Module bayesian

aprender

Module bayesian 

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Bayesian inference and probability methods.

This module implements classical and modern Bayesian methods including:

  • Conjugate priors (Beta-Binomial, Gamma-Poisson, Normal-InverseGamma, Dirichlet-Multinomial)
  • Bayesian regression and classification
  • Hierarchical models
  • MCMC methods (future)
  • Variational inference (future)
  • Gaussian processes (future)

§Design Philosophy

Following E.T. Jaynes’ “Probability Theory: The Logic of Science”, this module treats probability as an extension of logic under uncertainty. All methods provide:

  • Prior specification with sensible defaults
  • Exact posterior computation where tractable
  • Credible intervals (not confidence intervals)
  • Predictive distributions

§Example: Beta-Binomial (Binary Outcomes)

use aprender::bayesian::BetaBinomial;

// Prior: Beta(1, 1) = Uniform(0, 1)
let mut model = BetaBinomial::uniform();

// Observe 7 successes in 10 trials
model.update(7, 10);

// Posterior mean (point estimate)
let mean = model.posterior_mean();
assert!((mean - 0.6667).abs() < 0.001);

// 95% credible interval
let (lower, upper) = model.credible_interval(0.95).unwrap();
assert!(lower < mean && mean < upper);

// Predict next trial
let prob_success = model.posterior_predictive();

§Example: Gamma-Poisson (Count Data)

use aprender::bayesian::GammaPoisson;

// Prior: Gamma(0.001, 0.001) = weakly informative
let mut model = GammaPoisson::noninformative();

// Observe counts: [3, 5, 4, 6, 2] events per interval
model.update(&[3, 5, 4, 6, 2]);

// Posterior mean event rate
let mean = model.posterior_mean();
assert!((mean - 4.0).abs() < 0.5);

// 95% credible interval for rate
let (lower, upper) = model.credible_interval(0.95).unwrap();
assert!(lower < mean && mean < upper);

§Example: Normal-InverseGamma (Continuous Data)

use aprender::bayesian::NormalInverseGamma;

// Prior: weakly informative for both mean and variance
let mut model = NormalInverseGamma::new(0.0, 1.0, 3.0, 2.0).unwrap();

// Observe continuous data
model.update(&[4.2, 5.8, 6.1, 4.5, 5.0]);

// Posterior mean of μ (location)
let mean_mu = model.posterior_mean_mu();
assert!((mean_mu - 4.3).abs() < 0.3);

// Posterior mean of σ² (variance)
let mean_var = model.posterior_mean_variance().unwrap();
assert!(mean_var > 0.0);

// 95% credible interval for μ
let (lower, upper) = model.credible_interval_mu(0.95).unwrap();
assert!(lower < mean_mu && mean_mu < upper);

§Example: Dirichlet-Multinomial (Categorical Data)

use aprender::bayesian::DirichletMultinomial;

// Prior: uniform over 3 categories [A, B, C]
let mut model = DirichletMultinomial::uniform(3);

// Observe categorical data: 10 A's, 5 B's, 3 C's
model.update(&[10, 5, 3]);

// Posterior probabilities for each category
let probs = model.posterior_mean();
assert!((probs[0] - 11.0/21.0).abs() < 0.01); // P(A) ≈ 0.524
assert!((probs[1] - 6.0/21.0).abs() < 0.01);  // P(B) ≈ 0.286
assert!((probs[2] - 4.0/21.0).abs() < 0.01);  // P(C) ≈ 0.190

// Probabilities sum to 1
assert!((probs.iter().sum::<f32>() - 1.0).abs() < 1e-6);

// 95% credible intervals for each category
let intervals = model.credible_intervals(0.95).unwrap();
for i in 0..3 {
    assert!(intervals[i].0 < probs[i] && probs[i] < intervals[i].1);
}

Structs§

BayesianLinearRegression
Bayesian Linear Regression with analytical posterior.
BayesianLogisticRegression
Bayesian Logistic Regression with Laplace approximation.
BetaBinomial
Beta-Binomial conjugate prior for Bernoulli/Binomial likelihood.
DirichletMultinomial
Dirichlet-Multinomial conjugate prior for categorical data.
GammaPoisson
Gamma-Poisson conjugate prior for count data.
NormalInverseGamma
Normal-InverseGamma conjugate prior for normal data with unknown mean and variance.