CoLFI: Cosmological Likelihood-free Inference with Neural Density Estimators

TL;DR

This paper introduces CoLFI, a neural density estimator combining ANN and MDN, to efficiently estimate cosmological parameters with high accuracy using fewer simulations, especially when likelihoods are complex or intractable.

Contribution

The paper proposes the mixture neural network (MNN) and sampling in hyper-ellipsoids, improving parameter estimation efficiency and accuracy over previous neural network methods in cosmology.

Findings

Achieved high-fidelity posterior distributions with ~100 simulations.

Matched results of traditional MCMC methods with minimal numerical difference.

Extended applicability to higher-dimensional data.

Abstract

In previous works, we proposed to estimate cosmological parameters with the artificial neural network (ANN) and the mixture density network (MDN). In this work, we propose an improved method called the mixture neural network (MNN) to achieve parameter estimation by combining ANN and MDN, which can overcome shortcomings of the ANN and MDN methods. Besides, we propose sampling parameters in a hyper-ellipsoid for the generation of the training set, which makes the parameter estimation more efficient. A high-fidelity posterior distribution can be obtained using $\mathcal{O}(10^2)$ forward simulation samples. In addition, we develop a code-named CoLFI for parameter estimation, which incorporates the advantages of MNN, ANN, and MDN, and is suitable for any parameter estimation of complicated models in a wide range of scientific fields. CoLFI provides a more efficient way for parameter estimation, especially for cases where the likelihood function is intractable or cosmological models are complex and resource-consuming. It can learn the conditional probability density $p(\boldsymbol\theta|\boldsymbol{d})$ using samples generated by models, and the posterior distribution $p(\boldsymbol\theta|\boldsymbol{d}_0)$ can be obtained for a given observational data $\boldsymbol{d}_0$. We tested the MNN using power spectra of the cosmic microwave background and Type Ia supernovae and obtained almost the same result as the Markov Chain Monte Carlo method. The numerical difference only exists at the level of $\mathcal{O}(10^{-2}\sigma)$. The method can be extended to higher-dimensional data.