A complete reconstruction of the Monte Carlo Statistical Methods framework, implemented from scratch with clean visualizations, diagnostics, and mathematical writeups.
This repository builds intuition for the numerical engines behind Bayesian inference, probabilistic robotics, and reinforcement learning: sampling, importance sampling, variance reduction, Brownian motion, and advanced MCMC (MH, AM, DR, DRAM).
Everything is designed for clarity and insight, from foundational sampling methods to high-dimensional MCMC behavior on nonlinear targets.
Delayed Rejection Adaptive Metropolis sampling a nonlinear banana-shaped target.
This illustrates two-stage proposals, covariance adaptation, and real mixing behavior on warped geometries that appear in Bayesian robotics and RL when posterior surfaces are highly nonlinear.
This repository develops Monte Carlo techniques from first principles with a focus on:
- Inverse Transform
- Accept–Reject
- Exponential, Gamma, Logistic, Laplace sampling
- WLLN, SLLN, CLT
- Empirical versus theoretical convergence
- Weight behavior
- Tail mismatch
- Rare-event estimation
- Self-normalized IS
- Control variates
- Brownian motion scaling
- Metropolis–Hastings
- Adaptive Metropolis
- Delayed Rejection
- Delayed Rejection Adaptive Metropolis (DRAM)
- Burn-in
- Autocorrelation
- Integrated autocorrelation time
- Effective sample size (ESS)
- Mixing behavior
Each method includes animations, intuitive diagrams, code, and full mathematical analysis.
Transforms uniform draws (U \sim \text{Unif}(0,1)) through the inverse CDF (F^{-1}(U)) to generate exact Beta(10,3) samples.
Illustrates proposal mismatch, acceptance behavior, and envelope geometry.
Importance sampling is developed from basic motivation to full diagnostics, including weight stability, heavy-tail mismatch, and rare-event estimation.
Demonstrates how correlation structure can significantly reduce estimator variance.
From the scaling limit of random walks to full Brownian-motion sample paths.
Full implementations of:
- Metropolis–Hastings
- Adaptive Metropolis
- Delayed Rejection
- DRAM
Applied to Gaussian and banana-shaped targets.
Includes Laplace initialization, covariance adaptation, mixing behavior, autocorrelation diagnostics, and ESS analysis.
Laplace approximation guiding initialization.
- ch02_general_transforms.ipynb
- ch02_accept_reject.ipynb
- exponential.ipynb
- gamma.ipynb
- Cauchy tail motivation
- Rare-event estimation
- Self-normalized IS
- Control variates example
- Brownian motion
- MH diagnostics
- Adaptive Metropolis (Gaussian and Banana)
- Delayed Rejection
- DRAM
Mathematical writeups include:
- Sampling transformations
- Accept–Reject
- Importance Sampling
- Variance Reduction
- Brownian Motion
- Markov Chains
- Metropolis–Hastings
Written in a clean lecture-note style with derivations, proofs, and intuition.
MonteCarlo-Statistical-Methods/
│
├── animations/ # GIF/MP4 animations for visualizations
├── images/ # Figures for diagnostics and analysis
├── notebooks/ # Jupyter notebooks for each chapter
├── reports/ # Mathematical PDF writeups
├── src/ # Full code implementation
│ ├── sampling
│ ├── importance_sampling
│ ├── variance_reduction
│ ├── mcmc
│ ├── stochastic_processes
│ └── utils
└── README.md
I study and implement methods in optimization, control, robotics, Bayesian inference, and probabilistic reasoning.
- GitHub: https://github.com/SaiSampathKedari
- LinkedIn: https://linkedin.com/in/sai-sampath-kedari
- X: https://x.com/SSampathKedari
- Email: sampath@umich.edu