Monte Carlo–Markov Chains Statistical Methods

Welcome to the Monte Carlo–Markov Chains Statistical Methods series, where we explore the theory and practice of probabilistic inference and MCMC sampling.

Articles

1. What is Probability?
2. Random Variables and Sampling
   • Probability Density Function and Expectation
   • Sampling Methods for Simple Distributions
   • Introduction to Common Basic Sampling Algorithms
3. Monte Carlo Methods
   • Importance Sampling
   • Variance Reduction Techniques
4. Understanding Markov Chains
   • What is a Markov Process
   • Stationary Distribution and Convergence
   • Constructing Simple State Transition Processes
5. Introducing MCMC
   • Why do we need MCMC?
   • From Markov Chains to Sampling
   • Theory and Intuition
6. Metropolis Algorithm Explained: Implementation & Intuition
   • The Core Dilemma: Intractable Normalization Constants
   • Random Walk Metropolis Explained
   • Performance in High-Dimensional Distributions
7. The Metropolis-Hastings Algorithm: Breaking the Symmetry
   • Why do we need “asymmetric” proposals?
   • Derivation and intuition of the Hastings Correction
   • Practical Case: Solving boundary problems with Log-Normal proposals
8. Gibbs Sampling Explained: The Wisdom of Divide and Conquer
   • High-dimensional dilemmas and the “Manhattan Walk” intuition
   • Mathematical principle: Brook’s Lemma
   • Python implementation for discrete and continuous distributions
9. Deterministic Optimization Explained: The Mathematical Essence of Gradient Descent
   • Geometric Intuition of Convex vs. Non-Convex Optimization
   • Newton’s Method and Second-Order Approximation
   • Connection between Coordinate Descent and Gibbs Sampling
   • Pros and Cons of Steepest Descent
10. Markov Random Fields (MRF) and Image Denoising
    • From Time Series to Spatial Lattices: High-Dimensional Extension of Markov Property
    • Hammersley-Clifford Theorem and Gibbs Distribution
    • Ising Model and Image Denoising in Practice
11. Stochastic Optimization Explained: Simulated Annealing & Pincus Theorem
    • From Energy Minimization to Probability Maximization: The Physics of Annealing
    • High-Temp Exploration & Low-Temp Exploitation: Another Perspective on Metropolis
    • Pincus Theorem: Mathematical Proof of Convergence to Global Optimum
12. Bayesian Image Segmentation in Practice
    • The Game Between Likelihood and Prior: Modeling Homogeneity
    • MAP Estimation and Simulated Annealing
    • Hybrid Optimization Strategy: SA + ICM
13. Python Practical: MCMC Modeling