alpha : float Shape parameter (alpha > 0). beta : float Rate parameter (beta > 0). mu : float Alternative shape parameter (mu > 0). sigma : float Alternative scale parameter (sigma > 0).
logp(value)

Calculate log-probability of Gamma distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Gamma distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Weibull(alpha, beta, *args, **kwargs)

Weibull log-likelihood.

The pdf of this distribution is

$f(x \mid \alpha, \beta) = \frac{\alpha x^{\alpha - 1} \exp(-(\frac{x}{\beta})^{\alpha})}{\beta^\alpha}$
 Support $$x \in [0, \infty)$$ Mean $$\beta \Gamma(1 + \frac{1}{\alpha})$$ Variance $$\beta^2 \Gamma(1 + \frac{2}{\alpha} - \mu^2)$$
Parameters: alpha : float Shape parameter (alpha > 0). beta : float Scale parameter (beta > 0).
logcdf(value)

Compute the log CDF for the Weibull distribution

References

 [Machler20126] Martin Mächler (2012). “Accurately computing log(1-exp(-|a|)) Assessed by the Rmpfr package”
logp(value)

Calculate log-probability of Weibull distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Weibull distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.HalfStudentT(nu=1, sigma=None, lam=None, sd=None, *args, **kwargs)

Half Student’s T log-likelihood

The pdf of this distribution is

$f(x \mid \sigma,\nu) = \frac{2\;\Gamma\left(\frac{\nu+1}{2}\right)} {\Gamma\left(\frac{\nu}{2}\right)\sqrt{\nu\pi\sigma^2}} \left(1+\frac{1}{\nu}\frac{x^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}}$
 Support $$x \in [0, \infty)$$
Parameters: nu : float Degrees of freedom, also known as normality parameter (nu > 0). sigma : float Scale parameter (sigma > 0). Converges to the standard deviation as nu increases. (only required if lam is not specified) lam : float Scale parameter (lam > 0). Converges to the precision as nu increases. (only required if sigma is not specified)

Examples

# 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)

logp(value)

Calculate log-probability of HalfStudentT distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from HalfStudentT distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Lognormal(mu=0, sigma=None, tau=None, sd=None, *args, **kwargs)

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.

The pdf of this distribution is

$f(x \mid \mu, \tau) = \frac{1}{x} \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (\ln(x)-\mu)^2 \right\}$
 Support $$x \in [0, \infty)$$ Mean $$\exp\{\mu + \frac{1}{2\tau}\}$$ Variance $$(\exp\{\frac{1}{\tau}\} - 1) \times \exp\{2\mu + \frac{1}{\tau}\}$$
Parameters: mu : float Location parameter. sigma : float Standard deviation. (sigma > 0). (only required if tau is not specified). tau : float Scale parameter (tau > 0). (only required if sigma is not specified).
logp(value)

Calculate log-probability of Lognormal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Lognormal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.ChiSquared(nu, *args, **kwargs)

$$\chi^2$$ log-likelihood.

The pdf of this distribution is

$f(x \mid \nu) = \frac{x^{(\nu-2)/2}e^{-x/2}}{2^{\nu/2}\Gamma(\nu/2)}$
 Support $$x \in [0, \infty)$$ Mean $$\nu$$ Variance $$2 \nu$$
Parameters: nu : int Degrees of freedom (nu > 0).
class pymc3.distributions.continuous.HalfNormal(sigma=None, tau=None, sd=None, *args, **kwargs)

Half-normal log-likelihood.

The pdf of this distribution is

\begin{align}\begin{aligned}f(x \mid \tau) = \sqrt{\frac{2\tau}{\pi}} \exp\left(\frac{-x^2 \tau}{2}\right)\\f(x \mid \sigma) =\sigma \sqrt{\frac{2}{\pi}} \exp\left(\frac{-x^2}{2\sigma^2}\right)\end{aligned}\end{align}

Note

The parameters sigma/tau ($$\sigma$$/$$\tau$$) 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 $$\sigma^2 \equiv \frac{1}{\tau}$$ of a scale parameter

 Support $$x \in [0, \infty)$$ Mean $$\sqrt{\dfrac{2}{\tau \pi}}$$ or $$\dfrac{\sigma \sqrt{2}}{\sqrt{\pi}}$$ Variance $$\dfrac{1}{\tau}\left(1 - \dfrac{2}{\pi}\right)$$ or $$\sigma^2\left(1 - \dfrac{2}{\pi}\right)$$
Parameters: sigma : float Scale parameter $$sigma$$ (sigma > 0) (only required if tau is not specified). tau : float Precision $$tau$$ (tau > 0) (only required if sigma is not specified).

Examples

with pm.Model():
x = pm.HalfNormal('x', sigma=10)

with pm.Model():
x = pm.HalfNormal('x', tau=1/15)

logp(value)

Calculate log-probability of HalfNormal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from HalfNormal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Wald(mu=None, lam=None, phi=None, alpha=0.0, *args, **kwargs)

Wald log-likelihood.

The pdf of this distribution is

$f(x \mid \mu, \lambda) = \left(\frac{\lambda}{2\pi)}\right)^{1/2} x^{-3/2} \exp\left\{ -\frac{\lambda}{2x}\left(\frac{x-\mu}{\mu}\right)^2 \right\}$
 Support $$x \in (0, \infty)$$ Mean $$\mu$$ Variance $$\dfrac{\mu^3}{\lambda}$$

Wald distribution can be parameterized either in terms of lam or phi. The link between the two parametrizations is given by

$\phi = \dfrac{\lambda}{\mu}$
Parameters: mu : float, optional Mean of the distribution (mu > 0). lam : float, optional Relative precision (lam > 0). phi : float, optional Alternative shape parameter (phi > 0). alpha : float, optional 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

 [Tweedie19577] Tweedie, M. C. K. (1957). Statistical Properties of Inverse Gaussian Distributions I. The Annals of Mathematical Statistics, Vol. 28, No. 2, pp. 362-377
 [Michael19767] 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
 [Giner20167] Göknur Giner, Gordon K. Smyth (2016) statmod: Probability Calculations for the Inverse Gaussian Distribution
logp(value)

Calculate log-probability of Wald distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Wald distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Pareto(alpha, m, transform='lowerbound', *args, **kwargs)

Pareto log-likelihood.

Often used to characterize wealth distribution, or other examples of the 80/20 rule.

The pdf of this distribution is

$f(x \mid \alpha, m) = \frac{\alpha m^{\alpha}}{x^{\alpha+1}}$
 Support $$x \in [m, \infty)$$ Mean $$\dfrac{\alpha m}{\alpha - 1}$$ for $$\alpha \ge 1$$ Variance $$\dfrac{m \alpha}{(\alpha - 1)^2 (\alpha - 2)}$$ for $$\alpha > 2$$
Parameters: alpha : float Shape parameter (alpha > 0). m : float Scale parameter (m > 0).
logp(value)

Calculate log-probability of Pareto distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Pareto distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.InverseGamma(alpha=None, beta=None, mu=None, sigma=None, sd=None, *args, **kwargs)

Inverse gamma log-likelihood, the reciprocal of the gamma distribution.

The pdf of this distribution is

$f(x \mid \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\left(\frac{-\beta}{x}\right)$
 Support $$x \in (0, \infty)$$ Mean $$\dfrac{\beta}{\alpha-1}$$ for $$\alpha > 1$$ Variance $$\dfrac{\beta^2}{(\alpha-1)^2(\alpha - 2)}$$ for $$\alpha > 2$$
Parameters: alpha : float Shape parameter (alpha > 0). beta : float Scale parameter (beta > 0). mu : float Alternative shape parameter (mu > 0). sigma : float Alternative scale parameter (sigma > 0).
logp(value)

Calculate log-probability of InverseGamma distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from InverseGamma distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.ExGaussian(mu=0.0, sigma=None, nu=None, sd=None, *args, **kwargs)

Exponentially modified Gaussian log-likelihood.

Results from the convolution of a normal distribution with an exponential distribution.

The pdf of this distribution is

$f(x \mid \mu, \sigma, \tau) = \frac{1}{\nu}\; \exp\left\{\frac{\mu-x}{\nu}+\frac{\sigma^2}{2\nu^2}\right\} \Phi\left(\frac{x-\mu}{\sigma}-\frac{\sigma}{\nu}\right)$

where $$\Phi$$ is the cumulative distribution function of the standard normal distribution.

 Support $$x \in \mathbb{R}$$ Mean $$\mu + \nu$$ Variance $$\sigma^2 + \nu^2$$
Parameters: mu : float Mean of the normal distribution. sigma : float Standard deviation of the normal distribution (sigma > 0). nu : float Mean of the exponential distribution (nu > 0).

References

 [Rigby200510] Rigby R.A. and Stasinopoulos D.M. (2005). “Generalized additive models for location, scale and shape” Applied Statististics., 54, part 3, pp 507-554.
 [Lacouture200810] 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.
logcdf(value)

Compute the log CDF for the ExGaussian distribution

References

 [Rigby200512] R.A. Rigby (2005). “Generalized additive models for location, scale and shape” http://dx.doi.org/10.1111/j.1467-9876.2005.00510.x
logp(value)

Calculate log-probability of ExGaussian distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from ExGaussian distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.VonMises(mu=0.0, kappa=None, transform='circular', *args, **kwargs)

Univariate VonMises log-likelihood.

The pdf of this distribution is

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

where $$I_0$$ is the modified Bessel function of order 0.

 Support $$x \in [-\pi, \pi]$$ Mean $$\mu$$ Variance $$1-\frac{I_1(\kappa)}{I_0(\kappa)}$$
Parameters: mu : float Mean. kappa : float Concentration (frac{1}{kappa} is analogous to sigma^2).
logp(value)

Calculate log-probability of VonMises distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from VonMises distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.SkewNormal(mu=0.0, sigma=None, tau=None, alpha=1, sd=None, *args, **kwargs)

Univariate skew-normal log-likelihood.

The pdf of this distribution is
$f(x \mid \mu, \tau, \alpha) = 2 \Phi((x-\mu)\sqrt{\tau}\alpha) \phi(x,\mu,\tau)$
 Support $$x \in \mathbb{R}$$ Mean $$\mu + \sigma \sqrt{\frac{2}{\pi}} \frac {\alpha }{{\sqrt {1+\alpha ^{2}}}}$$ Variance $$\sigma^2 \left( 1-\frac{2\alpha^2}{(\alpha^2+1) \pi} \right)$$

Skew-normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

$\tau = \dfrac{1}{\sigma^2}$
Parameters: mu : float Location parameter. sigma : float Scale parameter (sigma > 0). tau : float Alternative scale parameter (tau > 0). alpha : float 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.

logp(value)

Calculate log-probability of SkewNormal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from SkewNormal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Triangular(lower=0, upper=1, c=0.5, *args, **kwargs)

Continuous Triangular log-likelihood

The pdf of this distribution is

$\begin{split}\begin{cases} 0 & \text{for } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c, \\[4pt] \frac{2}{b-a} & \text{for } x = c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b, \\[4pt] 0 & \text{for } b < x. \end{cases}\end{split}$
 Support $$x \in [lower, upper]$$ Mean $$\dfrac{lower + upper + c}{3}$$ Variance $$\dfrac{upper^2 + lower^2 +c^2 - lower*upper - lower*c - upper*c}{18}$$
Parameters: lower : float Lower limit. c: float mode upper : float Upper limit.
logp(value)

Calculate log-probability of Triangular distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Triangular distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Gumbel(mu=0, beta=1.0, **kwargs)
Univariate Gumbel log-likelihood

The pdf of this distribution is

$f(x \mid \mu, \beta) = \frac{1}{\beta}e^{-(z + e^{-z})}$

where

$z = \frac{x - \mu}{\beta}.$
 Support $$x \in \mathbb{R}$$ Mean $$\mu + \beta\gamma$$, where gamma is the Euler-Mascheroni constant Variance $$\frac{\pi^2}{6} \beta^2$$
Parameters: mu : float Location parameter. beta : float Scale parameter (beta > 0).
logp(value)

Calculate log-probability of Gumbel distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Gumbel distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Logistic(mu=0.0, s=1.0, *args, **kwargs)

Logistic log-likelihood.

The pdf of this distribution is

$f(x \mid \mu, s) = \frac{\exp\left(-\frac{x - \mu}{s}\right)}{s \left(1 + \exp\left(-\frac{x - \mu}{s}\right)\right)^2}$
 Support $$x \in \mathbb{R}$$ Mean $$\mu$$ Variance $$\frac{s^2 \pi^2}{3}$$
Parameters: mu : float Mean. s : float Scale (s > 0).
logcdf(value)

Compute the log CDF for the Logistic distribution

References

 [Machler201213] Martin Mächler (2012). “Accurately computing log(1-exp(-|a|)) Assessed by the Rmpfr package”
logp(value)

Calculate log-probability of Logistic distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Logistic distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.LogitNormal(mu=0, sigma=None, tau=None, sd=None, **kwargs)

Logit-Normal log-likelihood.

The pdf of this distribution is

$f(x \mid \mu, \tau) = \frac{1}{x(1-x)} \sqrt{\frac{\tau}{2\pi}} \exp\left\{ -\frac{\tau}{2} (logit(x)-\mu)^2 \right\}$
 Support $$x \in (0, 1)$$ Mean no analytical solution Variance no analytical solution
Parameters: mu : float Location parameter. sigma : float Scale parameter (sigma > 0). tau : float Scale parameter (tau > 0).
logp(value)

Calculate log-probability of LogitNormal distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from LogitNormal distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Interpolated(x_points, pdf_points, *args, **kwargs)

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.

 Support $$x \in [x\_points[0], x\_points[-1]]$$
Parameters: x_points : array-like A monotonically growing list of values pdf_points : array-like Probability density function evaluated on lattice x_points
logp(value)

Calculate log-probability of Interpolated distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(size=None)

Draw random values from Interpolated distribution.

Parameters: size : int, optional Desired size of random sample (returns one sample if not specified). array
class pymc3.distributions.continuous.Rice(nu=None, sigma=None, b=None, sd=None, *args, **kwargs)

Rice distribution.

$f(x\mid \nu ,\sigma )= {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right),$
 Support $$x \in (0, \infty)$$ Mean $$\sigma {\sqrt {\pi /2}}\,\,L_{{1/2}}(-\nu ^{2}/2\sigma ^{2})$$ Variance $$2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{{1/2}}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)$$
Parameters: nu : float noncentrality parameter. sigma : float scale parameter. b : float shape parameter (alternative to nu).

Notes

The distribution $$\mathrm{Rice}\left(|\nu|,\sigma\right)$$ is the distribution of $$R=\sqrt{X^2+Y^2}$$ where $$X\sim N(\nu \cos{\theta}, \sigma^2)$$, $$Y\sim N(\nu \sin{\theta}, \sigma^2)$$ are independent and for any real $$\theta$$.

The distribution is defined with either nu or b. The link between the two parametrizations is given by

$b = \dfrac{\nu}{\sigma}$
logp(value)

Calculate log-probability of Rice distribution at specified value.

Parameters: value : numeric Value(s) for which log-probability is calculated. If the log probabilities for multiple values are desired the values must be provided in a numpy array or theano tensor TensorVariable
random(point=None, size=None)

Draw random values from Rice distribution.

Parameters: point : dict, optional Dict of variable values on which random values are to be conditioned (uses default point if not specified). size : int, optional Desired size of random sample (returns one sample if not specified). array