zang-langyan / Markov-Chain-Monte-Carlo-MCMC

Markov Chain Monte Carlo MCMC methods are implemented in various languages (including R, Python, Julia, Matlab)

马尔可夫链蒙特卡洛方法 Markov Chain Monte Carlo（MCMC）

本程辑包仅为学习和练习之用

This repository is only for study and practice purposes

目录 Table of Contents

• 马尔可夫链蒙特卡洛方法 Markov Chain Monte Carlo（MCMC）
• 目录 Table of Contents
• 内容
• Overview
• Usage
  • Metropolis
    • Python
    • Julia
    • Matlab
    • R

内容

• Metropolis Algorithm
• Gibbs Sampling (待补充)
• Hamiltonian Monte Carlo (HMC) (目前只有python版本)

Overview

• Metropolis Algorithm
• Gibbs Sampling (to be added)
• Hamiltonian Monte Carlo (HMC) (only python version)

Usage

Metropolis

Python

>>> import metropolis
>>> from scipy.special import beta
>>> density = lambda x: x**14 * (1-x)**6 / beta(15,7) # Beta(15,7) distribution density function
>>> d = metropolis.MCMC(density, space = [0,1], burnin = 5, seed = 72)
>>> result = d.metropolis(chain = 50, theta_init = 0.1)
>>> result
[0.8176960093922774, 0.6965658096789994, 0.7918980615882665, 0.7742394795862185, 0.7742394795862185, 0.6215414421599866, 0.6215414421599866, 0.6468248283946754, 0.7523307146724492, 0.7523307146724492, 0.7680822171469903, 0.6717376155310788, 0.6717376155310788, 0.8189878864825765, 0.7533257832372013, 0.7648206889254298, 0.7648206889254298, 0.7648206889254298, 0.7648206889254298, 0.7648206889254298, 0.7967947965010628, 0.707139903476412, 0.707139903476412, 0.707139903476412, 0.707139903476412, 0.7949590848197712, 0.48018105229278457, 0.48018105229278457, 0.6163978253871556, 0.6163978253871556, 0.6163978253871556, 0.6241841537823003, 0.6241841537823003, 0.6241841537823003, 0.6241841537823003, 0.6922332333961135, 0.8204027203103916, 0.7521267229207833, 0.7521267229207833, 0.808566620237602, 0.808566620237602, 0.6460288022318678, 0.5452360032045938, 0.5452360032045938, 0.5934357613729606]

Julia

julia> p(θ) = Distributions.pdf(Beta(15,7),θ) # define a density function

julia> metropolis(p, 5000, 0.5; jump = Normal(0,0.2), space_min = 0, space_max = 1, burnin = 1000, rng = 42)
4000-element Vector{Float64}:
 0.7254870615709366
 0.5374079418869023
 0.5374079418869023
 0.5161750364065404
 0.7377560837277216
 0.7377560837277216
 0.8127557482035582
 ⋮
 0.7166820320569407
 0.7166820320569407
 0.7166820320569407
 0.7166820320569407
 0.7166820320569407
 0.7586353644413187

julia> @time metropolis(θ -> Distributions.pdf(Beta(15,7),θ), 10000, 0.5; jump = Normal(0,0.5), space_min = 0, space_max = 1, burnin = 1000, rng = nothing)
0.061715 seconds (103.38 k allocations: 3.814 MiB, 88.90% compilation time)
9000-element Vector{Float64}:
 0.6057362032299476
 0.6057362032299476
 0.6057362032299476
 0.6057362032299476
 0.6057362032299476
 0.6057362032299476
 0.6057362032299476
 ⋮
 0.4516105573548591
 0.4516105573548591
 0.4516105573548591
 0.5014099540247418
 0.6720304713998495
 0.6720304713998495

Matlab

>> pd = makedist('Weibull','A',5,'B',2)
>> chain = metropolis(@(x) pdf(pd,x),5000,0.8,makedist('Normal','mu',0,'sigma',0.9),[0,Inf],1000,'shuffle');

R

# Example
# posterior function 
# prior Beta(a = 1, b = 1)
# likelihood Bernoulli: N = 20, z = 14
posterior <- function(theta, a, b, N, z) {
  dbeta(theta, z + a, N - z + b)
}
a = 1
b = 1
N = 20 
z = 14
posterior_func <- function(theta) posterior(theta, a, b, N, z)

dposterior1 <- metropolis(chain = 5000, theta_cur = 0.1, dfunc = posterior_func, space_min = 0, space_max = 1, burnin = 2500)
dposterior2 <- metropolis(chain = 5000, theta_cur = 0.9, dfunc = posterior_func, space_min = 0, space_max = 1, burnin = 2500)
dposterior3 <- metropolis(chain = 5000, theta_cur = 0.5, dfunc = posterior_func, space_min = 0, space_max = 1, burnin = 2500)