amari-probabilistic

Probability theory on geometric algebra spaces for the Amari mathematical computing library.

Overview

amari-probabilistic provides probability distributions, stochastic processes, MCMC sampling, and Bayesian inference for multivector-valued random variables in Clifford algebras Cl(P,Q,R).

This crate builds on:

• amari-measure for measure-theoretic foundations
• amari-info-geom for Fisher-Riemannian geometry integration
• amari-core for the underlying Multivector type

Features

Distributions

Probability distributions over multivector spaces:

use amari_probabilistic::distribution::{Distribution, GaussianMultivector};
use amari_core::Multivector;

// Standard Gaussian on Cl(2,0,0)
let gaussian = GaussianMultivector::<2, 0, 0>::standard();

// Sample from the distribution
let mut rng = rand::thread_rng();
let sample: Multivector<2, 0, 0> = gaussian.sample(&mut rng);

// Evaluate log-probability
let log_p = gaussian.log_prob(&sample).unwrap();

MCMC Sampling

Metropolis-Hastings and Hamiltonian Monte Carlo:

use amari_probabilistic::sampling::{MetropolisHastings, Sampler};

// Create sampler with proposal standard deviation
let mut sampler = MetropolisHastings::new(&target_distribution, 0.1);

// Run MCMC with burnin
let samples = sampler.run(&mut rng, 1000, 100)?;

// Check diagnostics
let diag = sampler.diagnostics();
println!("Acceptance rate: {}", diag.acceptance_rate);

Stochastic Differential Equations

SDE solvers for multivector-valued stochastic processes:

use amari_probabilistic::stochastic::{
    GeometricBrownianMotion, EulerMaruyama, SDESolver
};

// Geometric Brownian motion: dX = μX dt + σX dW
let gbm = GeometricBrownianMotion::<2, 0, 0>::new(0.05, 0.2)?;

// Solve with Euler-Maruyama
let solver = EulerMaruyama::new();
let initial = Multivector::scalar(1.0);
let path = solver.solve(&gbm, initial, 0.0, 1.0, 100, &mut rng)?;

Bayesian Inference

Bayesian models on geometric algebra spaces:

use amari_probabilistic::bayesian::{BayesianGA, GaussianPrior};

// Define prior
let prior = GaussianPrior::<2, 0, 0>::diffuse();

// Define likelihood function
let likelihood = |theta: &Multivector<2, 0, 0>, data: &[Multivector<2, 0, 0>]| {
    // Compute log-likelihood
    Ok(log_lik)
};

// Create Bayesian model
let mut model = BayesianGA::new(prior, likelihood);
model.observe(observations);

// Sample from posterior
let posterior_samples = model.sample_posterior(&mut rng, 1000, 100, 0.1)?;

Uncertainty Propagation

Propagate uncertainty through geometric operations:

use amari_probabilistic::uncertainty::{UncertainMultivector, LinearPropagation};

// Create uncertain multivector with mean and covariance
let um = UncertainMultivector::diagonal(mean, &variances)?;

// Propagate through operations
let prop = LinearPropagation::new();
let result = prop.scalar_multiply(&um, 2.0);

Module Structure

• distribution: Core Distribution trait and implementations

  • GaussianMultivector: Gaussian distribution on multivector space
  • UniformMultivector: Uniform distribution on hypercube
  • GradeProjectedDistribution: Marginal distribution on specific grades

• random: Random variable traits and moment computation

  • GeometricRandomVariable: Trait for RVs with computable moments
  • CovarianceMatrix: Covariance representation
  • MomentComputer: Monte Carlo moment estimation

• sampling: MCMC and sampling algorithms

  • MetropolisHastings: Random-walk Metropolis
  • HamiltonianMonteCarlo: HMC with leapfrog integration
  • RejectionSampling: Accept-reject sampling

• stochastic: Stochastic processes and SDE solvers

  • StochasticProcess: Trait for SDEs
  • EulerMaruyama: First-order SDE solver
  • Milstein: Higher-order SDE solver
  • WienerProcess, GeometricBrownianMotion, OrnsteinUhlenbeck

• bayesian: Bayesian inference

  • BayesianGA: Bayesian model with prior and likelihood
  • JeffreysPrior: Non-informative prior
  • GaussianPrior: Conjugate Gaussian prior
  • PosteriorSampler: Posterior analysis utilities

• uncertainty: Uncertainty propagation

  • LinearPropagation: Jacobian-based propagation
  • UnscentedTransform: Sigma-point propagation

• monte_carlo: Monte Carlo integration extensions

• phantom: Compile-time distribution property markers

  • Bounded, LightTailed, RotorValued, etc.

Error Handling

The crate uses ProbabilisticError for error handling:

pub enum ProbabilisticError {
    NotNormalized { total: f64 },
    OutOfSupport { sample: String },
    SamplerNotConverged { iterations: usize, reason: String },
    SDEInstability { time: f64, details: String },
    InvalidParameters { description: String },
    // ...
}

Feature Flags

• std (default): Standard library support
• parallel: Parallel sampling with Rayon
• gpu: GPU-accelerated sampling via amari-gpu

Mathematical Background

Distributions on Cl(P,Q,R)

A probability distribution on the Clifford algebra Cl(P,Q,R) assigns probabilities to multivector-valued events. The dimension of the space is 2^(P+Q+R).

Stochastic Differential Equations

SDEs on multivector spaces take the form:

dX = μ(X,t) dt + σ(X,t) dW

where μ is the drift (a multivector), σ is the diffusion coefficient, and W is a Wiener process.

Bayesian Inference

For a parameter θ (a multivector), data D, prior p(θ), and likelihood p(D|θ):

p(θ|D) ∝ p(D|θ) p(θ)

License

MIT OR Apache-2.0