A justification of a negative binomial model for target sightings

Using an example drawn from the analysis of a campaign of aerial search for U-boats during WWII, this paper presents a heuristic argument that the number of sightings anticipated in a given period, e.g., the coming month, will be negative-binomially distributed. This result – normally found by assuming that the U-boat density is, for some reason, gamma-distributed – is then formally re-derived from the starting point of a reciprocal, or ‘Jeffreys’, prior distribution for the U-boat density and one or more months' worth of Poisson-distributed U-boat sightings, and heretofore distinct lines of reasoning regarding Bayesian updating and the fact that if the density of a Poisson distribution is itself gamma-distributed then the resulting distribution is the negative-binomial.