Combining Cosmological Data Sets

Cosmological likelihoods may be multiplied only when their data are conditionally independent given every shared cosmological and nuisance variable. Overlapping objects, sky modes, calibrators, simulations, selection corrections, or external priors can violate that condition even when two products carry different experiment names. The decision-relevant question is therefore not “Do the contours agree?” but “What joint data-generating model produced both products?” This assessment uses public evidence available through 2026-08-19 and keeps every verdict conditional on the stated model class.

Required background. Probability and Statistical Inference supplies conditional independence, hierarchical models, covariance, and posterior prediction.

Helpful background. Einstein Equations supplies the gravitational model being constrained; Relativistic Matter and Compact Objects supplies shared source and foreground physics; Relativistic Cosmology supplies probe responses and degeneracies; and Computational Relativity supplies simulation and convergence checks.

Conditional independence is the multiplying rule

For datasets $D_A$ and $D_B$, parameters $\boldsymbol\theta$, shared nuisances $\boldsymbol\lambda$, and dataset-specific nuisances $\boldsymbol\eta_A,\boldsymbol\eta_B$, the factorization

$$p(D_A,D_B\mid\boldsymbol\theta,\boldsymbol\lambda, \boldsymbol\eta_A,\boldsymbol\eta_B) =p(D_A\mid\boldsymbol\theta,\boldsymbol\lambda,\boldsymbol\eta_A) p(D_B\mid\boldsymbol\theta,\boldsymbol\lambda,\boldsymbol\eta_B)$$

is a scientific assumption, not an algebraic default. The joint likelihood after marginalizing shared calibration is

$$\mathcal L_{AB}(\boldsymbol\theta) =\int d\boldsymbol\lambda\,d\boldsymbol\eta_A\,d\boldsymbol\eta_B\, \pi(\boldsymbol\lambda,\boldsymbol\eta_A,\boldsymbol\eta_B) \mathcal L_A\mathcal L_B.$$

Multiplying separately marginalized likelihoods is generally different because it integrates the shared nuisance twice. It behaves as if each dataset had an independent copy of the same calibration uncertainty.

For Gaussian data, an explicit joint covariance makes the issue visible:

$$\mathsf C_{AB}= \begin{pmatrix} \mathsf C_A&\mathsf C_{AB}^{\rm cross}\\ (\mathsf C_{AB}^{\rm cross})^{\mathsf T}&\mathsf C_B \end{pmatrix}.$$

The product likelihood is recovered only when the cross block vanishes after conditioning on all modeled shared quantities. A small element-by-element correlation is not automatically negligible: many weakly correlated modes can shift an uncertainty or goodness-of-fit statistic materially.

Make the overlap map before combining

A release-level overlap map should answer five questions.

1. Samples and modes: Are any objects, sky area, Fourier modes, redshift bins, or maps reused? Does one “external” product already include the other?
2. Calibration and selection: Do the likelihoods share an absolute ruler, photometric zero point, beam, redshift calibration, completeness simulation, or target-selection correction?
3. Theory and nuisance definitions: Are common foregrounds, nonlinear prescriptions, neutrino assumptions, or fiducial transformations represented by the same parameter or by incompatible fixed corrections?
4. Simulation provenance: Were covariance, transfer functions, or emulator training sets estimated from common realizations?
5. Prior provenance: Is the same BBN, optical-depth, distance-ladder, or other external prior already present in both products?

The answer determines the combination. An explicit block covariance is natural for approximately Gaussian summaries. A shared-nuisance hierarchy is preferable when calibration or foreground uncertainty has a physical parameterization. A joint object-level likelihood may be necessary when selection depends on a latent population. “No known overlap” is weaker than a documented conditional-independence argument.

Posterior recycling is not posterior multiplication

Suppose independent datasets were analyzed under the same prior $\pi(\theta)$. Because $p(\theta\mid D_i)\propto\mathcal L_i(\theta)\pi(\theta)$,

$$p(\theta\mid D_A,D_B) \propto \frac{p(\theta\mid D_A)p(\theta\mid D_B)}{\pi(\theta)}.$$

The division removes one duplicated prior. With different priors, parameter bases, or model supports, the correction must include the full measures and Jacobians. Sample-based recycling also requires overlap: if the new joint posterior lies in a tail poorly populated by the proposal chain, importance weights collapse and a fresh run is required.

Compressed likelihoods introduce a separate boundary. A BAO distance vector can retain nearly all information for a declared smooth model while discarding broadband-shape, nuisance, or non-Gaussian information needed by an extension. A CMB “prior” on a few early-Universe combinations is not interchangeable with the full CMB likelihood for every model. Compression must be validated across the parameter regime in which it is used.

Consistency is a predictive question

A strong pre-combination check asks one dataset to predict the other. With shared nuisance parameters $\boldsymbol\lambda$ and conditional independence after all common variables are included,

$$p(D_B\mid D_A,M) =\int d\boldsymbol\theta\,d\boldsymbol\lambda\, p(D_B\mid\boldsymbol\theta,\boldsymbol\lambda,M) p(\boldsymbol\theta,\boldsymbol\lambda\mid D_A,M).$$

If residual dependence remains after conditioning—for example through shared modes not represented by $\boldsymbol\lambda$—the first factor must instead be $p(D_B\mid D_A,\boldsymbol\theta,\boldsymbol\lambda,M)$ or come from the full joint likelihood. Dropping $\boldsymbol\lambda$ would average the second dataset over the wrong calibration or foreground distribution.

The residual-correlation correction is explicit for a joint Gaussian. At fixed parameters, write $\mathsf X=\mathsf C_{AB}^{\rm cross}$. Completing the square in the joint density gives

$$\begin{aligned} \boldsymbol\mu_{B\mid A} &=\boldsymbol\mu_B+\mathsf X^{\mathsf T}\mathsf C_A^{-1} (\mathbf d_A-\boldsymbol\mu_A),\\ \mathsf C_{B\mid A} &=\mathsf C_B-\mathsf X^{\mathsf T}\mathsf C_A^{-1}\mathsf X. \end{aligned}$$

Shared fluctuations seen in $A$ therefore change both what is predicted for $B$ and how uncertain that prediction is. Setting $\mathsf X=0$ recovers the independent conditional density. These conditional moments must still be averaged over the parameter posterior from $A$; fixing its best fit would discard parameter uncertainty.

The observed $D_B$ can then be compared with replicated draws using a discrepancy fixed before inspecting the result. Reversing $A$ and $B$ tests a different predictive direction. Leave-one-dataset-out analysis exposes which product drives a joint posterior and whether a visually tight combination merely inherits one probe’s likelihood.

Parameter-difference statistics, posterior predictive checks, evidence ratios, and prior-volume-adjusted consistency measures answer different questions. Their calibration should be tested on simulations with the actual priors, boundaries, non-Gaussianity, and cross-covariance. No scalar “tension” statistic can repair an incorrect joint generative model.

Results and limitations through 2026-08-19

The evidence status as of 2026-08-19 supports versioned, model-conditional combination rather than a universal multiplication rule. The central unresolved issue is often not the algebra but whether public products expose every shared mode and systematic strongly enough to build the correct joint model.

Evidence or result	Implication for combination	Scope, limitation, or failure mode
Analytic correlated-Gaussian control	Product and block-covariance likelihoods agree only at zero conditional cross-covariance	Exact for the toy model; real non-Gaussian selection and parameter-dependent covariance require simulation
Planck 2018 spectra and likelihood package	Frequency spectra, masks, beams, calibration, foreground nuisances, cuts, and covariance define one versioned CMB likelihood	Planck Collaboration 2020, §§2–4 and appendices does not establish independence from every later lensing or external product
DESI DR2 BAO products	Released distances and full covariance permit an exact Gaussian combination within the declared compression	Abdul Karim et al. 2025, §§II–IV, Table IV, and §X requires tracer, redshift, reconstruction, fiducial, blinding, and systematic choices to remain attached
Cobaya 3.6.2 hierarchical pipeline	Multiple theory and likelihood components can share parameters and cached dependencies in one model graph	Torrado and Lewis 2021, §§II–IV supplies infrastructure, not proof that user-specified likelihoods are independent or correctly normalized
Separate-probe agreement or null discrepancy	Supports compatibility only for the chosen model, discrepancy, prior, and calibration domain	Weak signal-to-noise, shared priors, or broad degeneracies can make a null test uninformative
Leave-one-probe-out prediction	Reveals dominance and tests whether one release predicts another before joint fitting	Result depends on predictive direction, selections, model class, and whether cross-covariance is included

Trotta’s probability and model-comparison treatment makes the underlying distinction explicit: likelihoods are conditioned on hypotheses, priors are measures, and evidence uses normalized probability models. Trotta 2008, §§2–4 also explains why prior choice and unused parameter volume cannot be removed by relabeling a statistic.

Competing combination strategies

Conditional-independence products are simplest and correct when the factorization has been demonstrated. Explicit joint covariance retains linear shared fluctuations in fixed-dimensional summaries. Shared-nuisance hierarchies propagate common calibration or population uncertainty without inventing duplicate nuisances. Compressed likelihoods improve portability but can lose extension-relevant information. Posterior recycling is efficient only with exact prior correction and adequate support. Consistency or suspiciousness diagnostics can quantify disagreement after the data model is fixed, but they are not substitutes for it.

These are competing formulations rather than a ranking from primitive to sophisticated. A block Gaussian may be more transparent and reliable than a large hierarchy when the released summary is demonstrably sufficient; an object-level hierarchy may be indispensable when selection and calibration are coupled.

Independent synthetic and release checks

For two equal-variance estimates $d_A,d_B$ of a common scalar $\theta$, with variance $\sigma^2$ and correlation $\rho$, the joint estimator is their mean and

$$\operatorname{Var}(\widehat\theta_{\rm joint}) =\frac{\sigma^2(1+\rho)}{2}.$$

The independent product incorrectly returns $\sigma^2/2$. A blind simulation should generate many pairs at known $\theta$ for several $\rho$, recover intervals with both likelihoods, and verify coverage. The joint formulation should cover at its nominal rate; the product undercovers for positive correlation and overstates the information.

For real releases, reproduce each separate constraint first, then the joint result with exact data and code versions. Record shared absolute scales, photometric or redshift calibration, beam and selection calibration, common simulations, sample and footprint overlap, scale and redshift cuts, blinding state, prior provenance, likelihood normalization, and retrieval date. Finally perform leave-one-dataset-out prediction. This page does not claim to have rerun the full Planck–DESI–supernova joint posterior.

Exercise: one calibration, not two

Two probes obey $d_i=\theta+b+\epsilon_i$, where $\epsilon_i$ are independent with variance $1$ and the shared calibration $b$ has prior variance $0.25$. Both observe the same central value. What is the variance of the combined estimator after marginalizing $b$? What would result from treating the calibration error as independent between probes?

Solution

Marginalizing the common $b$ gives unit diagonal noise plus a rank-one covariance $0.25\,\mathbf1\mathbf1^{\mathsf T}$. The mean of the two observations has variance

$$\operatorname{Var}(\widehat\theta) =\frac{1}{2}+0.25=0.75.$$

If each probe is instead assigned its own independent calibration nuisance, each marginal variance is $1.25$ and their product gives $1.25/2=0.625$. That construction spuriously averages down an uncertainty that is common to both measurements. The correct alternatives are one shared nuisance parameter or the equivalent cross-covariance $C_{12}=0.25$.

Scope and open questions

The interpretation could change after a new covariance product or calibration revision changes a major overlap map, after evidence shows that a widely used combination double counts modes or priors, or after a validated compression fails for an important extension class. Each event can change which formulation is defensible.

The durable result is the conditional-independence criterion and its predictive checks. This page does not provide undated best-fit values, declare any current dataset combination universally consistent, or replace the instrument and collaboration documentation needed to build a release-specific joint likelihood.

References

• Abdul Karim, M., et al. (DESI Collaboration). 2025. “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints.” Physical Review D 112: 083515. arXiv:2503.14738v3. DOI. Open PDF.
• Planck Collaboration. 2020. “Planck 2018 Results. V. CMB Power Spectra and Likelihoods.” Astronomy & Astrophysics 641: A5. DOI. Open 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. Official v3.6.2 documentation.
• Trotta, Roberto. 2008. “Bayes in the Sky: Bayesian Inference and Model Selection in Cosmology.” Contemporary Physics 49: 71–104. DOI. Open PDF.