Computing flux upper-limits for non-detections

If a source is not detected above some flux threshold (when expected to be present at a target position), how can one assign a plausible upper limit to its flux such that (i) it has precise probabilistic meaning; (ii) has a degree of confidence that is not subjective or arbitrarily conservative; and (iii) has no dependence on any prior knowledge about the flux? Unfortunately, the astronomical literature is not consistent in how upper limits are derived, and there are many confusing interpretations. Although “safe” and irrefutable when compared to “real” detections (e.g., in a spectrum), the quoted upper limits often lack probabilistic power.