# Discrete¶

 Binomial(n, p, *args, **kwargs) Binomial log-likelihood. ZeroInflatedBinomial(psi, n, p, *args, **kwargs) Zero-inflated Binomial log-likelihood. BetaBinomial(alpha, beta, n, *args, **kwargs) Beta-binomial log-likelihood. Bernoulli([p, logit_p]) Bernoulli log-likelihood Poisson(mu, *args, **kwargs) Poisson log-likelihood. ZeroInflatedPoisson(psi, theta, *args, **kwargs) Zero-inflated Poisson log-likelihood. NegativeBinomial(mu, alpha, *args, **kwargs) Negative binomial log-likelihood. ZeroInflatedNegativeBinomial(psi, mu, alpha, …) Zero-Inflated Negative binomial log-likelihood. DiscreteUniform(lower, upper, *args, **kwargs) Discrete uniform distribution. Geometric(p, *args, **kwargs) Geometric log-likelihood. Categorical(p, *args, **kwargs) Categorical log-likelihood. DiscreteWeibull(q, beta, *args, **kwargs) Discrete Weibull log-likelihood Constant(c, *args, **kwargs) Constant log-likelihood. OrderedLogistic(eta, cutpoints, *args, **kwargs) Ordered Logistic 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. The pmf of this distribution is

$f(x \mid n, p) = \binom{n}{x} p^x (1-p)^{n-x}$
 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. The pmf of this distribution is

$f(x \mid \alpha, \beta, n) = \binom{n}{x} \frac{B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)}$
 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=None, logit_p=None, *args, **kwargs)

Bernoulli log-likelihood

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

$f(x \mid p) = p^{x} (1-p)^{1-x}$
 Support $$x \in \{0, 1\}$$ Mean $$p$$ Variance $$p (1 - p)$$
Parameters: p (float) – Probability of success (0 < p < 1). logit_p (float) – Logit of success probability. Only one of p and logit_p can be specified.
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. The pmf of this distribution is

$f(x \mid q, \beta) = q^{x^{\beta}} - q^{(x + 1)^{\beta}}$
 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. The pmf of this distribution is

$f(x \mid \mu) = \frac{e^{-\mu}\mu^x}{x!}$
 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. The pmf of this distribution is

$f(x \mid \mu, \alpha) = \binom{x + \alpha - 1}{x} (\alpha/(\mu+\alpha))^\alpha (\mu/(\mu+\alpha))^x$
 Support $$x \in \mathbb{N}_0$$ Mean $$\mu$$
Parameters: mu (float) – Poission distribution parameter (mu > 0). alpha (float) – Gamma distribution parameter (alpha > 0).
pymc3.distributions.discrete.ConstantDist
class pymc3.distributions.discrete.Constant(c, *args, **kwargs)

Constant log-likelihood.

Parameters: value (float or int) – Constant parameter.
class pymc3.distributions.discrete.ZeroInflatedPoisson(psi, theta, *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. The pmf of this distribution is

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

Zero-inflated Binomial log-likelihood.

The pmf of this distribution is

$\begin{split}f(x \mid \psi, n, p) = \left\{ \begin{array}{l} (1-\psi) + \psi (1-p)^{n}, \text{if } x = 0 \\ \psi {n \choose x} p^x (1-p)^{n-x}, \text{if } x=1,2,3,\ldots,n \end{array} \right.\end{split}$
 Support $$x \in \mathbb{N}_0$$ Mean $$(1 - \psi) n p$$ Variance $$(1-\psi) n p [1 - p(1 - \psi n)].$$
Parameters: psi (float) – Expected proportion of Binomial variates (0 < psi < 1) n (int) – Number of Bernoulli trials (n >= 0). p (float) – Probability of success in each trial (0 < p < 1).
class pymc3.distributions.discrete.ZeroInflatedNegativeBinomial(psi, mu, alpha, *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. The pmf of this distribution is

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

Discrete uniform distribution. The pmf of this distribution is

$f(x \mid lower, upper) = \frac{1}{upper-lower}$
 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. The pmf of this distribution is

$f(x \mid p) = p(1-p)^{x-1}$
 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. The pmf of this distribution is

$f(x \mid p) = p_x$
 Support $$x \in \{0, 1, \ldots, |p|-1\}$$
Parameters: p (array of floats) – p > 0 and the elements of p must sum to 1. They will be automatically rescaled otherwise.
class pymc3.distributions.discrete.OrderedLogistic(eta, cutpoints, *args, **kwargs)

Ordered Logistic log-likelihood.

Useful for regression on ordinal data values whose values range from 1 to K as a function of some predictor, $$\eta$$. The cutpoints, $$c$$, separate which ranges of $$\eta$$ are mapped to which of the K observed dependent variables. The number of cutpoints is K - 1. It is recommended that the cutpoints are constrained to be ordered.

$\begin{split}f(k \mid \eta, c) = \left\{ \begin{array}{l} 1 - \text{logit}^{-1}(\eta - c_1) \,, \text{if } k = 0 \\ \text{logit}^{-1}(\eta - c_{k - 1}) - \text{logit}^{-1}(\eta - c_{k}) \,, \text{if } 0 < k < K \\ \text{logit}^{-1}(\eta - c_{K - 1}) \,, \text{if } k = K \\ \end{array} \right.\end{split}$
Parameters: eta (float) – The predictor. c (array) – The length K - 1 array of cutpoints which break $$\eta$$ into ranges. Do not explicitly set the first and last elements of $$c$$ to negative and positive infinity.

Example

# Generate data for a simple 1 dimensional example problem
n1_c = 300; n2_c = 300; n3_c = 300
cluster1 = np.random.randn(n1_c) + -1
cluster2 = np.random.randn(n2_c) + 0
cluster3 = np.random.randn(n3_c) + 2

x = np.concatenate((cluster1, cluster2, cluster3))
y = np.concatenate((1*np.ones(n1_c),
2*np.ones(n2_c),
3*np.ones(n3_c))) - 1

# Ordered logistic regression
with pm.Model() as model:
cutpoints = pm.Normal("cutpoints", mu=[-1,1], sd=10, shape=2,
transform=pm.distributions.transforms.ordered)
y_ = pm.OrderedLogistic("y", cutpoints=cutpoints, eta=x, observed=y)
tr = pm.sample(1000)

# Plot the results
plt.hist(cluster1, 30, alpha=0.5);
plt.hist(cluster2, 30, alpha=0.5);
plt.hist(cluster3, 30, alpha=0.5);
plt.hist(tr["cutpoints"][:,0], 80, alpha=0.2, color='k');
plt.hist(tr["cutpoints"][:,1], 80, alpha=0.2, color='k');