Nuisance hardened data compression for fast likelihood-free inference (1903.01473v1)

Abstract: In this paper we show how nuisance parameter marginalized posteriors can be inferred directly from simulations in a likelihood-free setting, without having to jointly infer the higher-dimensional interesting and nuisance parameter posterior first and marginalize a posteriori. The result is that for an inference task with a given number of interesting parameters, the number of simulations required to perform likelihood-free inference can be kept (roughly) the same irrespective of the number of additional nuisances to be marginalized over. To achieve this we introduce two extensions to the standard likelihood-free inference set-up. Firstly we show how nuisance parameters can be re-cast as latent variables and hence automatically marginalized over in the likelihood-free framework. Secondly, we derive an asymptotically optimal compression from $N$ data down to $n$ summaries -- one per interesting parameter -- such that the Fisher information is (asymptotically) preserved, but the summaries are insensitive (to leading order) to the nuisance parameters. This means that the nuisance marginalized inference task involves learning $n$ interesting parameters from $n$ "nuisance hardened" data summaries, regardless of the presence or number of additional nuisance parameters to be marginalized over. We validate our approach on two examples from cosmology: supernovae and weak lensing data analyses with nuisance parameterized systematics. For the supernova problem, high-fidelity posterior inference of $\Omega_m$ and $w_0$ (marginalized over systematics) can be obtained from just a few hundred data simulations. For the weak lensing problem, six cosmological parameters can be inferred from $\mathcal{O}(10^3)$ simulations, irrespective of whether ten additional nuisance parameters are included in the problem or not.

• The paper introduces a novel method that recasts nuisance parameters as latent variables, streamlining likelihood-free inference.
• It derives an asymptotically optimal data compression scheme that preserves Fisher information while reducing the simulation overhead.
• Validation on supernova and weak lensing data demonstrates robust inference with fewer simulations, offering broad cross-disciplinary applications.

Insights on Nuisance Hardened Data Compression for Fast Likelihood-Free Inference

The paper authored by Alsing and Wandelt presents an innovative approach to streamline likelihood-free inference by directly marginalizing over nuisance parameters via refined data compression techniques. In the field of cosmological data analysis, researchers frequently confront the dilemma of needing to conduct inference over a limited number of interesting parameters while simultaneously grappling with a multitude of nuisance parameters. The approach delineated in this paper promises substantial computational efficiency, minimizing the dependency on the number of simulations typically required in high-dimensional spaces spanned by both interesting and nuisance parameters.

Key Contributions

1. Reconceptualizing Nuisance Parameters: The authors propose a paradigm shift where nuisance parameters are re-cast as latent variables in forward simulations. This transformation inherently allows marginalization over these nuisances within a likelihood-free framework, circumventing the conventional need for joint inference over interesting and nuisance parameters, followed by a posterior marginalization. This adjustment substantially cuts down on the complexity and simulation overhead.
2. Optimal Data Compression: A critical innovation is the derivation of an asymptotically optimal compression scheme. Large datasets are reduced to summary statistics that capture minimal yet sufficient (Fisher information preserving) essence of the interesting parameters. These summaries, identified as "nuisance hardened", are designed to be resilient to variations in the nuisance parameters, enabling them to enforce leading-order insensitivity to the nuisance factors.
3. Validation on Cosmological Data: The methodology is rigorously validated through examples involving cosmological data — specifically supernovae and weak lensing datasets. The supernova problem, characterized by systematics, demonstrates the capability to extract high-fidelity posterior inferences with reduced simulation counts in the order of hundreds, while the weak lensing problem achieves robust inference for multiple cosmological parameters with simulations numbering around a thousand, showing independence from the number of nuisance parameters involved.
4. Pseudo-Blackwell-Rao Estimator: The paper also suggests a method for reconstructing an approximate posterior for nuisance parameters post hoc, utilizing a pseudo-Blackwell-Rao estimator. This is accomplished without additional simulations, providing a method to approximate the nuisance posterior, though with recognition of the potential biases introduced due to assumptions in approximate likelihoods used in compression.

Implications and Speculations

The outlined paper presents practical and theoretical implications with the potential to significantly impact the practice of likelihood-free inference, particularly in domains with high-dimensional nuisance parameters and costly simulation requirements. The immediate implications are seen in expediting analyses where nuisance parameters proliferate, such as cosmological models.

The methodological rigor inherent in the compression scheme promises to be adaptable for various types of data beyond cosmology, possibly extending to fields like genomics and experimental physics where large numbers of nuisance parameters could otherwise impede comprehensive analyses. Future developments in this area may focus on further refining the compression schemes and exploring more adaptive methods that can dynamically identify compressions based on simulation outcomes, rather than predefined nuisance parameter structures.

In conclusion, by emphasizing both the reduction in computational resource allocation and preserving inferential robustness, this work by Alsing and Wandelt offers substantial contributions to the toolbox for cosmological inference, with clear pathways for cross-disciplinary application.