Bayesian Inference

The Bayesian page provides MCMC (Markov Chain Monte Carlo) inference with ArviZ diagnostic visualizations.

Overview

Bayesian inference provides:

  • Uncertainty quantification: Credible intervals for parameters

  • Full posterior: Complete probability distributions

  • Diagnostics: Convergence metrics (R-hat, ESS)

  • Model comparison: Evidence-based model selection

When to Use Bayesian

Use Bayesian inference when you need:

  • Parameter uncertainties

  • Confidence intervals for predictions

  • Robust handling of measurement noise

  • Publication-quality error bars

Configuration

MCMC Settings

Warmup Samples

Number of initial samples for adaptation (discarded). Recommended: 1000-2000

Posterior Samples

Number of samples for inference. Recommended: 2000-4000

Chains

Number of independent MCMC chains. Recommended: 4 (for R-hat calculation)

Random Seed

For reproducibility. Default: 42

Prior Configuration

Access via Prior Settings panel:

Default Priors

Automatic priors based on parameter bounds:

  • Uniform for bounded parameters

  • Log-uniform for scale parameters

Custom Priors

Specify distribution for each parameter:

  • Normal(mean, std)

  • LogNormal(mean, std)

  • Uniform(low, high)

  • HalfNormal(std)

Running Inference

Prerequisites

  1. Load data and select model

  2. Run NLSQ fit first (provides warm start)

  3. Configure MCMC settings

Starting MCMC

  1. Click “Run Bayesian Inference”

  2. Progress shows:

    • Warmup phase progress

    • Sampling phase progress

    • Current acceptance rate

  3. Wait for completion (typically 5-60 seconds)

Warm Start

Always run NLSQ first!

Warm start from NLSQ:

  • Initializes chains near optimum

  • Reduces warmup time

  • Improves convergence

  • Avoids poor initial samples

Diagnostics

ArviZ Plot Types

The ArviZ Canvas provides multiple diagnostic views:

Trace Plot (Default)

  • Left: Posterior density

  • Right: MCMC chain trace

  • Check: Chains should mix well

Pair Plot

  • Parameter correlations

  • Divergence markers

  • Check: No extreme correlations

Forest Plot

  • Credible intervals comparison

  • Point estimates with HDI

  • Check: Intervals don’t overlap zero (if significant)

Posterior Plot

  • Marginal distributions

  • HDI intervals

  • Check: Unimodal, well-defined peaks

Energy Plot

  • NUTS energy diagnostics

  • Check: Marginal and transition should overlap

Rank Plot

  • Chain rank statistics

  • Check: Uniform distribution across chains

ESS Plot

  • Effective sample size

  • Check: ESS > 400 for reliable estimates

Autocorrelation

  • Chain autocorrelation

  • Check: Quick decay to zero

Convergence Metrics

R-hat (Gelman-Rubin)

Potential scale reduction factor.

  • Good: R-hat < 1.01

  • Acceptable: R-hat < 1.1

  • Problematic: R-hat > 1.1 (more samples needed)

ESS (Effective Sample Size)

Independent samples equivalent.

  • Good: ESS > 400 for all parameters

  • Low ESS indicates autocorrelation

Divergences

Numerical integration issues.

  • Good: 0 divergences

  • Some: May indicate model issues

  • Many: Investigate model/data

Results

Posterior Summary

After inference completes:

  • Mean: Posterior mean

  • Std: Posterior standard deviation

  • HDI 3%/97%: 94% credible interval

  • MCSE: Monte Carlo standard error

Credible Intervals

Different credibility levels:

  • 50% HDI: Core distribution

  • 94% HDI: Standard reporting

  • 99% HDI: Conservative bounds

Prediction Intervals

Generate prediction uncertainty:

  1. Click “Plot Predictions”

  2. Shows fit with credible bands

  3. Inner band: Parameter uncertainty

  4. Outer band: Plus observation noise

Exporting Results

Posterior Samples

Export raw samples:

  1. Go to Export page

  2. Select Posterior Samples

  3. Choose format (CSV, HDF5)

ArviZ InferenceData

Export full ArviZ object:

  1. Select ArviZ InferenceData

  2. Save as NetCDF or pickle

  3. Load later for further analysis

Diagnostic Plots

Export diagnostic figures:

  1. Right-click any plot

  2. Select Export Plot

  3. Choose format (PNG, PDF, SVG)

Troubleshooting

High R-hat

R-hat > 1.1 indicates poor convergence:

  1. Increase warmup samples

  2. Increase total samples

  3. Check for multimodality

  4. Simplify model

Low ESS

Low effective samples:

  1. Increase total samples

  2. Check autocorrelation

  3. Tune step size (advanced)

Divergences

Many divergences indicate problems:

  1. Reparameterize model

  2. Adjust priors

  3. Check data quality

  4. Try non-centered parameterization

Slow Sampling

MCMC is slow:

  1. Ensure NLSQ warm start

  2. Reduce model complexity

  3. Use GPU if available

  4. Reduce number of chains

Best Practices

  1. Always warm start from NLSQ

  2. Use 4+ chains for diagnostics

  3. Check all diagnostic plots

  4. Report R-hat and ESS with results

  5. Use HDI, not mean±std for reporting

  6. Save InferenceData for reproducibility