= Solution
The rejection rule can use a tighter envelope than 1. For $k>0$, differentiate $\log(e^{-x}x^k/k!)=-x+k\log x-\log(k!)$. Its derivative $-1+k/x$ changes from positive to negative at $x=k$, so
$$
M_k=\sup_{x\ge0}e^{-x}\frac{x^k}{k!}=e^{-k}\frac{k^k}{k!},\qquad M_0=1.
$$
Keep the same joint prior proposal but accept with <probability> $w_k(\theta,T)/M_k$. Multiplication of the acceptance function by this constant leaves the accepted <probability density function> unchanged. The improved rate is
$$
\boxed{A_k^{\rm improved}=\frac{Z_k}{M_k}.}
$$
For $k>0$, $M_k<1$, so this is a genuine improvement; for large $k$, <Stirling's formula> gives $M_k\sim(2\pi k)^{-1/2}$. This is the best global constant envelope for an unrestricted positive prior proposal, because $L$ has support $(0,\infty)$ and $\theta L/2$ can approach $k$. For $k=0$ there is no improvement from this constant scaling. Further gains can come from proposals concentrating near $\theta L/2=k$, or integrating out the times first, but such changes require the appropriate proposal-density ratio and envelope rather than an uncorrected alteration of the sampling law.
Back to article page
