Cosmological Inference and Model Appraisal

Cosmological inference is the controlled passage from calibrated observations to conditional statements about a physical model. The durable challenge is to keep every link visible: the selected data and their response, the generative likelihood and nuisance structure, the prior measure, the numerical sampler, the derived-quantity map, and the criterion used to compare or stress-test models. This chapter develops that chain without turning a dated parameter table or tension significance into evergreen fact.

Helpful background. Einstein Equations supplies the gravitational models being tested; Relativistic Matter and Compact Objects supplies source, foreground, and nuisance physics; Relativistic Cosmology supplies background and perturbation observables; Probability and Statistical Inference supplies likelihood and predictive reasoning; and Computational Relativity supplies numerical validation and reproducibility checks.

One generative chain, six distinct questions

The chapter’s recurring structure is

$$\begin{aligned} \text{physical parameters} &\xrightarrow{\text{dynamics}}\text{predicted observables},\\ \text{predicted observables} &\xrightarrow{\text{response and selection}}p(D\mid\boldsymbol\theta),\\ p(D\mid\boldsymbol\theta),\ \pi(\boldsymbol\theta) &\xrightarrow{\text{observed data}}p(\boldsymbol\theta\mid D). \end{aligned}$$

Model appraisal adds a second branch: a normalized likelihood and prior define marginal evidence; held-out data define predictive performance; replicated data define posterior predictive adequacy; and prespecified discrepancies define consistency tests. These quantities share inputs, but they are not synonyms.

Five distinctions organize the chapter:

Distinction	Operational question	Common failure
Data product versus likelihood	Which vector, covariance, response, selection, and nuisance model define the random variable?	Treating a plotted error bar as the released probability model
Base versus derived parameter	Which coordinates are sampled, and which are pushforwards under a physical code?	Reporting a model-conditional derived posterior as a direct measurement
Likelihood versus posterior	Which information comes from data, and which from the prior measure?	Multiplying posteriors and duplicating a common prior
Fit versus prediction	Does flexibility only reduce residuals, or does it predict withheld observations?	Selecting an extension from maximum likelihood alone
Local versus global discrepancy	Was the statistic and search fixed before inspecting the anomaly?	Quoting the most discrepant split without a look-elsewhere correction

The 2026 Particle Data Group review illustrates why these distinctions matter: it separates fitted from derived parameters, lists astrophysical and instrumental nuisance dependence, and treats current cross-probe hints as model- and dataset-conditional. Cortês, Lahav, and Liddle 2026, §§25.1 and 25.3–25.5 provides the dated observational synthesis.

The six pages in order

1. Cosmological Parameters and Derived Quantities distinguishes sampled coordinates from deterministic pushforwards, carries prior Jacobians, and checks a public DESI DR2 distance transformation with its covariance.
2. Cosmological Likelihood Construction builds the generative density from response, calibration, selection, covariance, and nuisance parameters, using Pantheon+ as a release-level example.
3. Combining Cosmological Data Sets gives the conditional-independence criterion for multiplying likelihoods and compares cross-covariance, shared-nuisance, compression, and posterior-recycling formulations.
4. Model Selection and Bayesian Evidence separates marginal evidence, prior volume, information criteria, cross-validation, posterior predictive checks, and model averaging.
5. Cosmological Tensions and Robustness Tests distinguishes parameter differences, predictive failures, local and global probabilities, dataset splits, systematic alternatives, and genuine out-of-sample confirmation.
6. Reproducible Cosmological Inference turns provenance and convergence into an executable record and validates a correlated-Gaussian posterior by analytic, grid, and seeded Metropolis routes.

The order is deliberate. Combining likelihoods before understanding their construction can double count modes or priors. Comparing models before identifying the base coordinates hides prior-volume changes. Interpreting a tension before reproducing each separate posterior confounds physical disagreement with release or implementation drift.

Two routes through the chapter

The likelihood-builder route begins with parameter maps and likelihood construction, continues through data combination, and ends with reproducibility. It is the shortest route for someone implementing a new probe or public likelihood. The central checks are units, response normalization, selection conditioning, covariance ordering, nuisance provenance, injection recovery, and a clean-environment rerun.

The model-appraisal route begins with parameter maps, then reads data combination, model selection, and tensions before using the reproducibility page as a release checklist. It is designed for readers evaluating an extension or a claimed discrepancy. The central checks are prior and parameterization sensitivity, shared covariance, out-of-sample prediction, look-elsewhere calibration, expanded nuisance models, and evidence that would change the verdict.

A reader focused on current observations can follow the second route but should return to likelihood construction whenever two results use different calibration, selection, compression, or nuisance conventions. A reader focused on computation should run the final benchmark first, then use the earlier pages to understand what a production chain must contain beyond convergence statistics.

The correlated-Gaussian thread

A two-parameter Gaussian example recurs because it has an exact answer while exposing the same logic as a large analysis.

Chapter operation	Gaussian realization	Lesson retained in production
Derive a quantity	Apply a nonlinear map and propagate the full covariance	Keep weights, Jacobians, units, and model dependence
Construct a likelihood	Evaluate residual, quadratic form, determinant, and nuisance prior	A covariance and normalization are part of the probability model
Combine probes	Compare a product with a block covariance or shared nuisance	Independence must be conditional and demonstrated
Compare models	Balance peak likelihood gain against prior-to-posterior volume	A fit improvement alone is not evidence
Test a discrepancy	Form a difference with cross-covariance and calibrate its search	A local sigma is not a global or predictive verdict
Reproduce a posterior	Compare analytic moments with quadrature and independent chains	Convergence is necessary but cannot validate the target distribution by itself

The example also supplies counterchecks. Setting cross-covariance to zero recovers a product likelihood. Sending the prior covariance to infinity recovers a likelihood-dominated posterior only where the limit is proper. Increasing retained chain length reduces Monte Carlo error but cannot fix a wrong covariance sign. Broadening an extension prior changes its evidence even when its maximum likelihood is unchanged.

What a defensible result must state

A compact cosmological result should name the model and parameter basis; exact data and likelihood releases; calibration, units, selection, scale cuts, covariance, and nuisance priors; theory and sampler versions; interval convention; convergence and posterior predictive checks; and the evidence cutoff for mutable claims. A model comparison additionally states model prior odds or their omission, evidence-integration error, prior-sensitivity range, and predictive tests. A tension statement additionally states the discrepancy, covariance, search space, global calibration, dataset splits, and plausible competing explanations.

Trotta’s Bayesian review supplies the probability architecture behind these requirements, including prior transformations, marginal evidence, effective complexity, and the limitations of information criteria. Trotta 2008, §§2–4 is methodological rather than a current cosmological verdict. Cobaya illustrates how a production program can encode a hierarchical model graph, components, samplers, and metadata, while leaving scientific independence and correctness to the analyst. Torrado and Lewis 2021, §§II–V describes that separation.

Review the chapter

Two teams analyze probes $A$ and $B$ under the same prior. They share an absolute calibration and some simulations, but publish only separate posterior chains. A third team multiplies the chains, adds a two-parameter dark-energy extension with broad flat priors after seeing residuals, quotes the largest of several split discrepancies as “$3\sigma$,” and accepts the final chain because split-$\widehat R=1.001$. Identify the failures and give a defensible repair sequence.

Solution

Posterior chains cannot be multiplied directly: doing so duplicates the common prior, and the shared calibration and simulations violate unqualified independence. First reconstruct or obtain the two likelihoods, map sample and mode overlap, and represent shared effects by a joint covariance or one shared nuisance hierarchy. Reproduce each separate posterior before the joint run.

Next declare the extension, parameterization, physical prior measure, and discrepancy before inspecting its preference. Report prior-volume sensitivity and compare marginal evidence with blocked out-of-sample prediction and posterior predictive adequacy; a lower best-fit $\chi^2$ is insufficient. Calibrate the maximum over all inspected splits on end-to-end null simulations, retaining their correlations, rather than quoting the selected local significance.

Finally pin data, theory, likelihood, environment, configuration, seeds, and derived code. Split-$\widehat R$ is one mixing diagnostic, so also report effective sample size, Monte Carlo error, chain and proposal sensitivity, residual or likelihood reference values, and an independent benchmark. Only after these steps can the remaining discrepancy be weighed among new physics, calibration or selection shifts, unmodeled covariance, prior or likelihood effects, and fluctuation.

Chapter boundary

This chapter develops reproducible inference and model appraisal under correlated data and systematic uncertainty. It does not provide undated best-fit values, freeze a current tension ranking, reproduce collaboration pipelines without their artifacts, or prove that the best member of an incomplete model list is true. Instrument-specific reduction, mutable numerical summaries, and production-scale computing remain with their versioned releases and dedicated research records.

References

• Cortês, Marina, Ofer Lahav, and Andrew R. Liddle. 2026. “Cosmological Parameters.” In Review of Particle Physics, revised October 2025. Particle Data Group. Official PDF.
• Torrado, Jesús, and Antony Lewis. 2021. “Cobaya: Code for Bayesian Analysis of Hierarchical Physical Models.” Journal of Cosmology and Astroparticle Physics 05: 057. arXiv:2005.05290v2. DOI. Open PDF.
• Trotta, Roberto. 2008. “Bayes in the Sky: Bayesian Inference and Model Selection in Cosmology.” Contemporary Physics 49: 71–104. DOI. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Cosmological Parameter Tensions and Gravity