Posterior odds are monotone under a monotone likelihood ratio

For a family with a monotone likelihood ratio in $T$, a prior density $\pi$ and cutoff $a$, the posterior odds of $\theta >a$ against $\theta \le a$ are nondecreasing in $T$. To prove it, write the unnormalized masses as $A(t)$ and $B(t)$. For $t_2>t_1$, the difference $A(t_2)B(t_1)-A(t_1)B(t_2)$ is the double integral over $\theta >a\ge \eta$ of $\pi (\theta )\pi (\eta )[f_{\theta }(t_2)f_{\eta }(t_1)-f_{\theta }(t_1)f_{\eta }(t_2)]$. Each integrand is nonnegative by the likelihood-ratio ordering. Common factors in the full-data likelihood cancel from the odds, so the same argument works when $T$ is a statistic rather than the complete observation. Interpret zero denominators by extended odds whenever the posterior is defined.