I was thinking of this question lately, and I think I have an example where bayesian make sense, with the use a prior probability: the likelyhood ratio of a clinical test.

The example could be this one: the validity of the urine dipslide under daily practice conditions (Family Practice 2003;20:410-2). The idea is to see what a positive result of the urine dipslide imply on the diagnostic of urine infection. The likelyhood ratio of the positive result is:

$$LR(+) = \frac{test+|H+}{test+|H-} = \frac{Sensibility}{1-specificity} $$ with $H+$ the hypothesis of a urine infection, and $H-$ no urine infection. What Bayes tells us is

$$OR(+|test+) = LR(+) \times OR(+) $$ Where $OR$ is the odds ratio. $OR(+|test+)$ is the odd ratio of having a urine infection knowing that the test is positive, and $OR(+)$ the prior odd ratio.

The article gives that $LR(+) = 12.2$, and $LR(-) = 0.29$.

Here the prior knowledge is the probability to have a urine infection based on the clinical analysis of the potentially sick person before making the test. if the physician estimate that this probability is $p_{+} = 2/3$ based on observation, then a positive test leads the a post probability of $p_{+|test+} = 0.96$, and of $p_{+|test-} = 0.37$ if the test is negative.

Here the test is good to detect the infection, but not that good to discard the infection.