  We present Trinity, a flexible empirical model that self-consistently infers
the statistical connection between dark matter haloes, galaxies, and
supermassive black holes (SMBHs). Trinity is constrained by galaxy observables
from $0<z<10$ (galaxies' stellar mass functions, specific and cosmic SFRs,
quenched fractions, and UV luminosity functions) and SMBH observables from
$0<z<6.5$ (quasar luminosity functions, quasar probability distribution
functions, active black hole mass functions, local SMBH mass--bulge mass
relations, and the observed SMBH mass distributions of high redshift bright
quasars). The model includes full treatment of observational systematics (e.g.,
AGN obscuration and errors in stellar masses). From these data, Trinity infers
the average SMBH mass, SMBH accretion rate, merger rate, and Eddington ratio
distribution as functions of halo mass, galaxy stellar mass, and redshift. Key
findings include: 1) the normalization and the slope of the SMBH mass--bulge
mass relation increases mildly from $z=0$ to $z=10$; 2) The best-fitting AGN
radiative$+$kinetic efficiency is $\sim 0.05-0.06$, but can range from $\sim
0.035-0.07$ with alternative input assumptions; 3) AGNs show downsizing, i.e.,
the Eddington ratios of more massive SMBHs start to decrease earlier than those
of lower-mass objects; 4) The average ratio between average SMBH accretion rate
and SFR is $\sim 10^{-3}$ for low-mass galaxies, which are primarily
star-forming. This ratio increases to $\sim 10^{-1}$ for the most massive
haloes below $z\sim 1$, where star formation is quenched but SMBHs continue to
accrete.
