Fokker-Planck-Kolmogorov Equation

This section mostly comes from SDE book

In order to analyze the stationary distribution of MALA, we can use Fokker-Planck-Kolmogorov equation

\[\frac{\partial p(x,t)}{\partial t} = -\sum_{i}\frac{\partial}{\partial x_{i}}[f_{i}(x,t)p(x,t)] + \frac{1}{2}\sum_{i,j}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\{L(x,t)QL^{T}(x,t)]_{ij}p(x,t)\} \]

In general solving this equation is hard, but \(p(x)\) is tractable if the SDE has the special form

\[dx = -\frac{1}{2}\nabla v(x)dt + d\beta \]

where \(\beta(t)\) is a Brownian motion with diffusion matrix \(Q=qI\).

The corresponding stationary distribution of \(p(x)\) is

\[p(x)\propto \exp(\frac{-v(x)}{q}) \]

Metropolis-adjusted Langevin algorithm

SDE

The corresponding SDE of MALA can be written as below.

\[d\theta_{t} = \frac{\sigma^{2}}{2}\nabla\log\pi(\theta_{t})dt + \sigma d B_{t} \]

Stationary distribution

Observe that the above SDE can be written in the specific form

\[\begin{align*} \nabla v(x) &= -\sigma^{2}\nabla\log\pi(\theta)\\ q &= \sigma^{2} \end{align*} \]

Therefore, \(v(x)=-\sigma^{2}\log\pi(\theta)\), and the stationary distribution is

\[p(x)\propto \exp(\frac{\sigma^{2}\log\pi(\theta)}{\sigma^{2}}) = \exp\log\pi(\theta)=\pi(\theta) \]