  (Abridged) We examine the power source of 41 local Ultraluminous Infrared
Galaxies using archival infrared and optical photometry. We fit the observed
Spectral Energy Distributions (SEDs) with starburst and AGN components; each
component being drawn from a family of templates. We find all of the sample
require a starburst, whereas only half require an AGN. In 90% of the sample the
starburst provides over half the IR emission, with a mean fractional luminosity
of 82%. When combined with other galaxy samples we find that starburst and AGN
luminosities correlate over 6 decades in IR luminosity suggesting that a common
factor governs both luminosities, plausibly the gas masses in the nuclear
regions. We find that the mid-IR 7.7 micron line-continuum ratio is no
indication of the starburst luminosity, or the fractional AGN luminosity, and
therefore that this ratio is not a reliable diagnostic of the power source in
ULIRGs. We propose that the scatter in the radio-IR correlation in ULIRGs is
due to a skewed starburst IMF and/or relic relativistic electrons from a
previous starburst, rather than contamination from an obscured AGN. We show
that most ULIRGs undergo multiple starbursts during their lifetime, and by
inference that mergers between more than two galaxies may be common amongst
ULIRGs. Our results support the evolutionary model for ULIRGs proposed by
Farrah et al 2001, where they can follow many different evolutionary paths of
starburst and AGN activity in transforming merging spiral galaxies into
elliptical galaxies, but that most do not go through an optical QSO phase. The
lower level of AGN activity in our local sample than in z~1 HLIRGs implies that
the two samples are distinct populations. We postulate that different galaxy
formation processes at high-z are responsible for this difference.
