Estimating Cluster Masses: A Comparative Study between Machine Learning and Maximum Likelihood

Abstract We compare an autoencoder convolutional neural network (AE-CNN) with conventional maximum-likelihood estimation (MLE) for inferring cluster virial masses, M v , directly from the galaxy distribution around clusters, without identifying members or interlopers. The AE-CNN is trained on mock galaxy catalogues, whereas the MLE assumes that clusters of similar mass share the same phase-space galaxy profile. Conceptually, the MLE returns an unbiased estimate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:msub> </mml:math> at fixed true mass, whereas the AE-CNN approximates the posterior mean, so the true <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:msub> </mml:math> is unbiased at fixed estimate. Using MultiDark Planck 2 mock clusters with redshift space number density as input, the AE-CNN attains an rms scatter of 0.10 dex between predicted and true <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:msub> </mml:math> , compared with 0.16 dex for the MLE. With inputs based on mean peculiar velocities, binned in redshift space or observed distance, the AE-CNN achieves scatters of 0.12 and 0.16 dex, respectively, despite strong inhomogeneous Malmquist bias.