  The PRObabilistic Value-Added Bright Galaxy Survey (PROVABGS) catalog will
provide measurements of galaxy properties, such as stellar mass ($M_*$), star
formation rate (${\rm SFR}$), stellar metallicity ($Z_{\rm MW}$), and stellar
age ($t_{\rm age, MW}$), for >10 million galaxies of the DESI Bright Galaxy
Survey. Full posterior distributions of the galaxy properties will be inferred
using state-of-the-art Bayesian spectral energy distribution (SED) modeling of
DESI spectroscopy and Legacy Surveys photometry. In this work, we present the
SED model, Bayesian inference framework, and methodology of PROVABGS.
Furthermore, we apply the PROVABGS SED modeling on realistic synthetic DESI
spectra and photometry, constructed using the L-GALAXIES semi-analytic model.
We compare the inferred galaxy properties to the true galaxy properties of the
simulation using a hierarchical Bayesian framework to quantify accuracy and
precision. Overall, we accurately infer the true $M_*$, ${\rm SFR}$, $Z_{\rm
MW}$, and $t_{\rm age, MW}$ of the simulated galaxies. However, the priors on
galaxy properties induced by the SED model have a significant impact on the
posteriors. They impose a ${\rm SFR}{>}10^{-1} M_\odot/{\rm yr}$ lower bound on
${\rm SFR}$, a ${\sim}0.3$ dex bias on $\log Z_{\rm MW}$ for galaxies with low
spectral signal-to-noise, and $t_{\rm age, MW} < 8\,{\rm Gyr}$ upper bound on
stellar age. This work also demonstrates that a joint analysis of spectra and
photometry significantly improves the constraints on galaxy properties over
photometry alone and is necessary to mitigate the impact of the priors. With
the methodology presented and validated in this work, PROVABGS will maximize
information extracted from DESI observations and provide a probabilistic
value-added galaxy catalog that will extend current galaxy studies to new
regimes and unlock cutting-edge probabilistic analyses.
