bayesian-inference

name	bayesian-inference
description	Bayesian parameter estimation with MCMC (emcee) and probabilistic programming (PyMC). Posterior distributions, corner plots, model evidence, convergence diagnostics. Use when you need full posterior distributions, not just point estimates.
category	physics
version	1.0.0
author	Synthetic Sciences
license	MIT
tags	["Bayesian","MCMC","Posterior","emcee","PyMC","Inference","Uncertainty","Physics"]
dependencies	["emcee>=3.1.0","corner>=2.2.0","scipy>=1.11.0","numpy>=1.24.0","matplotlib>=3.7.0"]

Bayesian Inference

Overview

Full Bayesian parameter estimation using MCMC sampling. Get posterior distributions (not just best-fit values), compute credible intervals, compare models via Bayes factors, and diagnose sampler convergence.

When to Use

• You need full posterior distributions (not just point estimates)
• Parameters are correlated and you want the joint posterior
• You want to compare models using Bayesian evidence
• Prior information is available and should be incorporated
• Error bars from least-squares seem unreliable

Do NOT Use When

• A simple curve_fit with error bars is sufficient (use physics-fitting)
• You have > 20 parameters (consider variational inference instead)
• Likelihood is cheap and you want speed (least-squares is faster)

Core Workflows

1. MCMC with emcee (Ensemble Sampler)

import numpy as np
import emcee
import corner
import matplotlib.pyplot as plt

# Model and data
def model(t, A, gamma, omega):
    return A * np.exp(-gamma * t) * np.cos(omega * t)

t_data = np.linspace(0, 10, 50)
y_true = model(t_data, 2.0, 0.3, 1.5)
y_err = 0.1 * np.ones_like(t_data)
y_data = y_true + y_err * np.random.randn(len(t_data))

# Log-probability = log-prior + log-likelihood
def log_prior(theta):
    A, gamma, omega = theta
    if A < 0 or gamma < 0 or omega < 0:
        return -np.inf
     A >   gamma >   omega > :
         -np.inf
       

 ():
    A, gamma, omega = theta
    y_model = model(t, A, gamma, omega)
    chi2 = np.(((y - y_model) / yerr)**)
     - * chi2 - np.(np.log(yerr)) -  * (y) * np.log(*np.pi)

 ():
    lp = log_prior(theta)
      np.isfinite(lp):
         -np.inf
     lp + log_likelihood(theta, t, y, yerr)


ndim = 
nwalkers = 
p0 = np.array([, , ]) +  * np.random.randn(nwalkers, ndim)


sampler = emcee.EnsembleSampler(nwalkers, ndim, log_probability,
                                 args=(t_data, y_data, y_err))
sampler.run_mcmc(p0, , progress=)


flat_samples = sampler.get_chain(discard=, thin=, flat=)
()