= 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.
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