When inferring parameters from a Gaussian-distributed data set by computing a likelihood, a covariance matrix is needed that describes the data errors and their correlations. If the covariance matrix is not known a priori, it may be estimated and thereby becomes a random object with some intrinsic uncertainty itself. We show how to infer parameters in the presence of such an estimated covariance matrix, by marginalising over the true covariance matrix, conditioned on its estimated value. This leads to a likelihood function that is no longer Gaussian, but rather an adapted version of a multivariate t-distribution, which has the same numerical complexity as the multivariate Gaussian. As expected, marginalisation over the true covariance matrix improves inference when compared with Hartlap et al.'s method, which uses an unbiased estimate of the inverse covariance matrix but still assumes that the likelihood is Gaussian.

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 Baryon Oscillation Spectroscopic Survey (BOSS) and the 2-degree Field Lensing Survey (2dFLenS). Cross-correlations between BOSS and 2dFLenS galaxy positions and source galaxy ellipticities have been incorporated into the analysis, necessitating the development of a hybrid model of non-linear scales that blends perturbative and non-perturbative approaches, and an assessment of signal contributions by astrophysical effects. All weak lensing signals were 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 was propagated into the likelihood. A dedicated suite of more than 20 000 mocks was 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 χ2 goodness-of-fit statistic have been validated, with proposed changes for calculating the effective number of degrees of freedom. The prior volume was explicitly mapped, and a more conservative, wide top-hat prior on the key structure growth parameter S8 = σ8 (Ωm/0.3)1/2 was introduced. The prevalent custom of reporting S8 weak lensing constraints via point estimates derived from its marginal posterior is highlighted to be easily misinterpreted as yielding systematically low values of S8, and an alternative estimator and associated credible interval are proposed. Known systematic effects pertaining to weak lensing modelling and inference are shown to bias S8 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, S8 constraints are expected to improve by 20% for weak lensing alone and by 29% for the joint analysis.

We compute the Bayesian evidence for models considered in the main analysis of Planck cosmic microwave background data. By utilizing carefully defined nearest-neighbor distances in parameter space, we reuse the Monte Carlo Markov chains already produced for parameter inference to compute Bayes factors B for many different model-data set combinations. The standard 6-parameter flat cold dark matter model with a cosmological constant (ΛCDM) is favored over all other models considered, with curvature being mildly favored only when cosmic microwave background lensing is not included. Many alternative models are strongly disfavored by the data, including primordial correlated isocurvature models (lnB=-7.8), nonzero scalar-to-tensor ratio (lnB=-4.3), running of the spectral index (lnB=-4.7), curvature (lnB=-3.6), nonstandard numbers of neutrinos (lnB=-3.1), nonstandard neutrino masses (lnB=-3.2), nonstandard lensing potential (lnB=-4.6), evolving dark energy (lnB=-3.2), sterile neutrinos (lnB=-6.9), and extra sterile neutrinos with a nonzero scalar-to-tensor ratio (lnB=-10.8). Other models are less strongly disfavored with respect to flat ΛCDM. As with all analyses based on Bayesian evidence, the final numbers depend on the widths of the parameter priors. We adopt the priors used in the Planck analysis, while performing a prior sensitivity analysis. Our quantitative conclusion is that extensions beyond the standard cosmological model are disfavored by Planck data. Only when newer Hubble constant measurements are included does ΛCDM become disfavored, and only mildly, compared with a dynamical dark energy model (lnB∼+2).

We present the new method DALI (Derivative Approximation for LIkelihoods) for reconstructing and forecasting posteriors. DALI extends the Fisher Matrix formalism but allows for a much wider range of posterior shapes. While the Fisher Matrix formalism is limited to yield ellipsoidal confidence contours, our method can reproduce the often observed flexed, deformed or curved shapes of known posteriors. This gain in shape fidelity is obtained by expanding the posterior to higher order in derivatives with respect to parameters, such that non-Gaussianity in the parameter space is taken into account. The resulting expansion is positive definite and normalizable at every order. Here, we present the new technique, highlight its advantages and limitations, and show a representative application to a posterior of dark energy parameters from supernovae measurements.