【Advanced】Markov Chain Monte Carlo (MCMC) Fitting in MATLAB

# [Advanced篇] Fitting with Markov Chain Monte Carlo (MCMC) in MATLAB

Markov Chain Monte Carlo (MCMC) is a powerful sampling method used to generate samples from complex probability distributions. It is based on the theory of Markov chains, where each state depends only on the previous state but is independent of all others. In MCMC, Markov chains are utilized to explore probability distributions and generate samples through Monte Carlo simulations.

The advantage of MCMC algorithms is their ability to handle high-dimensional and complex distributions that might be infeasible for other sampling methods. Additionally, MCMC samples can be used to approximate posterior distributions, which is particularly useful in Bayesian statistics.

# 2. Implementing MCMC in MATLAB

### 2.1 MCMC Toolbox in MATLAB

#### 2.1.1 Basic Functions and Usage

MATLAB offers a robust MCMC toolbox containing a set of functions and tools for MCMC sampling. These tools simplify the implementation of MCMC algorithms and provide efficient sampling methods.

To use the MCMC toolbox, you must first install the Statistics and Machine Learning Toolbox. Once installed, you can use the toolbox by following these steps:

1. Load the toolbox: `>> toolbox('stats')`
2. Create an MCMC object: `>> mcmc = mcmc('name', 'MetropolisHastings')`
3. Set sampling parameters: `>> mcmc.Parameters.NumIterations = 1000`
4. Run the sampling: `>> samples = mcmc.sample()`

#### 2.1.2 Common Sampling Algorithms

The MCMC toolbox provides various common sampling algorithms, including:

- Metropolis-Hastings algorithm
- Gibbs sampling algorithm
- Independent Metropolis-Hastings algorithm
- Adaptive Metropolis algorithm

These algorithms can be selected by setting the `'name'` parameter. For instance, to use the Metropolis-Hastings algorithm, you can set `>> mcmc = mcmc('name', 'MetropolisHastings')`.

### 2.2 Customizing MCMC Algorithms

Aside from using the MCMC toolbox, you can also customize your own MCMC algorithms. This offers greater flexibility and enables the implementation of more complex sampling schemes.

#### 2.2.1 Metropolis-Hastings Algorithm

The Metropolis-Hastings algorithm is a general-purpose MCMC algorithm applicable to various distributions. Its basic steps are:

1. Generate a candidate state `x'` from the current state `x`.
2. Calculate the acceptance probability of the candidate state: `alpha = min(1, p(x'|x) / p(x|x'))`.
3. Accept or reject the candidate state based on the acceptance probability.

The MATLAB code for this is:

function [samples, acceptanceRate] = metropolisHastings(targetPDF, initialState, numIterations)
     % Initialization
     samples = zeros(numIterations, 1);
     acceptanceRate = 0;
     % Current state
     x = initialState;
     for i = 1:numIterations
         % Generate candidate state
         xPrime = generateCandidate(x);
         % Calculate acceptance probability
         alpha = min(1, targetPDF(xPrime) / targetPDF(x));
         % Accept or reject candidate state
         if rand() < alpha
             x = xPrime;
             acceptanceRate = acceptanceRate + 1;
         end
         % Store sampling value
         samples(i) = x;
     end
end

#### 2.2.2 Gibbs Sampling Algorithm

The Gibbs sampling algorithm is a special type of MCMC algorithm used for sampling from mult