MCMC
Introduction
Markov Chain Monte-Carlo (MCMC) is a method of sampling from a distribution, when it is difficult to sample from the distribution directly. It can be used with Monte-Carlo Integration for numerical calculation of integrals.
It explores the state space ⚠ $X$ using a Markov Chain mechanism, and produces samples ⚠ $x_i$ which mimic samples drawn from the distribution ⚠ $p(x)$ directly. We assume that we can evaluate ⚠ $p(x)$ up to a normalizing constant, but it is difficult to draw samples from it.
Let us consider the situation when the number of possible states in a Markov chain is ⚠ $n$. Then we can choose an initial distribution ⚠ $q(x) \in \mathbb{R}^n$ and have a transition matrix ⚠ $T \in \mathbb{R}^{n \times n}$. Following the transition matrix at the first step, we get the distribution ⚠ $qT$. If we repeat the process several times and it converges to a distribution ⚠ $p(x)$, we say it converged to an invariant distribution. The process converges if the matrix ⚠ $T$ has properties of irreducibility and aperiodicity.
Definition 1. A transition matrix ⚠ $T$ is irreducible if for any state of the Markov chain there is a positive probability of visiting all other states.
Definition 2. A transition matrix ⚠ $T$ is aperiodical if the Markov chain does not get trapped into cycles.
A sufficient condition for the transition matrix to satisfy these requirenments is the reversibility condition:
⚠ $$p(x_i) T(x_j|x_i) = p(x_j) T(x_i|x_j).⚠ $$
In this case ⚠ $p(x)$ here is the invariant distribution. Indeed, summing both sides over ⚠ $x_j$ gives us
⚠ $$p(x_i) = \sum _{x_j} p(x_j) T(x_i|x_j).⚠ $$
In continuous state spaces the transition matrix ⚠ $T$ becomes an integral kernel ⚠ $K$ and ⚠ $p(x)$ becomes the corresponding eigenfunction.
MCMC samples are organized in a way that the desired ⚠ $p(x)$ is the invariant distribution:
The Metropolis-Hastings algorithm
Let us have a proposal distribution ⚠ $q(x^* | x)$ and invariant distribution ⚠ $p(x)$. An MH step involves sampling from the proposal distribution a candidate value ⚠ $x^*$. The Markov chain then moves towards ⚠ $x^*$ with acceptance probability ⚠ $A(x,x^*) = \min\{1,\frac{p(x^*)q(x|x^*)}{p(x)q(x^*|x)}\}$, otherwise it remains at ⚠ $x$.
⚠ $x_0$;
⚠ $i=0,...,N-1$}
⚠ $u \sim U_{[0,1]}$ from the uniform distribution.
⚠ $x^* \sim q(x^* |x_i)$ from the proposal distribution.
⚠ $u < A(x_i,x^*)$ then ⚠ $x_{i+1} = x^*$ else ⚠ $x_{i+1}=x_i$.
If ⚠ $q(x^* | x) = q(x | x^*)$ the MH algorithm transforms to the Metropolis algorithm.
Integration using MCMC
To calculate the integral ⚠ $\int f(x) p(x)dx$ it is possible to sample from the uniform distribution and calculate the integral using Monte-Carlo integration. But MCMC method of sampling from ⚠ $p(x)$ may lead to much better convergence. First one needs to perform burn-in state, when a Markov chain finds a way to sample from a distribution close to ⚠ $p(x)$ using MH algorithm. Then search for the distribution continues, but the sum ⚠ $f(x_i)$ is calculated. The resulting approximation of the integral is ⚠ $1/N_2 \sum_{j=N_1}^{N_1+N_2} f(x_j)$, where ⚠ $N_1$ is the length of the burn-in stage, and ⚠ $N_2$ is the length of the sampling stage.
MCMC is also used to solve optimization problems like ⚠ $\hat x = \arg\max_{x \in X} p(x)$ by applying a so-called simulated annealing technique.
Bibliography
- Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E. Equations of State Calculations by Fast Computing Machines. Journal of Chemical Physics, 21(6): 1087–1092 (1953).
- Hastings, W.K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97–109 (1970).
- Christophe Andrieu et al. An Introduction to MCMC for Machine Learning, Machine Learning, 50: 5–43 (2003).