Alleviating prior dependencies for DESI DR1 clustering fits through reparameterization

Technical Summary: Alleviating Prior Dependencies for DESI DR1 Clustering Fits through Reparameterization

Problem Statement
This work addresses a specific systematic issue in Bayesian analyses of the full-shape clustering of Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1). When analyzing the power spectrum multipoles using the Effective Field Theory of Large Scale Structure (EFTofLSS), weakly constrained nuisance parameters (related to galaxy bias, counterterms, and shot noise) exhibit strong degeneracies with cosmological parameters. In standard analyses using broad, uniform, or Gaussian priors, these degeneracies lead to "prior-volume projection effects." Consequently, the marginalized posterior distributions for key cosmological parameters—specifically the Hubble constant ($H_0$) and the dark energy equation of state parameters ($w_0, w_a$)—are systematically shifted away from the Maximum a Posteriori (MAP) estimate. This phenomenon results in multi-$\sigma$ discrepancies between the reported expectation values and the true likelihood maximum, particularly in datasets with weak external anchoring (e.g., DESI combined only with Big Bang Nucleosynthesis and a weak spectral index constraint).

Methodology
The authors reanalyze DESI DR1 power spectrum multipoles using two complementary strategies designed to mitigate these projection effects without relying on external astrophysical modeling assumptions (such as Halo Occupation Distribution models):

1. Nonlinear Orthogonal Reparameterization:

   • The authors implement a technique to decorrelate nuisance parameters ($N$) from cosmological parameters ($C$).
   • Instead of the Generalized Additive Models (GAMs) used in previous work, they employ a fully connected neural network to learn the conditional expectation $f(C) \approx E[N|C]$ from an initial posterior sample.
   • Nuisance parameters are transformed into an orthogonal basis $N' = N - f(C)$, rendering them approximately uncorrelated with cosmology.
   • The analysis is then refitted in this transformed space, with priors rescaled to maintain physical variance in the original basis.

2. Reparameterization-Invariant Jeffreys Prior:

   • The authors implement a fully reparameterization-invariant Jeffreys prior over all EFTofLSS coefficients. This prior is constructed from the square root of the determinant of the Fisher information matrix ($p_J(\theta) \propto \sqrt{\det I(\theta)}$).
   • To overcome computational bottlenecks associated with high-dimensional Fisher matrices, they derive closed-form analytical expressions for the Jacobian of the power spectrum multipoles with respect to the EFT parameters.
   • These Jacobians are evaluated on-the-fly during Markov Chain Monte Carlo (MCMC) sampling using the Effort.jl emulator and Turing.jl framework, allowing for efficient updates of the Jeffreys factor at every step.
   • The authors test two configurations: a "pure" Jeffreys prior and a "hybrid" approach (Jeffreys + Baseline Gaussian), where the Jeffreys prior is combined with the original baseline Gaussian widths to control residual over-broad tails in physical parameters like the cold dark matter density ($\omega_c$).

Key Contributions

• Algorithmic Implementation: The paper extends the application of Jeffreys priors in EFTofLSS analyses from linear subsets of parameters to the full set of nonlinear nuisance parameters, utilizing analytical Jacobians to maintain computational efficiency.
• Neural Network Orthogonalization: The work updates the orthogonalization methodology by replacing GAMs with neural networks, improving the handling of multivariate, highly non-linear degeneracies and ensuring numerical stability in high-dimensional spaces.
• Cross-Framework Validation: The study provides a rigorous comparison between three distinct inference frameworks:
  1. The proposed Bayesian approach with Jeffreys-based volume correction (specifically the hybrid Jeffreys+Baseline).
  2. A simulation-calibrated approach using HOD-informed priors (HIP).
  3. A frequentist profile likelihood analysis.

Results

• Mitigation of Projection: In the weakly anchored "DESI + BBN + $n_s$" configuration, the standard baseline prior (BLP) exhibits significant projection effects, shifting $H_0$, $w_0$, and $w_a$ by several standard deviations from the MAP. The orthogonalization method partially mitigates these shifts, but the Jeffreys prior (particularly the hybrid variant) successfully recenters the posteriors, enclosing the MAP within the 68% credible regions for these parameters.
• Parameter Constraints: The hybrid Jeffreys+Baseline approach is identified as the fiducial method. It controls the over-broad tails in $\omega_c$ observed with the pure Jeffreys prior while preserving the volume correction benefits for expansion parameters.
• Consistency Across Frameworks: When external constraints are added (CMB and Type Ia Supernovae), the differences between the baseline, orthogonalized, Jeffreys, and HOD-informed methods diminish. However, even in the weakly constrained regime, the hybrid Jeffreys approach and the HOD-informed approach yield consistent central values and degeneracy directions in the $w_0$–$w_a$ plane, despite their fundamentally different assumptions (geometric volume correction vs. astrophysical modeling).
• Frequentist-Bayesian Agreement: The credible intervals derived using the hybrid Jeffreys approach show excellent agreement with frequentist profile likelihood confidence intervals in terms of central values and degeneracy directions, suggesting that the underlying likelihood information is being extracted consistently once projection effects are controlled.

Significance and Claims
The paper claims that by properly controlling prior-volume projection effects through reparameterization-invariant measures (specifically the Jeffreys prior), robust inferences about late-time cosmological expansion can be made independent of the specific statistical framework (Bayesian vs. Frequentist) or the adoption of specific astrophysical assumptions (HOD models).

The authors emphasize that their methodology demonstrates that the discrepancies previously observed between different analysis pipelines were largely driven by projection artifacts rather than fundamental inconsistencies in the data or models. The convergence of results across the Jeffreys-based Bayesian analysis, the HOD-calibrated Bayesian analysis, and the frequentist profile likelihood analysis validates the reliability of DESI DR1 full-shape clustering constraints on $H_0$, $w_0$, and $w_a$. The work establishes a methodological foundation for future large-scale structure analyses (e.g., with DESI and Euclid) where increased statistical power will require rigorous control of such systematic effects.