### Discrete

Bernoulli: Any random variable taking values in $$\{0, 1\}$$.

• param RealVar probability: Probability $$p \in [0, 1]$$ that the realization is one.

Binomial: A sum of $$n$$ iid Bernoulli variables. Values in $$\{0, 1, 2, \dots, n\}$$.

• param IntVar numberOfTrials: The number $$n$$ of Bernoulli variables being summed. $$n > 0$$
• param RealVar probabilityOfSuccess: The parameter $$p \in [0, 1]$$ shared by all the Bernoulli variables (probability that they be equal to 1).

Categorical: Any random variable over a finite set $$\{0, 1, 2, \dots, n-1\}$$.

• param Simplex probabilities: Vector of probabilities $$(p_0, p_1, \dots, p_{n-1})$$ for each of the $$n$$ integers.

DiscreteUniform: Uniform random variable over the contiguous set of integers $$\{m, m+1, \dots, M-1\}$$.

• param IntVar minInclusive: The left point of the set (inclusive). $$m \in (-\infty, M)$$
• param IntVar maxExclusive: The right point of the set (exclusive). $$M \in (m, \infty)$$

Poisson: Poisson random variable. Values in $$0, 1, 2, \dots$$

• param RealVar mean: Mean parameter $$\lambda$$. $$\lambda > 0$$

NegativeBinomial: Number of successes in a sequence of iid Bernoulli until $$r$$ failures occur. Values in $$\{0, 1, 2, \dots\}$$.

• param RealVar r: Number of failures until experiment is stopped (generalized to the reals). $$r > 0$$
• param RealVar p: Probability of success of each experiment. $$p \in (0, 1)$$

YuleSimon: An exponential-geometric mixture.

• param RealVar rho: The rate of the mixing exponential distribution.

Geometric: The number of unsuccessful Bernoulli trials until a success. Values in $$\{0, 1, 2, \dots\}$$

• param RealVar p: The probability of success for each Bernoulli trial.

HyperGeometric: Hyper-geometric distribution with population N and population satisfying certain condition K and drawing n samples

• param IntVar numberOfDraws: number of samples (n)
• param IntVar population: number of population (N)
• param IntVar populationConditioned: number of population satisfying condition (K)

BetaBinomial: A sum of $$n$$ iid Bernoulli variables, with a marginalized Beta prior on the success probability. Values in $$\{0, 1, 2, \dots, n\}$$.

• param IntVar numberOfTrials: The number $$n$$ of Bernoulli variables being summed. $$n > 0$$
• param RealVar alpha: Higher values brings mean closer to one. $$\alpha > 0$$
• param RealVar beta: Higher values brings mean closer to zero. $$\beta > 0$$

### Continuous

ContinuousUniform: Uniform random variable over a close interval $$[m, M]$$.

• param RealVar min: The left end point $$m$$ of the interval. $$m \in (\infty, M)$$
• param RealVar max: The right end point of the interval. $$M \in (m, \infty)$$

Exponential: Exponential random variable. Values in $$(0, \infty)$$

• param RealVar rate: The rate $$\lambda$$, inversely proportional to the mean. $$\lambda > 0$$

Normal: Normal random variables. Values in $$\mathbb{R}$$

• param RealVar mean: Mean $$\mu$$. $$\mu \in \mathbb{R}$$
• param RealVar variance: Variance $$\sigma^2$$. $$\sigma^2 > 0$$

Beta: Beta random variable on the open interval $$(0, 1)$$.

• param RealVar alpha: Higher values brings mean closer to one. $$\alpha > 0$$
• param RealVar beta: Higher values brings mean closer to zero. $$\beta > 0$$

Gamma: Gamma random variable. Values in $$(0, \infty)$$.

• param RealVar shape: The shape $$\alpha$$ is proportional to the mean and variance. $$\alpha > 0$$
• param RealVar rate: The rate $$\beta$$ is inverse proportional to the mean and quadratically inverse proportional to the variance. $$\beta > 0$$

StudentT: Student T random variable. Values in $$\mathbb{R}$$

• param RealVar nu: The degrees of freedom $$\nu$$. $$\nu > 0$$
• param RealVar mu

HalfStudentT: HalfStudentT random variable. Values in $$(0, \infty)$$

• param RealVar nu: A degree of freedom parameter $$\nu$$. $$\nu > 0$$
• param RealVar sigma: A scale parameter $$\sigma$$. $$\sigma > 0$$.

ChiSquared: Chi Squared random variable. Values in $$(0, \infty)$$.

• param IntVar nu: The degrees of freedom $$\nu$$. $$\nu > 0$$

Laplace: The Laplace Distribution over $$\mathbb{R}$$

• param RealVar location: The mean parameter.
• param RealVar scale: The scale parameter $$b$$, equal to the square root of half of the variance. $$b > 0$$

Logistic: A random variable with a logistic probability distribution function. Values in $$\mathbb{R}$$

• param RealVar location: The center of the PDF. Also the mean, mode and median. $$\mu \in \mathbb{R}$$
• param RealVar scale: The scale parameter. $$\s > 0$$

LogLogistic: A log-logistic distribution is the probability distribution of a random variable

• param RealVar scale: The scale parameter $$\alpha$$ and also the median. $$\alpha > 0$$
• param RealVar shape: The shape parameter $$\beta$$. $$\beta > 0$$

F: The F-distribution. Also known as Fisher-Snedecor distribution. Values in $$(0, +\infty)$$

• param RealVar d1, d2: The degrees of freedom $$d_1$$ and $$d_2$$ . $$d_1, d_2 > 0$$

Weibull: The Weibull Distribution. Values \in $$(0, \infty)$$.

• param RealVar scale: The scale parameter $$\lambda$$. $$\lambda \in (0, +\infty)$$
• param RealVar shape: The shape parameter $$k$$. $$k \in (0, +\infty)$$

Gumbel: The Gumbel Distribution. Values in $$\mathbb{R}$$

• param RealVar location: The location parameter $$\mu$$. $$\mu \in \mathbb{R}$$
• param RealVar scale: The scale parameter $$\beta$$. $$\beta > 0$$

Gompertz: The Gompertz distribution. Values in $$[0, \infty)$$.

• param RealVar shape: The shape parameter $$\nu$$. $$\nu > 0$$
• param RealVar scale: The scale parameter $$b$$. $$b > 0$$

### Multivariate

MultivariateNormal: Arbitrary linear transformations of $$n$$ iid standard normal random variables.

• param Matrix mean: An $$n \times 1$$ vector $$\mu$$. $$\mu \in \mathbb{R}^n$$
• param CholeskyDecomposition precision: Inverse covariance matrix $$\Lambda$$, a positive definite $$n \times n$$ matrix.

NormalField: A mean-zero normal, sparse-precision Markov random field.

• param Precision precision: Precision matrix structure.

Dirichlet: The Dirichlet distribution over vectors of probabilities $$(p_0, p_1, \dots, p_{n-1})$$. $$p_i \in (0, 1), \sum_i p_i = 1.$$

• param Matrix concentrations: Vector $$(\alpha_0, \alpha_1, \dots, \alpha_{n-1})$$ such that increasing the $$i$$th component increases the mean of entry $$p_i$$.

SymmetricDirichlet: $$n$$ dimensional Dirichlet with all concentrations equal to $$\alpha / n$$.

• param Integer dim: The dimensionality $$n$$. $$n > 0$$
• param RealVar concentration: The shared concentration parameter $$\alpha$$ before normalization by the dimensionality. $$\alpha > 0$$

SimplexUniform: $$n$$ dimensional Dirichlet with all concentrations equal to one.

• param Integer dim: The dimensionality $$n$$. $$n > 0$$

### Misc

LogPotential: Not really a distribution, but rather a way to handle undirected model (AKA random fields). See Ising under the Examples page.

• param RealVar logPotential: The log of the current value of this potential.