Why do you maximize specificity for circumstantial findings? Tests for a medical condition can be classified into two categories:


*

*Tests on an asymptomatic (no reason to suspect condition) patient where the result is a circumstantial finding; and

*Tests on a symptomatic (prior reason to suspect condition) patient where the result is a diagnostic finding.


In the first case, the emphasis of any test should be on maximum specificity. For the second case, the emphasis of any test should be on maximum sensitivity. 
Intuitively, I understand why. However, can somebody give me a formal run-through, preferably from a Bayesian perspective, as to why this is the case?
References:
Slide presentation, "Categorizing variants after whole 
genome sequencing", J.S. Berg
Lang, E., Naraghi, R. (2005) Neurovascular relationship at the trigeminal root entry zone in persistent idiopathic facial pain: findings from MRI 3D visualisation. J Neurol Neurosurg Psychiatry 76:1506-1509.  Full text
 A: The simple way to work it out is that a test with high specificity has a high proportion of disease negative patients resulting in test negatives. Therefore high specificity implies low false positive rate. When you are not very convinced a patient has a condition you want to make sure a test is only likely to give a positive result if the patient has the condition.
From a Bayesian perspective, let $P(D+)$ be your prior belief that the patient has the disease. Let $T+$ and $T-$ denote the outcomes test positive and test negative respectively.
$sensitivity = P(T+\mid D+)$ and $specificity = P(T- \mid D-)$
By Bayes' rule
$P(D+ \mid T+) = \frac{P(T+ \mid D+)P(D+)}{P(T+)} = \frac{sensitivity \times P(D+)}{P(T+)} = \frac{sensitivity \times P(D+)}{P(T+ \mid D+)P(D+)+P(T+ \mid D-)P(D-)} = \frac{sensitivity \times P(D+)}{sensitivity \times P(D+)+(1-specificity)\times P(D-)}$
A similar formula can be constructed for $P(T- \mid D-)$.
You should observe that when the prior belief that the patient has the disease is small the denominator is strongly affected by $specificity$
