Suppose you compute the posterior probability distribution of the intensity of a source, s, when the data include counts in a source region (N_S) and counts in a background region (N_B). When N_S=0, i.e., no counts are observed in the source region,

p(s|N_S=0, N_B) = (1+b)^a/Gamma(a) * s^a-1 * e^-s*(1+b),

where a,b are the parameters of a gamma prior.

Why does N_B have no effect? Because when you have zero counts, the entire effect of the background is going towards evaluating how good the actual chosen model is (so it is become a model comparison problem, not a parameter estimation one), and not into estimating the parameter of interest, the source intensity. That is, into the normalization factor of the probability distribution, p(N_S,N_B). Those parts that depend on N_B cancel out when the expression for p(s|N_S,N_B) is written out because the shape is independent of N_B and the pdf must integrate to 1.

No doubt this is obvious, but I hadn’t noticed it before.

PS: Also shows why upper limits should not be identified with upper confidence bounds.