  We apply methods from Bayesian inferencing and graph theory to a dataset of
102 mid-infrared spectra, and archival data from the optical to the millimeter,
to construct an evolutionary paradigm for z<0.4 infrared-luminous galaxies
(ULIRGs). We propose that the ULIRG lifecycle consists of three phases. The
first phase lasts from the initial encounter until approximately coalescence.
It is characterized by homogeneous mid-IR spectral shapes, and IR emission
mainly from star formation, with a contribution from an AGN in some cases. At
the end of this phase, a ULIRG enters one of two evolutionary paths depending
on the dynamics of the merger, the available quantities of gas, and the masses
of the black holes in the progenitors. On one branch, the contributions from
the starburst and the AGN to the total IR luminosity decline and increase
respectively. The IR spectral shapes are heterogeneous, likely due to feedback
from AGN-driven winds. Some objects go through a brief QSO phase at the end. On
the other branch, the decline of the starburst relative to the AGN is less
pronounced, and few or no objects go through a QSO phase. We show that the 11.2
micron PAH feature is a remarkably good diagnostic of evolutionary phase, and
identify six ULIRGs that may be archetypes of key stages in this lifecycle.
