Discrete

Binomial(n, p, \*args, \*\*kwargs) Binomial log-likelihood.
BetaBinomial(alpha, beta, n, \*args, \*\*kwargs) Beta-binomial log-likelihood.
Bernoulli(p, \*args, \*\*kwargs) Bernoulli log-likelihood
Poisson(mu, \*args, \*\*kwargs) Poisson log-likelihood.
NegativeBinomial(mu, alpha, \*args, \*\*kwargs) Negative binomial log-likelihood.
ConstantDist(\*args, \*\*kwargs)
ZeroInflatedPoisson(theta, psi, \*args, \*\*kwargs) Zero-inflated Poisson log-likelihood.
DiscreteUniform(lower, upper, \*args, \*\*kwargs) Discrete uniform distribution.
Geometric(p, \*args, \*\*kwargs) Geometric log-likelihood.
Categorical(p, \*args, \*\*kwargs) Categorical log-likelihood.
class pymc3.distributions.discrete.Binomial(n, p, *args, **kwargs)

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.

\[f(x \mid n, p) = \binom{n}{x} p^x (1-p)^{n-x}\]
Support \(x \in \{0, 1, \ldots, n\}\)
Mean \(n p\)
Variance \(n p (1 - p)\)
Parameters:
  • n (int) – Number of Bernoulli trials (n >= 0).
  • p (float) – Probability of success in each trial (0 < p < 1).
class pymc3.distributions.discrete.BetaBinomial(alpha, beta, n, *args, **kwargs)

Beta-binomial log-likelihood.

Equivalent to binomial random variable with success probability drawn from a beta distribution.

\[f(x \mid \alpha, \beta, n) = \binom{n}{x} \frac{B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)}\]
Support \(x \in \{0, 1, \ldots, n\}\)
Mean \(n \dfrac{\alpha}{\alpha + \beta}\)
Variance \(n \dfrac{\alpha \beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}\)
Parameters:
  • n (int) – Number of Bernoulli trials (n >= 0).
  • alpha (float) – alpha > 0.
  • beta (float) – beta > 0.
class pymc3.distributions.discrete.Bernoulli(p, *args, **kwargs)

Bernoulli log-likelihood

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

\[f(x \mid p) = p^{x} (1-p)^{1-x}\]
Support \(x \in \{0, 1\}\)
Mean \(p\)
Variance \(p (1 - p)\)
Parameters:p (float) – Probability of success (0 < p < 1).
class pymc3.distributions.discrete.DiscreteWeibull(q, beta, *args, **kwargs)

Discrete Weibull log-likelihood

The discrete Weibull distribution is a flexible model of count data that can handle both over- and under-dispersion.

\[f(x \mid q, \beta) = q^{x^{\beta}} - q^{(x + 1)^{\beta}}\]
Support \(x \in \mathbb{N}_0\)
Mean \(\mu = \sum_{x = 1}^{\infty} q^{x^{\beta}}\)
Variance \(2 \sum_{x = 1}^{\infty} x q^{x^{\beta}} - \mu - \mu^2\)
class pymc3.distributions.discrete.Poisson(mu, *args, **kwargs)

Poisson log-likelihood.

Often used to model the number of events occurring in a fixed period of time when the times at which events occur are independent.

\[f(x \mid \mu) = \frac{e^{-\mu}\mu^x}{x!}\]
Support \(x \in \mathbb{N}_0\)
Mean \(\mu\)
Variance \(\mu\)
Parameters:mu (float) – Expected number of occurrences during the given interval (mu >= 0).

Notes

The Poisson distribution can be derived as a limiting case of the binomial distribution.

class pymc3.distributions.discrete.NegativeBinomial(mu, alpha, *args, **kwargs)

Negative binomial log-likelihood.

The negative binomial distribution describes a Poisson random variable whose rate parameter is gamma distributed.

\[f(x \mid \mu, \alpha) = \frac{\Gamma(x+\alpha)}{x! \Gamma(\alpha)} (\alpha/(\mu+\alpha))^\alpha (\mu/(\mu+\alpha))^x\]
Support \(x \in \mathbb{N}_0\)
Mean \(\mu\)
Parameters:
  • mu (float) – Poission distribution parameter (mu > 0).
  • alpha (float) – Gamma distribution parameter (alpha > 0).
class pymc3.distributions.discrete.Constant(c, *args, **kwargs)

Constant log-likelihood.

Parameters:value (float or int) – Constant parameter.
class pymc3.distributions.discrete.ZeroInflatedPoisson(theta, psi, *args, **kwargs)

Zero-inflated Poisson log-likelihood.

Often used to model the number of events occurring in a fixed period of time when the times at which events occur are independent.

\[\begin{split}f(x \mid \theta, \psi) = \left\{ \begin{array}{l} (1-\psi) + \psi e^{-\theta}, \text{if } x = 0 \\ \psi \frac{e^{-\theta}\theta^x}{x!}, \text{if } x=1,2,3,\ldots \end{array} \right.\end{split}\]
Support \(x \in \mathbb{N}_0\)
Mean \(\psi\theta\)
Variance \(\theta + \frac{1-\psi}{\psi}\theta^2\)
Parameters:
  • theta (float) – Expected number of occurrences during the given interval (theta >= 0).
  • psi (float) – Expected proportion of Poisson variates (0 < psi < 1)
class pymc3.distributions.discrete.ZeroInflatedNegativeBinomial(mu, alpha, psi, *args, **kwargs)

Zero-Inflated Negative binomial log-likelihood.

The Zero-inflated version of the Negative Binomial (NB). The NB distribution describes a Poisson random variable whose rate parameter is gamma distributed.

\[\begin{split}f(x \mid \mu, \alpha, \psi) = \left\{ \begin{array}{l} (1-\psi) + \psi \left (\frac{\alpha}{\alpha+\mu} \right) ^\alpha, \text{if } x = 0 \\ \psi \frac{\Gamma(x+\alpha)}{x! \Gamma(\alpha)} \left (\frac{\alpha}{\mu+\alpha} \right)^\alpha \left( \frac{\mu}{\mu+\alpha} \right)^x, \text{if } x=1,2,3,\ldots \end{array} \right.\end{split}\]
Support \(x \in \mathbb{N}_0\)
Mean \(\psi\mu\)
Var \(\psi\mu + \left (1 + \frac{\mu}{\alpha} + \frac{1-\psi}{\mu} \right)\)
Parameters:
  • mu (float) – Poission distribution parameter (mu > 0).
  • alpha (float) – Gamma distribution parameter (alpha > 0).
  • psi (float) – Expected proportion of NegativeBinomial variates (0 < psi < 1)
class pymc3.distributions.discrete.DiscreteUniform(lower, upper, *args, **kwargs)

Discrete uniform distribution.

\[f(x \mid lower, upper) = \frac{1}{upper-lower}\]
Support \(x \in {lower, lower + 1, \ldots, upper}\)
Mean \(\dfrac{lower + upper}{2}\)
Variance \(\dfrac{(upper - lower)^2}{12}\)
Parameters:
  • lower (int) – Lower limit.
  • upper (int) – Upper limit (upper > lower).
class pymc3.distributions.discrete.Geometric(p, *args, **kwargs)

Geometric log-likelihood.

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

\[f(x \mid p) = p(1-p)^{x-1}\]
Support \(x \in \mathbb{N}_{>0}\)
Mean \(\dfrac{1}{p}\)
Variance \(\dfrac{1 - p}{p^2}\)
Parameters:p (float) – Probability of success on an individual trial (0 < p <= 1).
class pymc3.distributions.discrete.Categorical(p, *args, **kwargs)

Categorical log-likelihood.

The most general discrete distribution.

\[f(x \mid p) = p_x\]
Support \(x \in \{1, 2, \ldots, |p|\}\)
Parameters:p (array of floats) – p > 0 and the elements of p must sum to 1. They will be automatically rescaled otherwise.