For implementing the MH algorithm, we must provide a “transition kernel”, Q. A transition kernel is simply a way of moving, randomly, to a new position in space (y say), given a current position (x say). That is, Q is a distribution on y given x, and we will write it Q(y|x).
The MH algorithm for sampling from a target distribution $\pi$, using transition kernel $Q$, consists of the following steps:
- Initialize, $X_{1}=x_{1}$ say.
- For $t=1,2, \ldots$
- sample $y$ from $Q\left(y \mid x_{t}\right)$. Think of $y$ as a "proposed" value for $x_{t+1}$.
- Compute $$ A=\min \left(1, \frac{\pi(y) Q\left(x_{t} \mid y\right)}{\pi\left(x_{t}\right) Q\left(y \mid x_{t}\right)}\right) $$ $A$ is often called the "acceptance probabilty". o with probability $A$ "accept" the proposed value, and set $x_{t+1}=y .$ Otherwise set $x_{t+1}=x_{t} .$
In all examples I find just the case when Q is a symmetric kernel. I want somes examples of a random walk with a non symmetric kernel. and how can we do the simulation in this case?