Inverting Fisher biases for fast systematics exploration

1. Problem Statement

Upcoming Stage-IV cosmological surveys (e.g., LSST, Euclid) aim for sub-percent precision in constraining cosmological parameters. However, achieving this precision requires that systematic biases be kept below statistical errors.

• The Challenge: When tensions arise between different cosmological probes (e.g., Weak Lensing vs. CMB), researchers must determine if these discrepancies are due to new physics or unmodeled systematic effects.
• Current Limitation: The standard approach involves running expensive likelihood sampling algorithms (e.g., MCMC, Nested Sampling) repeatedly to test various systematic models and see if they can explain the observed tension. This is computationally prohibitive, especially during the "blinding" phases of surveys where rapid iteration is needed.
• The Goal: Develop a computationally efficient method to quickly assess whether a specific missing or mis-modeled systematic effect can explain a given tension between datasets without re-running full likelihood samplers.

2. Methodology

The authors propose a novel application of the Fisher Information Matrix formalism. While traditionally used to forecast parameter constraints or estimate biases forward (from systematics to parameters), this paper introduces an inverted approach.

Theoretical Framework

• Standard Fisher Bias: Traditionally, if a systematic parameter $\psi$ is fixed at a wrong value ($\psi_{fixed}$) instead of its fiducial value ($\psi_{fid}$), the best-fit cosmological parameters $\theta$ shift by $\Delta\theta$. This is approximated by:
  $$\Delta\theta = -(F^{\theta\theta})^{-1} F^{\theta\psi} \Delta\psi$$
  Where $F^{\theta\theta}$ is the Fisher matrix for cosmological parameters, $F^{\theta\psi}$ is the cross-term between cosmological and systematic parameters, and $\Delta\psi = \psi_{fixed} - \psi_{fid}$.
• Inversion: The authors invert this relation to solve for the systematic parameter shift required to explain an observed cosmological shift ($\Delta\theta_{obs}$):
  $$F^{\theta\psi} \Delta\psi = -F^{\theta\theta} \Delta\theta_{obs}$$
• Generalization: The method is generalized to $m$ cosmological parameters and $n$ systematic parameters.
  • Case $m \geq n$: The system is solvable (over-determined or square). The authors use this to find the specific value of $\Delta\psi$ that would generate the observed tension.
  • Case $m < n$: The system is under-determined and has no unique solution (not applicable in this specific formulation without further constraints).
• Validity: The derivation relies on a first-order Taylor expansion, assuming small offsets and Gaussian likelihoods. However, the authors argue that even for larger offsets, the method provides a robust indication of the magnitude and direction of the bias.

Implementation

The authors implemented this framework using a mock analysis of Stage-IV Weak Lensing data (simulating 1 year of LSST observations):

• Data Vector: Angular power spectra ($C_{\ell}$) computed using the Limber approximation and the Core Cosmology Library (CCL).
• Parameters: Cosmological parameters $\theta = (\Omega_m, A_s, w_0)$.
• Systematics Tested:
  1. Intrinsic Alignments (IA): Modeled via the Non-Linear Alignment (NLA) model with amplitude $A_{IA}$. This is a linear systematic in the data vector.
  2. Baryonic Feedback: Modeled via the HM-CODE2020 halo model, parametrized by the AGN feedback strength $\log(T_{AGN})$. This is a non-linear systematic.
• Validation: They generated synthetic data with a "true" systematic value, then analyzed it with a model where the systematic was fixed to a "wrong" value. They compared the Fisher-inverted prediction against the actual bias found via Nautilus (a nested sampler).

3. Key Contributions

1. Inverted Fisher Formalism: A new mathematical application of the Fisher matrix that allows researchers to work backward from an observed cosmological tension to estimate the required systematic parameter value.
2. Computational Efficiency: The method eliminates the need for expensive likelihood sampling when screening potential systematic causes for data tensions. It allows for rapid exploration of the systematic parameter space.
3. Robustness to Non-Linearity: While derived for linear regimes, the paper demonstrates that the method remains effective even for non-linear systematics (like baryonic feedback) and large biases, correctly predicting the direction and approximate magnitude of the shift.
4. Blinding Utility: The approach is particularly valuable during the blinding phases of surveys, where quick assessments of systematic impacts are required before unblinding the data.

4. Results

The authors validated their method using two case studies:

• Case 1: Intrinsic Alignments (Linear Systematic)

  • Setup: Synthetic data generated with $A_{IA} = 0.05$; analyzed with $A_{IA} = 0$.
  • Outcome: The Fisher prediction for the required $A_{IA}$ to explain the bias matched the Nautilus nested sampler results almost perfectly. The linear approximation held well because the data vector depends linearly on $A_{IA}$.
  • Conclusion: The method is highly accurate for linear systematics and small-to-moderate biases.

• Case 2: Baryonic Feedback (Non-Linear Systematic)

  • Setup: Synthetic data generated with strong feedback ($\log(T_{AGN}) = 8.0$); analyzed with weaker feedback ($\log(T_{AGN}) = 7.7$).
  • Outcome: Despite the non-linear nature of baryonic feedback and the resulting large bias (outside the $1\sigma$ contour), the Fisher-inverted prediction aligned closely with the Nautilus posterior contours.
  • Conclusion: Even when the linear approximation is stretched, the method successfully identifies the correct direction and magnitude of the systematic shift needed to resolve the tension.

5. Significance and Future Applications

• Decision Making: This tool allows cosmologists to quickly rule out (or prioritize) specific systematic effects as the cause of tensions between datasets (e.g., $S_8$ tension) without waiting for days of MCMC runs.
• Model Selection: It helps determine if a tension is "plausible" under a specific systematic model. If the required systematic shift is physically impossible (e.g., outside prior bounds), that systematic can be discarded as the cause.
• Generalizability: The method is agnostic to the specific systematic and can be applied to various scenarios, including redshift uncertainties, magnification bias, and higher-order perturbation models (e.g., TATT models).
• Limitations: The method is mathematically rigorous only for small biases and smooth likelihoods. For highly non-Gaussian likelihoods or very large biases, higher-order Fisher terms (e.g., using DALI) or full sampling may eventually be required, though the inverted Fisher approach still serves as a valuable "sanity check" for the direction of the bias.

In summary, this paper provides a powerful, low-cost diagnostic tool for the next generation of cosmological surveys, enabling faster and more robust identification of systematic errors that could mimic new physics.