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Tphysicsletters/6879/10/1490/784031tpl/Bayesian and frequentist investigation of prior effects in EFTofLSS analyses of full-shape BOSS and eBOSS data
Monday, September 11, 2023 at 6:45:00 AM UTC
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Bayesian and frequentist investigation of prior effects in EFTofLSS analyses of full-shape BOSS and eBOSS data
Emil Brinch Holm,1 Laura Herold,2 Th´eo Simon,3 Elisa G. M.
Ferreira,4, 5 Steen Hannestad,1 Vivian Poulin,3 and Thomas Tram1
1Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
2Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
3Laboratoire Univers & Particules de Montpellier (LUPM),
CNRS & Universit´e de Montpellier (UMR-5299),
Place Eug`ene Bataillon, F-34095 Montpellier Cedex 05, France
4Kavli Institute for the Physics and Mathematics of the Universe (WPI),
UTIAS, The University of Tokyo, Chiba 277-8583, Japan 5
Instituto de F´ısica, Universidade de S˜ao Paulo - C.P. 66318, CEP: 05315-970, S˜ao Paulo, Brazil
We thank Pierre Zhang for his comments and insights throughout the project, and Eiichiro Komatsu and Luisa Lucie-Smith for helpful discussions. EBH and LH would like to thank the Laboratoire Univers & Particules de Montpellier for their hospitality, where part of this work was conducted. We acknowledge computing resources from the Centre for Scientific Computing Aarhus (CSCAA). These results have also been made possible thanks to LUPM’s cloud computing infrastructure founded by Ocevu labex, and France-Grilles. E.B.H. and T.T. were supported by a research grant (29337) from VILLUM FONDEN. This project has received support from the European Union’s Horizon 2020 research and innovation program under the Marie Skodowska-Curie grant agreement No 860881-HIDDeN. This project has also received funding from the European Research Council (ERC) under the European Union’s HORIZON-ERC2022 (Grant agreement No. 101076865).
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Abstract
Previous studies based on Bayesian methods have shown that the constraints on cosmological parameters from the Baryonic Oscillation Spectroscopic Survey (BOSS) full-shape data using the Effective Field Theory of Large Scale Structure (EFTofLSS) depend on the choice of prior on the EFT nuisance parameters. In this work, we explore this prior dependence by adopting a frequentist approach based on the profile likelihood method, which is inherently independent of priors, considering data from BOSS, eBOSS and Planck. We find that the priors on the EFT parameters in the Bayesian inference are informative and that prior volume effects are important. This is reflected in shifts of the posterior mean compared to the maximum likelihood estimate by up to 1.0 σ (1.6 σ) and in a widening of intervals informed from frequentist compared to Bayesian intervals by factors of up to 1.9 (1.6) for BOSS (eBOSS) in the baseline configuration, while the constraints from Planck are unchanged. Our frequentist confidence intervals give no indication of a tension between BOSS/eBOSS and Planck. However, we find that the profile likelihood prefers extreme values of the EFT parameters, highlighting the importance of combining Bayesian and frequentist approaches for a fully nuanced cosmological inference. We show that the improved statistical power of future data will reconcile the constraints from frequentist and Bayesian inference using the EFTofLSS.
Introduction
In the last decades, the increasing precision of measurements of the cosmic microwave background (CMB) temperature fluctuations has reduced the experimental uncertainties to such an extent, that they are now dominated by cosmic variance [1]. This places an unavoidable limit on the amount of information extractable from the CMB and, therefore, additional cosmological probes are emerging, predominantly from large-scale structure (LSS) measurements. The Baryon Oscillation Spectroscopic survey (BOSS) of the Sloan Digital Sky survey [2] is an example of a modern LSS probe, which will soon be joined by ambitious missions such as the Dark Energy Spectroscopic Instrument (DESI, [3]), the Vera Rubin Observatory [4] and the Euclid space telescope [5], providing exciting new information about the LSS of the Universe. As the accuracy of the surveys increases, so does the demand for accurate theoretical model predictions. In particular, efficient computations of the statistics of inhomogeneities at small scales are crucial for drawing robust conclusions based on the upcoming data. N-body calculations, while giving accurate predictions, suffer from high demand for computational resources which usually make them unfeasible for full cosmological parameter inferences (although recent approaches based on machine learning may remedy this [6–8]). Instead, by compromising accuracy at the smallest scales, semi-analytic approaches based on perturbation theory (see e.g. [9, 10], and references therein) may provide a computationally efficient alternative to N-body simulations. The recently developed effective field theory of large-scale structure (EFTofLSS) employs an effective field theory approach to predict the biased power spectrum up to mildly nonlinear scales [11–15]. The one-loop prediction of the EFTofLSS has allowed the determination of the ΛCDM parameters from the full-shape analysis of BOSS and eBOSS data at a precision higher than that from conventional baryon acoustic oscillation (BAO) and redshiftspace distortion (RSD) analyses, and for some parameters even comparable to that of CMB experiments (see e.g., Refs. [16–27]). Furthermore, the EFTofLSS may provide competitive and interesting constraints on models beyond ΛCDM (see e.g., Refs. [28–37]).
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Conclusion
Motivated by previous Bayesian studies that found a prior dependence of the inferred cosmological parameters from BOSS full-shape data using the EFTofLSS [25, 34, 41], in this work, we present frequentist profile likelihood constraints to view this matter from a different statistical point of view. In particular, two of the commonly used parametrizations of the EFTofLSS, the WC [19] and EC parametrizations [78], give different constraints on the cosmological parameters of up to ∼ 1 σ in a Bayesian analysis [25]. Using the profile likelihood, we find that the WC and EC parametrizations yield the same confidence interval for σ8, confirming that the two parametrizations are mathematically equivalent, i.e., they describe the same space of model predictions for the galaxy power spectrum multipoles (see Fig. 1 in Sec. IV A).9 However, we find that the profile likelihood gives constraints on σ8 that are factors of > 2 wider than the constraints based on the MCMC posterior. Moreover, we observed that several of the EFT parameters take on extreme values during the profile likelihood analysis, indicating that the frequentist analysis takes into account parts of the EFT parameter space beyond the intended use of the theory, in which the perturbative nature might be broken. This issue is addressed in the Bayesian case by imposing narrow Gaussian priors on the EFT parameters. If these priors were well founded, e.g., motivated from theory, simulations, or other observations, the priors could in principle be promoted to data likelihoods in the frequentist analysis. Although the priors on the EFT parameters are not rigorously motivated, we explore the effect of including Gaussian data likelihoods in the frequentist analysis, which correspond to the priors in the Bayesian analysis. We find that the inclusion of the Gaussian likelihoods on the EFT parameters reduces the width of the constraints almost to the level of the ones inferred from the MCMC posterior and keeps the EFT parameters in the intended range (see Fig. 2 in Sec. IV B). However, it also leads to a shift of the confidence interval of σ8. This demonstrates that the priors on the EFT parameters in the Bayesian analysis are informative and influence the inferred cosmological parameters. As a way forward, we explore the impact that data from future surveys like DESI [3] will have by considering BOSS+BAO data with a data covariance matrix rescaled by 16 (see Fig. 3 in Sec. IV C). We find that the constraints from Bayesian and frequentist approaches converge to the same interval for σ8 as the likelihood dominates over the prior information, suggesting that the issues discussed above will subside with more data. Finally, we construct frequentist confidence intervals for five selected ΛCDM parameters, σ8, h, Ωm, ns, ln 1010As, and compare the constraints from different data sets, including BOSS, eBOSS and Planck (see Sec. V). With the profile likelihood, we find that the constraints from BOSS and Planck for all five parameters are within 1.4 σ, finding no indication of a tension. In particular, while the MCMC posterior prefers intervals for σ8, which are 1.4 σ (2.5 σ) lower than the Planck value for the WC (EC) EFT parametrization, the intervals from the profile likelihood are only 0.5 σ (0.3 σ) lower than the Planck constraint. The reduction of the σ-distances can be mainly attributed to the wide confidence intervals from the profile likelihood, but in the case of σ8, also to shifts of the MLE closer to the Planck value than the posterior mean. In line with previous studies [24, 25], we find that the parameter σ8 is most subject to prior effects. This indicates that the slight “σ8 discrepancy” seen in the Bayesian results using the EC parametrization is due to the particular choice of priors. On the other hand, although our main profile likelihood analysis makes use of the WC baseline parametrization of the EFTofLSS without priors, we do not expect major changes in our conclusions regarding the state of the σ8 tension from resorting to the use of “priors” or a different parametrization. Our results clearly show the advantages and disadvantages of frequentist and Bayesian parameter inference. Since the frequentist inference does not include priors that confine the EFT parameters to the regime intended by the theory, we observe that the data prefers several EFT parameters to take on extreme values, possibly breaking the perturbativeness of the theory. The lack of prior further leads to significantly wider confidence intervals. This loss of constraining power reflects the purely data driven frequentist approach, which is completely agnostic about which model parameters are deemed more likely a priori. On the other hand, the priors in the Bayesian inference are informative and have an impact on the inferred cosmological parameters. This is important since it is not straightforward to define well motivated priors on the EFT parameters, which is reflected in the fact that the WC and EC parametrizations use different standard configurations for the EFT priors. Looking towards the future, which will bring more constraining data sets, we can expect these points of discussion to subside as the data will dominate over any subjective preference introduced by the analysis setup. While waiting for better data, our results indicate that the use of frequentist along with Bayesian methods are valuable in order to obtain a fully nuanced view of the data.
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Abstract
Introduction
Conclusion
References
