Slice sampling algorithm A single transition $(x,u) \rightarrow (x',u')$ of a one-dimensional slice sampling algorithm has the following steps.
(1). evaluate $P^* (x)$
(2). draw a vertical coordinate $u' \sim$ Uniform$(0,P^* (x))$
(3). create a horizontal interval $(x_l, x_r)$ enclosing $x$
3a. draw $r \sim$ Uniform$(0,1)$
3b. $x_l := x-rw$
3c. $x_r := x+(1-r)w$
3d. while $(P^* (x_l) > u')$ ${x_l := x-rw}$
3e. while $(P^* (x_r) > u')$ ${x_r:= x+w}$
