PyMC Distributions Reference

Targets PyMC 6.3.2, reviewed 2026-10-01. This reference provides a selected catalog of probability distributions available in PyMC, organized by category. Use this to select appropriate distributions for priors and likelihoods when building Bayesian models.

Continuous Distributions

Continuous distributions define probability densities over real-valued domains.

Common Continuous Distributions

**pm.Normal(name, mu, sigma)**

• Normal (Gaussian) distribution
• Parameters: mu (mean), sigma (standard deviation)
• Support: (-∞, ∞)
• Common uses: Default prior for unbounded parameters, likelihood for continuous data with additive noise

**pm.HalfNormal(name, sigma)**

• Half-normal distribution (positive half of normal)
• Parameters: sigma (underlying Normal scale; not the half-normal standard deviation)
• Support: [0, ∞)
• Common uses: Prior for scale/standard deviation parameters

**pm.Uniform(name, lower, upper)**

• Uniform distribution
• Parameters: lower, upper (bounds)
• Support: [lower, upper]
• Common uses: Weakly informative prior when parameter must be bounded

**pm.Beta(name, alpha, beta)**

• Beta distribution
• Parameters: alpha, beta (shape parameters)
• Support: [0, 1]
• Common uses: Prior for probabilities and proportions

**pm.Gamma(name, alpha, beta)**

• Gamma distribution
• Parameters: alpha (shape), beta (rate)
• Support: (0, ∞)
• Common uses: Prior for positive parameters, rate parameters

**pm.Exponential(name, lam)**

• Exponential distribution
• Parameters: lam (rate parameter)
• Support: [0, ∞)
• Common uses: Prior for scale parameters, waiting times

**pm.LogNormal(name, mu, sigma)**

• Log-normal distribution
• Parameters: mu, sigma (parameters of underlying normal)
• Support: (0, ∞)
• Common uses: Prior for positive parameters with multiplicative effects

**pm.StudentT(name, nu=nu, mu=mu, sigma=sigma)**

• Student's t-distribution
• Parameters: nu (degrees of freedom), mu (location), sigma (scale)
• Support: (-∞, ∞)
• Common uses: Robust alternative to normal for outlier-resistant models

**pm.Cauchy(name, alpha, beta)**

• Cauchy distribution
• Parameters: alpha (location), beta (scale)
• Support: (-∞, ∞)
• Common uses: Heavy-tailed alternative to normal

**pm.HalfStudentT(name, nu, sigma)**

• Positive half-Student-t distribution
• Common uses: Heavy-tailed prior for scale parameters

Specialized Continuous Distributions

**pm.Laplace(name, mu, b)** - Laplace (double exponential) distribution

**pm.AsymmetricLaplace(name, kappa, mu, b)** - Asymmetric Laplace distribution

**pm.InverseGamma(name, alpha, beta)** - Inverse gamma distribution

**pm.Weibull(name, alpha, beta)** - Weibull distribution for reliability analysis

**pm.Logistic(name, mu, s)** - Logistic distribution

**pm.LogitNormal(name, mu, sigma)** - Logit-normal distribution for (0,1) support

**pm.Pareto(name, alpha, m)** - Pareto distribution for power-law phenomena

**pm.ChiSquared(name, nu)** - Chi-squared distribution

**pm.ExGaussian(name, mu=mu, sigma=sigma, nu=nu)** - Exponentially modified Gaussian

**pm.VonMises(name, mu, kappa)** - Von Mises (circular normal) distribution

**pm.SkewNormal(name, mu=mu, sigma=sigma, alpha=alpha)** - Skew-normal distribution

**pm.Triangular(name, lower=lower, c=c, upper=upper)** - Triangular distribution

**pm.Gumbel(name, mu, beta)** - Gumbel distribution for extreme values

**pm.PolyaGamma(name, h, z)** - Polya-gamma distribution for data augmentation patterns

**pm.Rice(name, nu, sigma)** - Rice (Rician) distribution

**pm.Moyal(name, mu, sigma)** - Moyal distribution

**pm.Kumaraswamy(name, a, b)** - Kumaraswamy distribution (Beta alternative)

**pm.Wald(name, mu, lam)** - Wald / inverse Gaussian distribution

**pm.Interpolated(name, x_points, pdf_points)** - Custom distribution from interpolation

Discrete Distributions

Discrete distributions define probabilities over integer-valued domains.

Common Discrete Distributions

**pm.Bernoulli(name, p)**

• Bernoulli distribution (binary outcome)
• Parameters: p (success probability)
• Support: {0, 1}
• Common uses: Binary classification, coin flips

**pm.Binomial(name, n, p)**

• Binomial distribution
• Parameters: n (number of trials), p (success probability)
• Support: {0, 1, ..., n}
• Common uses: Number of successes in fixed trials

**pm.Poisson(name, mu)**

• Poisson distribution
• Parameters: mu (rate parameter)
• Support: {0, 1, 2, ...}
• Common uses: Count data, rates, occurrences

**pm.Categorical(name, p)**

• Categorical distribution
• Parameters: p (probability vector)
• Support: {0, 1, ..., K-1}
• Common uses: Multi-class classification

**pm.DiscreteUniform(name, lower, upper)**

• Discrete uniform distribution
• Parameters: lower, upper (bounds)
• Support: {lower, ..., upper}
• Common uses: Uniform prior over finite integers

**pm.NegativeBinomial(name, mu, alpha)**

• Negative binomial distribution
• Parameters: mu (mean), alpha (dispersion)
• Support: {0, 1, 2, ...}
• Common uses: Overdispersed count data

**pm.Geometric(name, p)**

• Geometric distribution
• Parameters: p (success probability)
• Support: {1, 2, 3, ...}
• Common uses: Trial count through the first success (includes that success)

Specialized Discrete Distributions

**pm.BetaBinomial(name, alpha, beta, n)** - Beta-binomial (overdispersed binomial)

**pm.HyperGeometric(name, N, k, n)** - Hypergeometric distribution

**pm.DiscreteWeibull(name, q, beta)** - Discrete Weibull distribution

**pm.OrderedLogistic(name, eta, cutpoints)** - Ordered logistic for ordinal data

**pm.OrderedProbit(name, eta, cutpoints)** - Ordered probit for ordinal data

Multivariate Distributions

Multivariate distributions define joint probability distributions over vector-valued random variables.

Common Multivariate Distributions

**pm.MvNormal(name, mu, cov)**

• Multivariate normal distribution
• Parameters: mu (mean vector), cov (covariance matrix)
• Common uses: Correlated continuous variables, Gaussian processes

**pm.Dirichlet(name, a)**

• Dirichlet distribution
• Parameters: a (concentration parameters)
• Support: Simplex (sums to 1)
• Common uses: Prior for probability vectors, topic modeling

**pm.Multinomial(name, n, p)**

• Multinomial distribution
• Parameters: n (number of trials), p (probability vector)
• Common uses: Count data across multiple categories

**pm.DirichletMultinomial(name, n, a)**

• Dirichlet-multinomial distribution
• Common uses: Overdispersed categorical counts

**pm.MvStudentT(name, nu=nu, mu=mu, scale=scale)**

• Multivariate Student's t-distribution
• Parameters: nu (degrees of freedom), mu (location), scale (scale matrix; covariance is nu/(nu-2) times this matrix for nu > 2)
• Common uses: Robust multivariate modeling

Specialized Multivariate Distributions

**pm.LKJCorr(name, n, eta)** - LKJ correlation matrix prior (for correlation matrices)

**pm.LKJCholeskyCov(name, n=n, eta=eta, sd_dist=sd_dist)** - LKJ prior with Cholesky decomposition

**pm.OrderedMultinomial(name, eta, cutpoints, n)** - Ordered multinomial outcomes

**pm.StickBreakingWeights(name, alpha, K)** - Stick-breaking weights for mixture models

**pm.ZeroSumNormal(name, sigma)** - Normal prior constrained to sum to zero

**pm.Wishart(name, nu, V)** - Wishart distribution (for covariance matrices; prefer LKJ-based priors for most covariance models)

There is no public pm.InverseWishart in PyMC 6.3.2. Prefer an explicit scientifically justified covariance model, usually LKJ with marginal scales.

**pm.WishartBartlett(name, S, nu)** - Wishart with Bartlett decomposition

**pm.MatrixNormal(name, mu=mu, rowcov=rowcov, colcov=colcov)** - Matrix normal distribution

**pm.KroneckerNormal(name, mu=mu, covs=covs, sigma=sigma)** - Kronecker-structured normal

**pm.CAR(name, mu, W, alpha, tau)** - Conditional autoregressive (spatial)

**pm.ICAR(name, W, sigma)** - Intrinsic conditional autoregressive (spatial)

Mixture Distributions

Mixture distributions combine multiple component distributions.

**pm.Mixture(name, w, comp_dists)**

• General mixture distribution
• Parameters: w (weights), comp_dists (component distributions)
• Common uses: Clustering, multi-modal data

**pm.NormalMixture(name, w, mu, sigma)**

• Mixture of normal distributions
• Common uses: Mixture of Gaussians clustering

Zero-Inflated and Hurdle Models

**pm.ZeroInflatedPoisson(name, psi, mu)** - Excess zeros in count data

**pm.ZeroInflatedBinomial(name, psi, n, p)** - Zero-inflated binomial

**pm.ZeroInflatedNegativeBinomial(name, psi, mu, alpha)** - Zero-inflated negative binomial

**pm.HurdlePoisson(name, psi, mu)** - Hurdle Poisson (two-part model)

**pm.HurdleNegativeBinomial(name, psi, mu, alpha)** - Hurdle negative binomial for overdispersed counts with structural zeros

**pm.HurdleGamma(name, psi, alpha, beta)** - Hurdle gamma

**pm.HurdleLogNormal(name, psi, mu, sigma)** - Hurdle log-normal

Time Series Distributions

Distributions designed for temporal data and sequential modeling.

**pm.AR(name, rho=rho, sigma=sigma, init_dist=init_dist)**

• Autoregressive process
• Parameters: rho (AR coefficients), sigma (innovation std), init_dist (initial distribution)
• Common uses: Time series modeling, sequential data

**pm.GaussianRandomWalk(name, mu=mu, sigma=sigma, init_dist=init_dist)**

• Gaussian random walk
• Parameters: mu (drift), sigma (step size), init_dist (initial value)
• Common uses: Cumulative processes, random walk priors

**pm.MvGaussianRandomWalk(name, mu=mu, cov=cov, init_dist=init_dist)**

• Multivariate Gaussian random walk

**pm.MvStudentTRandomWalk(name, nu=nu, mu=mu, scale=scale, init_dist=init_dist)**

• Heavy-tailed multivariate random walk

**pm.GARCH11(name, omega=omega, alpha_1=alpha_1, beta_1=beta_1, initial_vol=initial_vol)**

• GARCH(1,1) volatility model
• Common uses: Financial time series, volatility modeling

**pm.EulerMaruyama(name, dt, sde_fn, sde_pars, init_dist=init_dist)**

• Stochastic differential equation via Euler-Maruyama discretization
• Common uses: Continuous-time processes

Special Distributions

**pm.Deterministic(name, var)**

• Deterministic transformation (not a random variable)
• Use for computed quantities derived from other variables

**pm.Potential(name, logp)**

• Add arbitrary log-probability contribution; ignored by prior/posterior forward sampling, so a Potential alone does not define a generative likelihood
• Use for custom likelihood components or constraints

**pm.Flat(name)**

• Improper flat prior (constant density); cannot draw prior predictive samples
• Use sparingly; can cause sampling issues

**pm.HalfFlat(name)**

• Improper flat prior on positive reals; cannot draw prior predictive samples
• Use sparingly; can cause sampling issues

Distribution Modifiers

**pm.Truncated(name, dist, lower, upper)**

• Truncate a supported univariate .dist() distribution to specified bounds; support and inverse-CDF/rejection-sampling limitations depend on the base distribution

**pm.Censored(name, dist, lower, upper)**

• Handle censored observations (observed bounds, not exact values)

**pm.CustomDist(name, ..., logp, random)**

• Define custom distributions with user-specified log-probability and random sampling functions

**pm.Simulator(name, fn, params, ...)**

• Custom distributions via simulation (for likelihood-free inference)

Usage Tips

Choosing Priors

1. Scale parameters (σ, τ): Use HalfNormal, HalfCauchy, Exponential, or Gamma
2. Probabilities: Use Beta or Uniform(0, 1)
3. Unbounded parameters: Use Normal or StudentT (for robustness)
4. Positive parameters: Use LogNormal, Gamma, or Exponential
5. Correlation/covariance matrices: Prefer LKJCholeskyCov for covariance models; use LKJCorr when only correlations are needed
6. Count data: Use Poisson or NegativeBinomial (for overdispersion)

Shape Broadcasting

PyMC distributions support NumPy-style broadcasting. Use the shape parameter to create vectors or arrays of random variables:

# Vector of 5 independent normals
beta = pm.Normal('beta', mu=0, sigma=1, shape=5)

# 3x4 matrix of independent gammas
tau = pm.Gamma('tau', alpha=2, beta=1, shape=(3, 4))

Using dims for Named Dimensions

Instead of shape, use dims for more readable models:

with pm.Model(coords={'predictors': ['age', 'income', 'education']}) as model:
    beta = pm.Normal('beta', mu=0, sigma=1, dims='predictors')

Parameterization and scientific caveats

• Entries list important parameters, not exhaustive positional signatures. Prefer keywords: several distributions have keyword-only parameters. Catalog names and constructor signatures were checked against released source; specialized distributions were not all fit or sampled in this review.
• NegativeBinomial has variance mu + mu**2 / alpha. InverseGamma beta is a scale, while Gamma beta is a rate. PolyaGamma needs the optional polyagamma package for relevant numerical operations.
• In zero-inflated models psi is the weight on the ordinary count component, not the probability of a structural zero. In hurdle models it is the weight on the positive component. HurdleGamma/HurdleLogNormal are intended as observed likelihoods or forward-sampled RVs; ordinary MCMC does not correctly sample their mixed point-mass/continuous latent support.
• LKJCholeskyCov(..., compute_corr=True) returns (chol, corr, stds) by default. LKJCorr ordinarily returns packed upper-triangle correlations, not a full correlation matrix. eta=1 is uniform over correlation matrices, not uniform independent marginal correlations for arbitrary dimension.
• pm.Censored is not pm.Truncated: censoring puts probability mass at recorded limits; truncation conditions on being included inside limits. Selection and measurement processes decide the correct likelihood.
• A density definition without a random/generative implementation can support fitting but not meaningful forward simulation. Check CustomDist event/batch shapes, log-density normalization and gradient behavior independently.
• Ordinary dims names annotate arrays; they do not align symbolic arithmetic. Multivariate event dimensions and batch dimensions have different meanings.

Sources: distribution catalog, released distribution source, forward sampling source.