 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 $x=k$, soKeep 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 isFor $k>0$, $M_k<1$, 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.