Fig 1. The biologically hierarchical problem of detecting a pathogen in its hosts.

A subject drawn from a population is infected with the pathogen with probability Ψ (which reflects the prevalence of infection in that population). A sample drawn from a subject contains the pathogen with probability θ, which is conditional on Ψ. Therefore, θ represents sample-level pathogen (or, more broadly, target) availability, given subject-level infection. Finally, pathogen detection tests run on samples drawn from a subject detect the target with a probability p that is conditional on θ; therefore, p represents test-level probability of detection, given sample-level availability (and, hence, subject-level infection). This shows that the biological problem of detecting a pathogen in its hosts involves a hierarchy of nested levels: subjects (within populations), samples within subjects, and tests within samples. To understand how the system works, we therefore need to estimate 3 probabilities: test-level detection (p), sample-level availability (θ), and subject-level infection (Ψ). Replicate tests inform on the value of p (which is, strictly speaking, the sensitivity of the test) and replicate samples (possibly encompassing different tissues or bodily fluids, as suggested by black/white drops) drawn from the same subject inform on the value of θ; this opens the possibility of estimating Ψ and, therefore, prevalence in the population. Note that all the parameters above (Ψ, θ, and p) are probabilities and hence can take on any value between 0 and 1.