  Obtaining accurately calibrated redshift distributions of photometric samples
is one of the great challenges in photometric surveys like LSST, Euclid, HSC,
KiDS, and DES. We combine the redshift information from the galaxy photometry
with constraints from two-point functions, utilizing cross-correlations with
spatially overlapping spectroscopic samples. Our likelihood framework is
designed to integrate directly into a typical large-scale structure and weak
lensing analysis based on two-point functions. We discuss efficient and
accurate inference techniques that allow us to scale the method to the large
samples of galaxies to be expected in LSST. We consider statistical challenges
like the parametrization of redshift systematics, discuss and evaluate
techniques to regularize the sample redshift distributions, and investigate
techniques that can help to detect and calibrate sources of systematic error
using posterior predictive checks. We evaluate and forecast photometric
redshift performance using data from the CosmoDC2 simulations, within which we
mimic a DESI-like spectroscopic calibration sample for cross-correlations.
Using a combination of spatial cross-correlations and photometry, we show that
we can provide calibration of the mean of the sample redshift distribution to
an accuracy of at least $0.002(1+z)$, consistent with the LSST-Y1 science
requirements for weak lensing and large-scale structure probes.
