KiDS-1000 Methodology: Modelling and inference for joint weak
  gravitational lensing and spectroscopic galaxy clustering analysis
  We present the methodology for a joint cosmological analysis of weak
gravitational lensing from the fourth data release of the ESO Kilo-Degree
Survey (KiDS-1000) and galaxy clustering from the partially overlapping BOSS
and 2dFLenS surveys. Cross-correlations between galaxy positions and
ellipticities have been incorporated into the analysis, necessitating a hybrid
model of non-linear scales that blends perturbative and non-perturbative
approaches, and an assessment of contributions by astrophysical effects. All
weak lensing signals are measured consistently via Fourier-space statistics
that are insensitive to the survey mask and display low levels of mode mixing.
The calibration of photometric redshift distributions and multiplicative
gravitational shear bias has been updated, and a more complete tally of
residual calibration uncertainties is propagated into the likelihood. A
dedicated suite of more than 20000 mocks is used to assess the performance of
covariance models and to quantify the impact of survey geometry and spatial
variations of survey depth on signals and their errors. The sampling
distributions for the likelihood and the $\chi^2$ goodness-of-fit statistic
have been validated, with proposed changes to the number of degrees of freedom.
Standard weak lensing point estimates on $S_8=\sigma_8\,(\Omega_{\rm
m}/0.3)^{1/2}$ derived from its marginal posterior are easily misinterpreted to
be biased low, and an alternative estimator and associated credible interval
have been proposed. Known systematic effects pertaining to weak lensing
modelling and inference are shown to bias $S_8$ by no more than 0.1 standard
deviations, with the caveat that no conclusive validation data exist for models
of intrinsic galaxy alignments. Compared to the previous KiDS analyses, $S_8$
constraints are expected to improve by 20% for weak lensing alone and by 29%
for the joint analysis. [abridged]
