9. Probability distributions¶

PyMC provides 35 built-in probability distributions. For each distribution, it provides:

A function that evaluates its log-probability or log-density: normal_like().
A function that draws random variables: rnormal().
A function that computes the expectation associated with the distribution: normal_expval().
A Stochastic subclass generated from the distribution: Normal.

This section describes the likelihood functions of these distributions.

9.1. Discrete distributions¶

pymc.distributions.bernoulli_like(x, p)¶

Bernoulli log-likelihood

The Bernoulli distribution describes the probability of successes (x=1) and failures (x=0).

$f(x \mid p) = p^{x} (1-p)^{1-x}$

Parameters:	x : Series of successes (1) and failures (0). $x=0,1$ p : Probability of success. $0 < p < 1$ .
Example:	>>> bernoulli_like([0,1,0,1], .4) -2.8542325496673584

Note

$E(x)= p$
$Var(x)= p(1-p)$

pymc.distributions.binomial_like(x, n, p)¶

binomial_like(x, n, p)

Binomial log-likelihood. The discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

$f(x \mid n, p) = \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x}$

Parameters:	x : [int] Number of successes, > 0. n : [int] Number of Bernoulli trials, > x. p : Probability of success in each trial, $p \in [0,1]$ .

Note

$E(X)=np$
$Var(X)=np(1-p)$

pymc.distributions.categorical_like(x, p)¶

categorical_like(x,p)

Categorical log-likelihood. The most general discrete distribution.

$f(x=i \mid p) = p_i$

for $i \in 0 \ldots k-1$ .

Parameters:	x : [int] $x \in 0\ldots k-1$ p : [float] $p > 0$ , $\sum p = 1$

pymc.distributions.discrete_uniform_like(x, lower, upper)¶

discrete_uniform_like(x, lower, upper)

Discrete uniform log-likelihood.

$f(x \mid lower, upper) = \frac{1}{upper-lower}$

Parameters:	x : [int] $lower \leq x \leq upper$ lower : Lower limit. upper : Upper limit (upper > lower).

pymc.distributions.geometric_like(x, p)¶

geometric_like(x, p)

Geometric log-likelihood. The probability that the first success in a sequence of Bernoulli trials occurs on the x’th trial.

$f(x \mid p) = p(1-p)^{x-1}$

Parameters:	x : [int] Number of trials before first success (x > 0). p : Probability of success on an individual trial, $p \in [0,1]$

Note

$E(X)=1/p$
$Var(X)=\frac{1-p}{p^2}$

pymc.distributions.hypergeometric_like(x, n, m, N)¶

hypergeometric_like(x, n, m, N)

Hypergeometric log-likelihood. Discrete probability distribution that describes the number of successes in a sequence of draws from a finite population without replacement.

$f(x \mid n, m, N) = \frac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}}$

Parameters:	x : [int] Number of successes in a sample drawn from a population. $\max(0, draws-failures) \leq x \leq \min(draws, success)$ n : [int] Size of sample drawn from the population. m : [int] Number of successes in the population. N : [int] Total number of units in the population.

Note

$E(X) = \frac{n n}{N}$

pymc.distributions.negative_binomial_like(x, mu, alpha)¶

negative_binomial_like(x, mu, alpha)

Negative binomial log-likelihood. The negative binomial distribution describes a Poisson random variable whose rate parameter is gamma distributed. PyMC’s chosen parameterization makes this mixture interpretation more convenient to work with.

$f(x \mid \mu, \alpha) = \frac{\Gamma(x+\alpha)}{x! \Gamma(\alpha)} (\alpha/(\mu+\alpha))^\alpha (\mu/(\mu+\alpha))^x$

x > 0, mu > 0, alpha > 0

Note

math: E[x]=mu
In Wikipedia’s parameterization, :math: r=alpha :math: p=alpha/(mu+alpha) :math: mu=r(1-p)/p

pymc.distributions.poisson_like(x, mu)¶

poisson_like(x,mu)

Poisson log-likelihood. The Poisson is a discrete probability distribution. It is often used to model the number of events occurring in a fixed period of time when the times at which events occur are independent. The Poisson distribution can be derived as a limiting case of the binomial distribution.

$f(x \mid \mu) = \frac{e^{-\mu}\mu^x}{x!}$

Parameters:	x : [int] $x \in {0,1,2,...}$ mu : Expected number of occurrences during the given interval, $\mu \geq 0$ .

Note

$E(x)=\mu$
$Var(x)=\mu$

9.2. Continuous distributions¶

pymc.distributions.beta_like(x, alpha, beta)¶

beta_like(x, alpha, beta)

Beta log-likelihood. The conjugate prior for the parameter :math: p of the binomial distribution.

$f(x \mid \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}$

Parameters:	x : 0 < x < 1 alpha : alpha > 0 beta : beta > 0
Example:	>>> beta_like(.4,1,2) 0.18232160806655884

Note

$E(X)=\frac{\alpha}{\alpha+\beta}$
$Var(X)=\frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

pymc.distributions.cauchy_like(x, alpha, beta)¶

cauchy_like(x, alpha, beta)

Cauchy log-likelihood. The Cauchy distribution is also known as the Lorentz or the Breit-Wigner distribution.

$f(x \mid \alpha, \beta) = \frac{1}{\pi \beta [1 + (\frac{x-\alpha}{\beta})^2]}$

Parameters:	alpha : Location parameter. beta : Scale parameter > 0.

Note

Mode and median are at alpha.

pymc.distributions.chi2_like(x, nu)¶

chi2_like(x, nu)

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

$f(x \mid \nu) = \frac{x^{(\nu-2)/2}e^{-x/2}}{2^{\nu/2}\Gamma(\nu/2)}$

Parameters:	x : > 0 nu : [int] Degrees of freedom ( nu > 0 )

Note

$E(X)=\nu$
$Var(X)=2\nu$

pymc.distributions.degenerate_like(x, k)¶

degenerate_like(x, k)

Degenerate log-likelihood.

$f(x \mid k) = \left\{ \begin{matrix} 1 \text{ if } x = k \\ 0 \text{ if } x \ne k\end{matrix} \right.$

Parameters:	x : Input value. k : Degenerate value.

pymc.distributions.exponential_like(x, beta)¶

exponential_like(x, beta)

Exponential log-likelihood.

The exponential distribution is a special case of the gamma distribution with alpha=1. It often describes the time until an event.

$f(x \mid \beta) = \frac{1}{\beta}e^{-x/\beta}$

Parameters:	x : x > 0 beta : Survival parameter (beta > 0).

Note

$E(X) = \beta$
$Var(X) = \beta^2$

pymc.distributions.exponweib_like(x, alpha, k, loc=0, scale=1)¶

exponweib_like(x,alpha,k,loc=0,scale=1)

Exponentiated Weibull log-likelihood.

The exponentiated Weibull distribution is a generalization of the Weibull family. Its value lies in being able to model monotone and non-monotone failure rates.

$f(x \mid \alpha,k,loc,scale) & = \frac{\alpha k}{scale} (1-e^{-z^k})^{\alpha-1} e^{-z^k} z^{k-1} \\ z & = \frac{x-loc}{scale}$

Parameters:	x : x > 0 alpha : Shape parameter k : k > 0 loc : Location parameter scale : Scale parameter (scale > 0).

pymc.distributions.gamma_like(x, alpha, beta)¶

gamma_like(x, alpha, beta)

Gamma log-likelihood.

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

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

Parameters:	x : float $x \ge 0$ alpha : float Shape parameter $\alpha > 0$ . beta : float Scale parameter $\beta > 0$ .

Note

$E(X) = \frac{\alpha}{\beta}$
$Var(X) = \frac{\alpha}{\beta^2}$

pymc.distributions.half_normal_like(x, tau)¶

half_normal_like(x, tau)

Half-normal log-likelihood, a normal distribution with mean 0 and limited to the domain $x \in [0, \infty)$ .

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

Parameters:	x : $x \ge 0$ tau : tau > 0

pymc.distributions.hypergeometric_like(x, n, m, N)

hypergeometric_like(x, n, m, N)

Hypergeometric log-likelihood. Discrete probability distribution that describes the number of successes in a sequence of draws from a finite population without replacement.

$f(x \mid n, m, N) = \frac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}}$

Parameters:	x : [int] Number of successes in a sample drawn from a population. $\max(0, draws-failures) \leq x \leq \min(draws, success)$ n : [int] Size of sample drawn from the population. m : [int] Number of successes in the population. N : [int] Total number of units in the population.

Note

$E(X) = \frac{n n}{N}$

pymc.distributions.inverse_gamma_like(x, alpha, beta)¶

inverse_gamma_like(x, alpha, beta)

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

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

Parameters:	x : x > 0 alpha : Shape parameter (alpha > 0). beta : Scale parameter (beta > 0).

Note

$E(X)=\frac{1}{\beta(\alpha-1)}$ for $\alpha > 1$ .

pymc.distributions.laplace_like(x, mu, tau)¶

laplace_like(x, mu, tau)

Laplace (double exponential) log-likelihood.

The Laplace (or double exponential) distribution describes the difference between two independent, identically distributed exponential events. It is often used as a heavier-tailed alternative to the normal.

$f(x \mid \mu, \tau) = \frac{\tau}{2}e^{-\tau |x-\mu|}$

Parameters:	x : $-\infty < x < \infty$ mu : Location parameter :math: -infty < mu < infty tau : Scale parameter $\tau > 0$

Note

$E(X) = \mu$
$Var(X) = \frac{2}{\tau^2}$

pymc.distributions.logistic_like(x, mu, tau)¶

logistic_like(x, mu, tau)

Logistic log-likelihood.

The logistic distribution is often used as a growth model; for example, populations, markets. Resembles a heavy-tailed normal distribution.

$f(x \mid \mu, tau) = \frac{\tau \exp(-\tau[x-\mu])}{[1 + \exp(-\tau[x-\mu])]^2}$

Parameters:	x : $-\infty < x < \infty$ mu : Location parameter :math: -infty < mu < infty tau : Scale parameter (tau > 0)

Note

$E(X) = \mu$
$Var(X) = \frac{\pi^2}{3\tau^2}$

pymc.distributions.lognormal_like(x, mu, tau)¶

lognormal_like(x, mu, tau)

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.

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

Parameters:	x : x > 0 mu : Location parameter. tau : Scale parameter (tau > 0).

Note

$E(X)=e^{\mu+\frac{1}{2\tau}}$

pymc.distributions.normal_like(x, mu, tau)¶

normal_like(x, mu, tau)

Normal log-likelihood.

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

Parameters:	x : Input data. mu : Mean of the distribution. tau : Precision of the distribution, which corresponds to 1/sigma**2 (tau > 0).

Note

$E(X) = \mu$
$Var(X) = 1/\tau$

pymc.distributions.skew_normal_like(x, mu, tau, alpha)¶

skew_normal_like(x, mu, tau, alpha)

Azzalini’s skew-normal log-likelihood

$f(x \mid \mu, \tau, \alpha) = 2 \Phi((x-\mu)\sqrt{\tau}\alpha) \phi(x,\mu,\tau)$

where :math: Phi is the normal CDF and :math: phi is the normal PDF.

Parameters:	x : float Input data. mu : float Mean of the distribution. tau : float Precision of the distribution, > 0. alpha : float Shape parameter of the distribution.

Note

See http://azzalini.stat.unipd.it/SN/

pymc.distributions.t_like(x, nu)¶

t_like(x, nu)

Student’s T log-likelihood. Describes a zero-mean normal variable whose precision is gamma distributed. Alternatively, describes the mean of several zero-mean normal random variables divided by their sample standard deviation.

$f(x \mid \nu) = \frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2}) \sqrt{\nu\pi}} \left( 1 + \frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}$

Parameters:	x : float Input data. nu : float Degrees of freedom.

pymc.distributions.truncnorm_like(x, mu, tau, a, b)¶

truncnorm_like(x, mu, tau, a, b)

Truncated normal log-likelihood.

$f(x \mid \mu, \tau, a, b) = \frac{\phi(\frac{x-\mu}{\sigma})} {\Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})},$

where $\sigma^2=1/\tau$ .

Parameters:	x : Input data. mu : Mean of the distribution. tau : Precision of the distribution, which corresponds to 1/sigma**2 (tau > 0). a : Left bound of the distribution. b : Right bound of the distribution.

pymc.distributions.uniform_like(x, lower, upper)¶

uniform_like(x, lower, upper)

Uniform log-likelihood.

$f(x \mid lower, upper) = \frac{1}{upper-lower}$

Parameters:	x : $lower \leq x \leq upper$ lower : Lower limit. upper : Upper limit (upper > lower).

pymc.distributions.von_mises_like(x, mu, kappa)¶

von_mises_like(x, mu, kappa)

von Mises log-likelihood.

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

where I_0 is the modified Bessel function of order 0.

Parameters:	x : Input data. mu : Mean of the distribution. kappa : Dispersion of the distribution

Note

$E(X) = \mu$

pymc.distributions.weibull_like(x, alpha, beta)¶

weibull_like(x, alpha, beta)

Weibull log-likelihood

$f(x \mid \alpha, \beta) = \frac{\alpha x^{\alpha - 1} \exp(-(\frac{x}{\beta})^{\alpha})}{\beta^\alpha}$

Parameters:	x : $x \ge 0$ alpha : alpha > 0 beta : beta > 0

Note

$E(x)=\beta \Gamma(1+\frac{1}{\alpha})$
$Var(x)=\beta^2 \Gamma(1+\frac{2}{\alpha} - \mu^2)$

9.3. Multivariate discrete distributions¶

pymc.distributions.multivariate_hypergeometric_like(x, m)¶

multivariate_hypergeometric_like(x, m)

The multivariate hypergeometric describes the probability of drawing x[i] elements of the ith category, when the number of items in each category is given by m.

$\frac{\prod_i \binom{m_i}{x_i}}{\binom{N}{n}}$

where $N = \sum_i m_i$ and $n = \sum_i x_i$ .

Parameters:	x : [int sequence] Number of draws from each category, (x < m). m : [int sequence] Number of items in each categoy.

pymc.distributions.multinomial_like(x, n, p)¶

multinomial_like(x, n, p)

Multinomial log-likelihood. Generalization of the binomial distribution, but instead of each trial resulting in “success” or “failure”, each one results in exactly one of some fixed finite number k of possible outcomes over n independent trials. ‘x[i]’ indicates the number of times outcome number i was observed over the n trials.

$f(x \mid n, p) = \frac{n!}{\prod_{i=1}^k x_i!} \prod_{i=1}^k p_i^{x_i}$

Parameters:	x : [(ns, k) int] Random variable indicating the number of time outcome i is observed, $\sum_{i=1}^k x_i=n$ , $x_i \ge 0$ . n : [int] Number of trials. p : (k,) Probability of each one of the different outcomes, $\sum_{i=1}^k p_i = 1)$ , $p_i \ge 0$ .

Note

$E(X_i)=n p_i$
$var(X_i)=n p_i(1-p_i)$
$cov(X_i,X_j) = -n p_i p_j$

9.4. Multivariate continuous distributions¶

pymc.distributions.dirichlet_like(x, theta)¶

dirichlet_like(x, theta)

Dirichlet log-likelihood.

This is a multivariate continuous distribution.

$f(\mathbf{x}) = \frac{\Gamma(\sum_{i=1}^k \theta_i)}{\prod \Gamma(\theta_i)} \prod_{i=1}^k x_i^{\theta_i - 1}$

Parameters:	x : An (n,k-1) array where n is the number of samples and k the dimension. $0 < x_i < 1$ , $\sum_{i=1}^{k-1} x_i < 1$ theta : An (n,k) or (1,k) array > 0.

pymc.distributions.inverse_wishart_like(X, n, Tau)¶

inverse_wishart_like(X, n, Tau)

Inverse Wishart log-likelihood. The inverse Wishart distribution is the conjugate prior for the covariance matrix of a multivariate normal distribution.

$f(X \mid n, T) = \frac{{\mid T \mid}^{n/2}{\mid X \mid}^{(n-k-1)/2} \exp\left\{ -\frac{1}{2} Tr(TX^{-1}) \right\}}{2^{nk/2} \Gamma_p(n/2)}$