🔗 GitHub Repository

Project Page:
github.com/SaiSampathKedari/MonteCarlo-Statistical-Methods

Monte Carlo Statistical Methods

Last Commit

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.

DRAM on a Banana Distribution

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.

Overview

This repository develops Monte Carlo techniques from first principles with a focus on:

Sampling

Inverse Transform
Accept–Reject
Exponential, Gamma, Logistic, Laplace sampling

Monte Carlo Estimation

WLLN, SLLN, CLT
Empirical versus theoretical convergence

Importance Sampling

Weight behavior
Tail mismatch
Rare-event estimation
Self-normalized IS

Variance Reduction

Control variates
Brownian motion scaling

Markov Chain Monte Carlo

Metropolis–Hastings
Adaptive Metropolis
Delayed Rejection
Delayed Rejection Adaptive Metropolis (DRAM)

Diagnostics

Burn-in
Autocorrelation
Integrated autocorrelation time
Effective sample size (ESS)
Mixing behavior

Each method includes animations, intuitive diagrams, code, and full mathematical analysis.

Sampling Visualizations

Inverse Transform Sampling for Beta Distribution

Transforms uniform draws (U \sim \text{Unif}(0,1)) through the inverse CDF (F^{-1}(U)) to generate exact Beta(10,3) samples.

Accept–Reject Sampling (Gaussian Target, Laplace Proposal)

Illustrates proposal mismatch, acceptance behavior, and envelope geometry.

Importance Sampling

Importance sampling is developed from basic motivation to full diagnostics, including weight stability, heavy-tail mismatch, and rare-event estimation.

Control Variates

Demonstrates how correlation structure can significantly reduce estimator variance.

Brownian Motion

From the scaling limit of random walks to full Brownian-motion sample paths.

MCMC Gallery

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.

Notebook Gallery

Sampling

ch02_general_transforms.ipynb
ch02_accept_reject.ipynb
exponential.ipynb
gamma.ipynb

Importance Sampling

Cauchy tail motivation
Rare-event estimation
Self-normalized IS

Variance Reduction

Control variates example

Stochastic Processes

Brownian motion

MCMC

MH diagnostics
Adaptive Metropolis (Gaussian and Banana)
Delayed Rejection
DRAM

PDF Chapter Library

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.

Project Structure

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

About Me

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

👤 About Me

Interested in robotics, control, and probabilistic methods for intelligent systems.

📫 Email: sampath@umich.edu
: linkedin.com/in/sai-sampath-kedari
𝕏 : @SaiSampathK
: github.com/SaiSampathKedari