84 PREDICTIVE RANDOM SETS
by the probability model X ∼ N(µ,1) and the constraint µ ≥ 0. The Gaussian model
for X allows any real-valued µ. For this unrestricted case, many inference methods
have proven to be simple and produce practically the same results for µ. Somewhat
surprisingly, when µ is known to belong to a restricted interval, the same problem
becomes challenging; see, for example, [81]. As discussed in [174], this problem
arises when measuring particle masses, which must be non-negative and are expected
to be relatively small, if nonzero.
In the Poisson example, the observed count, Y , is known to be comprised of signal
and background events, each coming from their own independent Poisson distribu-
tions. Suppose the background rate, b,