gh-k-dense-ai-claude-scientific-skills-scientific-skills/skills/pymc/references/distributions.md

PyMC Distributions Reference

This reference provides a comprehensive 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 (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, mu, 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

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, sigma, nu) - Exponentially modified Gaussian

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

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

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

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

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.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: {0, 1, 2, ...}
• Common uses: Number of failures before first 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.MvStudentT(name, nu, mu, cov)

• Multivariate Student's t-distribution
• Parameters: nu (degrees of freedom), mu (location), cov (scale matrix)
• Common uses: Robust multivariate modeling

Specialized Multivariate Distributions

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

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

pm.Wishart(name, nu, V) - Wishart distribution (for covariance matrices)

pm.InverseWishart(name, nu, V) - Inverse Wishart distribution

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

pm.KroneckerNormal(name, mu, covs, 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.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, sigma, 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, sigma, 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, cov, init_dist)

• Multivariate Gaussian random walk

pm.GARCH11(name, omega, alpha_1, beta_1)

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

pm.EulerMaruyama(name, dt, sde_fn, sde_pars, 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
• Use for custom likelihood components or constraints

pm.Flat(name)

• Improper flat prior (constant density)
• Use sparingly; can cause sampling issues

pm.HalfFlat(name)

• Improper flat prior on positive reals
• Use sparingly; can cause sampling issues

Distribution Modifiers

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

• Truncate any distribution to specified bounds

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 matrices: Use LKJCorr
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')