Travis: Segment 39

To Calculate

1. Suppose the domain of a model are the five integers $x = \{1,2,3,4,5\}$, and that your proposal distribution is: "When $x_1 = 2,3,4$, choose with equal probability $x_2 = x_1 \pm 1$. For $x_1=1$ always choose $x_2 =2$. For $x_1=5$ always choose $x_2 =4$. What is the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$ for all the possible values of $x_1$ and $x_2$?

$q(x2|x1) =$

 0 1 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 1 0

and $q(x1|x2) = q(x2|x1)^{T}$

So, the ratio of q's is just the element wise division of $\frac{q(x1|x2)}{q(x2|x1)} = \frac{q(x1|x2)}{q(x1|x2)^T}$ which is

 0 2 0 0 0 0.5 0 1 0 0 0 1 0 1 0 0 0 1 0 0.5 0 0 0 1 0

2. Suppose the domain of a model is $-\infty < x < \infty$ and your proposal distribution is (perversely),

$q(x_2|x_1) = \begin{cases}\tfrac{7}{2}\exp[-7(x_2-x_1)],\quad & x_2 \ge x_1 \\ \tfrac{5}{2}\exp[-5(x_1-x_2)],\quad & x_2 < x_1 \end{cases}$

Sketch this distribution as a function of $x_2-x_1$. Then, write down an expression for the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$.