Likelihood-free Inference with the Mixture Density Network
Guo-Jian Wang, Cheng Cheng, Yin-Zhe Ma, Jun‐Qing Xia
Abstract
Abstract In this work, we propose using the mixture density network (MDN) to estimate cosmological parameters. We test the MDN method by constraining parameters of the ΛCDM and w CDM models using Type Ia supernovae and the power spectra of the cosmic microwave background. We find that the MDN method can achieve the same level of accuracy as the Markov Chain Monte Carlo method, with a slight difference of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Furthermore, the MDN method can provide accurate parameter estimates with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> forward simulation samples, which are useful for complex and resource-consuming cosmological models. This method can process either one data set or multiple data sets to achieve joint constraints on parameters, extendable for any parameter estimation of complicated models in a wider scientific field. Thus, the MDN provides an alternative way for likelihood-free inference of parameters.