  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 present an inference methodology that combines the redshift
information from the galaxy photometry with constraints from two-point
functions, utilizing cross-correlations with spatially overlapping
spectroscopic samples, and illustrate the approach on CosmoDC2 simulations. 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.
