  Artificial broadcasts from extraterrestrial intelligences (ETIs) are a
hypothetical class of celestial phenomena. Unlike known astrophysical objects,
the societies that generate them may be able to replicate on galactic scales
through interstellar travel. Different galaxies could thus have drastically
different populations, with abundance variations of many orders of magnitude. I
present a probabilistic formalism to treat this shared history, in which
societies and their broadcasts are described by distributions over basic
properties like lifespan and energy released. The framework contains a
hierarchy of objects related by a tree structure. Discrete societies, the
sources of broadcasts, are organized into potentially interstellar
"metasocieties." The population of each type of object is represented by a
random point process in an abstract parameter hyperspace, a "haystack." When a
selection like an observation draws a sample, the point process is thinned.
Given assumptions of interchangeability and independence, observables are
modeled with compound Poisson random variables. I present an example of how
selection bias can favor sampling longer-lived objects. I rederive the Drake
Equation for societies in the limit of no expansion. When interstellar
replication is present, however, the mean number of detected broadcasts can
depend quadratically on stellar mass, suggesting a search strategy favoring
large galaxies.
