Cosmological likelihoods and parameter degeneracy

Let d be a measured summary vector and m its predicted mean for cosmological parameters theta and nuisance parameters nu. Nuisance quantities can include detector calibration, galaxy bias, intrinsic alignments, foreground amplitudes, and redshift distributions. The residual is d minus m. A Gaussian data likelihood is useful when its sampling approximation is justified, but sparse counts, low-multipole powers, selection-truncated samples, or non-Gaussian summaries may require different distributions. Gaussianity of a plotted posterior is not a derivation of the data likelihood.

For a Gaussian vector, inverse covariance weights combinations of residuals rather than each entry separately. The determinant term supplies normalization and cannot be discarded when covariance depends on fitted parameters. If covariance is fixed throughout a fit, that term is a parameter-independent constant and does not move the maximum. The covariance must also match the estimator and survey window; errors for an ideal full-sky spectrum cannot simply be copied into a masked correlation analysis.

Consider n measurements of the same mean, each with independent zero-mean noise of variance sigma squared, plus one common offset delta. Assign delta a zero-mean Gaussian calibration prior with variance tau squared. Integrating out that offset yields a covariance with tau squared in every entry. The sample mean then has independent-noise variance divided by n plus the unchanged calibration variance. This floor follows from a specified statistical model, not from labeling every unexplained discrepancy systematic.

Profiling chooses the best nuisance value for each cosmology, whereas marginalizing integrates nuisance values with their prior measure. Fixing the nuisance to its nominal value is a third operation and can yield overly narrow constraints. Under a linear Gaussian model some profiled and marginalized shapes coincide up to normalization, but that is not a general equivalence. Calibration information should enter either through an explicit nuisance prior or its properly marginalized covariance representation, avoiding duplicate use of the same information.

A degeneracy occurs when different parameter combinations produce nearly indistinguishable predicted data. Local derivatives identify weakly constrained directions, but a Fisher ellipse can miss curved ridges, boundaries, and multiple modes. Priors or complementary observations may close such directions; the resulting precision must then be attributed to that added information. A joint analysis needs cross-covariance for shared sky or calibration, and should check data-level residuals instead of relying solely on overlapping one-dimensional posterior intervals.

Covariance estimated from finitely many simulations is itself uncertain. For independent Gaussian simulation vectors and a specified covariance prior, integrating that uncertainty leads to a modified multivariate t likelihood, as derived by Sellentin and Heavens. This result is conditional on those assumptions; it does not make every simulated covariance exact. Convergence checks, simulation independence, model discrepancy, and posterior-predictive diagnostics remain necessary. A precise numerical sampler cannot repair an incorrect likelihood, an unmodeled selection, or a hidden reused datum.