Poisson-uniform posterior mean ($\mathbb{E}[\Lambda \mid x_1,\dots ,x_n]=\frac{{\int }_l^u{t}^{s+1}{e}^{-nt}dt}{{\int }_l^u{t}^s{e}^{-nt}dt}$)

For a uniform prior distribution on $(l,u)$ with $0<l<u$, conditionally independent Poisson distribution observations have sufficient count $s={\sum }_i{x}_i$. Their Bayesian posterior density is proportional to $t^s{e}^{-nt}$ on that bounded interval. The displayed posterior mean is the Bayes estimator under squared error loss, and is generally not the affine Bühlmann credibility estimate.