[Paper Review] Agnostic cosmology in the CAMEL framework

S. Henrot–Versillé, O. Perdereau+4 more|arXiv (Cornell University)|Jul 11, 2016

Cosmology and Gravitation Theories13 references7 citations

TL;DR

This paper introduces CAMEL, a C++-based cosmological analysis framework that enables agnostic comparison of Bayesian and frequentist methods for parameter estimation using Planck data. It demonstrates that profile-likelihoods and MCMC posteriors can differ significantly due to 'likelihood volume effects,' particularly in the optical depth τ, where the posterior shifts to lower values (τ ≈ 0.15) compared to profile-likelihoods (τ ≈ 0.17), revealing tension with low-ℓ data that is masked in Bayesian analyses.

ABSTRACT

Cosmological parameter estimation is traditionally performed in the Bayesian context. By adopting an "agnostic" statistical point of view, we show the interest of confronting the Bayesian results to a frequentist approach based on profile-likelihoods. To this purpose, we have developed the Cosmological Analysis with a Minuit Exploration of the Likelihood ("CAMEL") software. Written from scratch in pure C++, emphasis was put in building a clean and carefully-designed project where new data and/or cosmological computations can be easily included. CAMEL incorporates the latest cosmological likelihoods and gives access from the very same input file to several estimation methods: (i) A high quality Maximum Likelihood Estimate (a.k.a "best fit") using MINUIT ; (ii) profile likelihoods, (iii) a new implementation of an Adaptive Metropolis MCMC algorithm that relieves the burden of reconstructing the proposal distribution. We present here those various statistical techniques and roll out a full use-case that can then used as a tutorial. We revisit the $Λ$CDM parameters determination with the latest Planck data and give results with both methodologies. Furthermore, by comparing the Bayesian and frequentist approaches, we discuss a "likelihood volume effect" that affects the optical reionization depth when analyzing the high multipoles part of the Planck data. The software, used in several Planck data analyzes, is available from http://camel.in2p3.fr. Using it does not require advanced C++ skills.

Motivation & Objective

• To develop a robust, modular cosmological analysis framework that supports both Bayesian and frequentist inference for direct comparison.
• To address the lack of frequentist methods in cosmology due to numerical noise in Boltzmann solvers by enabling precise profile-likelihood estimation.
• To investigate the impact of Bayesian priors and likelihood geometry on parameter constraints, especially in high-dimensional parameter spaces.
• To provide a unified, user-accessible tool (CAMEL) that requires no advanced C++ knowledge but supports extensibility for new data and models.
• To demonstrate that volume effects in likelihood shape can lead to systematic differences between Bayesian credible intervals and frequentist confidence intervals.

Proposed method

• CAMEL is implemented in pure C++ with object-oriented design and clear abstractions to ensure long-term maintainability and extensibility.
• It integrates state-of-the-art cosmological likelihoods (e.g., Hillipop, Plik) and Boltzmann solvers (e.g., CLASS, CAMB) via a modular 'Engine' interface.
• The framework supports multiple estimation methods: MINUIT-based Maximum Likelihood Estimation (MLE), profile-likelihoods, and a novel Adaptive Metropolis MCMC algorithm with automatic proposal tuning.
• The Adaptive Metropolis MCMC avoids manual proposal distribution design by learning from the chain’s past, improving convergence and sampling efficiency.
• Profile-likelihoods are computed by maximizing the likelihood over nuisance parameters for each fixed value of the parameter of interest.
• All methods use the same input configuration file, enabling consistent and reproducible cross-comparison of results.

Experimental results

Research questions

• RQ1 How do Bayesian posterior distributions and frequentist profile-likelihoods compare in constraining ΛCDM parameters using Planck high-ℓ data?
• RQ2 To what extent do 'likelihood volume effects' alter the inferred optical depth τ when comparing Bayesian and frequentist methods?
• RQ3 Can a unified software framework like CAMEL enable robust, agnostic cosmological inference without favoring one statistical paradigm?
• RQ4 How does the choice of statistical method affect the detection of tension between high-ℓ and low-ℓ Planck data for τ?
• RQ5 What is the impact of marginalization in Bayesian inference versus maximization in profile-likelihoods on the shape and location of credible/confidence intervals?

Key findings

• The profile-likelihood for τ from Planck high-ℓ data alone yields τ = 0.17 ± 0.04 (68% CL), indicating a higher optical depth estimate.
• The Bayesian posterior from the same high-ℓ data, obtained via Adaptive Metropolis MCMC, peaks at τ ≈ 0.15 and has a 68% credible interval of [0.11, 0.18], showing a systematic shift toward lower values.
• This discrepancy arises from 'likelihood volume effects', where the posterior is influenced by the geometric volume of the likelihood surface, while profile-likelihoods focus on the maximum at each point.
• The tension between high-ℓ and low-ℓ data is more pronounced in the profile-likelihood result (2.2σ) than in the posterior (1.8σ), suggesting Bayesian methods may obscure such tensions.
• The Hessian matrix from MINUIT provides an optimal starting covariance for the Adaptive Metropolis MCMC, improving convergence and reliability.
• CAMEL enables direct comparison of Bayesian and frequentist results using the same input, demonstrating that statistical methodology choice can significantly affect cosmological parameter inference.