### Discrete

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

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

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. $\beta > 0$

BetaNegativeBinomial: Negative Binomial Distribution with a marginalized Beta prior. Values in $\{0, 1, 2, \dots\}$.

• param RealVar r: Number of failures until experiment is stopped (generalized to the reals). $r > 0$
• param RealVar alpha: Higher values brings mean accept probability closer to one. $\alpha > 0$
• param RealVar beta: Higher values brings mean accept probabilitycloser to zero. $\beta > 0$

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). $M \in (m, \infty)$

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: A population of size $N$, $K$ of which are marked, and drawing without replacement $n$ samples from the population; the HyperGeometric models the number in the sample that are marked.

• param IntVar numberOfDraws: Number sampled. $n$
• param IntVar population: Population size. $N$
• param IntVar populationConditioned: Number marked in the population. $K$

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. $p \in (0, 1)$

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

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

YuleSimon: An exponential-geometric mixture.

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

### Continuous

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. $\beta > 0$

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

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

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 $M$ 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. $\lambda > 0$

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 $d_2$ . $d_1, d_2 > 0$

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. $\beta > 0$

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

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

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 $\beta$. $\beta > 0$

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 $\sigma$. $\sigma > 0$.

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 cumulative distribution given by the logistic function. Values in $\mathbb{R}$

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

LogLogistic: A log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. Values in $[0, +\infty)$

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

LogUniform: The random variable $X = b^Y$ where $Y \sim \text{ContinuousUniform}[m, M]$.

• param RealVar min: The left end point $m$ of the interval. $m \in (-\infty, M)$
• param RealVar max: The right end point $M$ of the interval. $M \in (m, \infty)$
• param RealVar base: The base $b$. $b > 0$

LnUniform: The random variable $X = e^Y$ where $Y \sim \text{ContinuousUniform}[m, M]$.

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

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

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

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

• param RealVar nu: The degrees of freedom $\nu$. $\nu > 0$
• param RealVar mu: Location parameter $\mu$. $\mu \in \mathbb{R}$
• param RealVar sigma: Scale parameter $\sigma$. $\sigma > 0$

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 $k$. $k \in (0, +\infty)$

### Multivariate

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 $p_i$.

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.

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

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

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. $\alpha > 0$

### 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.