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Samplers

MCMC and nested sampling algorithms for posterior exploration and model comparison.

Overview

Sampler Type Use Case
MetropolisHastings MCMC Posterior exploration, uncertainty quantification
DynamicNestedSampling Nested Model evidence, Bayes factors

Samplers complement optimisers by providing full posterior distributions rather than point estimates.

Metropolis-Hastings

MCMC sampling for exploring parameter posterior distributions.

import diffid as chron

# Will be available in a future release
sampler = (
    diffid.MetropolisHastings()
    .with_max_iter(10000)
    .with_step_size(0.1)
    .with_burn_in(1000)
    .with_seed(42)
)

result = sampler.run(problem, initial_guess)

# Access samples
print(result.samples.shape)  # (n_samples, n_params)
print(result.acceptance_rate)  # Target: 0.2-0.4

When to Use

Advantages:

  • Provides full posterior distribution
  • Quantifies parameter uncertainty
  • Captures correlations between parameters
  • Enables credible intervals

Limitations:

  • Requires many function evaluations
  • Need to assess convergence
  • Requires likelihood (GaussianNLL cost metric)

Typical Use Cases:

  • Uncertainty quantification
  • Confidence intervals
  • Posterior predictive distributions
  • Parameter correlations

Dynamic Nested Sampling

Nested sampling for calculating model evidence (marginal likelihood) for model comparison.

import diffid as chron

# Will be available in a future release
sampler = (
    diffid.DynamicNestedSampling()
    .with_max_iter(5000)
    .with_n_live_points(500)
    .with_seed(42)
)

result = sampler.run(problem, initial_guess)

# Access evidence
print(f"Log evidence: {result.log_evidence:.2f}")
print(f"Error: {result.evidence_error:.2f}")

# Posterior samples also available
print(result.samples.shape)

When to Use

Advantages:

  • Calculates marginal likelihood (evidence)
  • Enables model comparison via Bayes factors
  • Provides posterior samples as byproduct
  • Efficient for multi-modal posteriors

Limitations:

  • More expensive than MCMC
  • Requires careful tuning of live points
  • Needs likelihood (GaussianNLL cost metric)

Typical Use Cases:

  • Model comparison
  • Bayes factors
  • Evidence calculation
  • Multi-modal posteriors

Optimisers vs Samplers

Optimisers Samplers
Output Single best parameters Distribution of parameters
Use case Point estimate Uncertainty quantification
Cost metric SSE, RMSE, GaussianNLL GaussianNLL (required)
Computational cost Lower Higher
Uncertainty None Full posterior

When to Use Which

graph TD
    A[Start] --> B{Need uncertainty?}
    B -->|No| C[Use Optimiser]
    B -->|Yes| D{Need model comparison?}
    D -->|No| E[Use MCMC]
    D -->|Yes| F[Use Nested Sampling]

Cost Metric Requirement

Warning

Samplers require a probabilistic cost metric (typically GaussianNLL):

# Required for samplers
builder = (
    diffid.DiffsolBuilder()
    .with_diffsl(dsl)
    .with_data(data)
    .with_parameter("k", 1.0)
    .with_cost_metric(diffid.GaussianNLL())  # Required!
)

SSE and RMSE cannot be used with samplers as they lack probabilistic interpretation.

Workflow Example

Typical workflow: optimise first, then sample for uncertainty:

import diffid as chron

# 1. Build problem with GaussianNLL
builder = (
    diffid.DiffsolBuilder()
    .with_diffsl(dsl)
    .with_data(data)
    .with_parameter("k", 1.0)
    .with_cost_metric(diffid.GaussianNLL())
)
problem = builder.build()

# 2. Find MAP estimate with optimiser
optimiser = diffid.CMAES().with_max_iter(1000)
opt_result = optimiser.run(problem, [1.0])

print(f"MAP estimate: {opt_result.x}")

# 3. Sample around MAP for uncertainty (future API)
# sampler = diffid.MetropolisHastings().with_max_iter(10000)
# sample_result = sampler.run(problem, opt_result.x)
# print(f"Posterior mean: {sample_result.samples.mean(axis=0)}")
# print(f"Posterior std: {sample_result.samples.std(axis=0)}")

Tutorials

The followings present sampler functionality:

See Also