Introduction to ARES

The Algorithm for REconstruction and Sampling (ARES) is a full Bayesian large scale structure inference method targeted at precision recovery of cosmological power-spectra from three dimensional galaxy redshift surveys. Specifically it performs joint inferences of three dimensional density fields, cosmological power spectra as well as luminosity dependent galaxy biases and corresponding noise levels for different galaxy populations in the survey.

In order to provide full Bayesian uncertainty quantification the algorithm explores the joint posterior distribution of all these quantities via an efficient implementation of high dimensional Markov Chain Monte Carlo methods in a block sampling scheme. In particular the sampling consists in generating from a Wiener posterior distribution random realizations of three dimensional density fields constrained by data in the form of galaxy number counts. Following each generation, we produce conditioned random realizations of the power-spectrum, galaxy biases and noise levels through several sampling steps. Iterating these sampling steps correctly yields random realizations from the joint posterior distribution. In this fashion the ARES algorithm accounts for all joint and correlated uncertainties between all inferred quantities and allows for accurate inferences from galaxy surveys with non-trivial survey geometries. Classes of galaxies with different biases are treated as separate sub samples, allowing even for combined analyses of more than one galaxy survey.

For further information please consult our publications that are listed here.

Implementation: the ARES3 code

The ARES3 package comes with a basic flavour within the binary program “ares3”. “ares3” is an implementation of the algorithm outlined in the paper “Matrix free Large scale Bayesian inference” (Jasche & Lavaux 2014)

The ARES3 serves as a basis for number of extensions and modules. The minimal extension is the foreground sampler mechanism, that allows to fit some model of foreground contamination in large scale structure data. The second main module is the HADES sampler, which incorporates the HMC base definition and implementation alongside some likelihood models. The third module is the BORG sampler. It is a much more advanced likelihood analysis which incorporates non-linear dynnamics of the Large scale structures.

ARES model

The model implemented in ARES is the most simple ‘linear’ model. The density field is supposed to be a pure Gaussian random field, which linearly biased, selected and with a Gaussian error model. For a single catalog, the forward model corresponds to:

\(N^\mathrm{g}_p = \bar{N} R_p (1 + b \delta_p) + n_p\) with \(\langle n_p n_{p'} \rangle = R_p \bar{N} \delta^K_{p, p'}\)

\(\delta^K\) is the Kronecker symbol, \(R_p\) is the linear response of the survey, i.e. the 3d completeness, \(b\) the linear bias and \(\bar{N}\) the mean number of galaxies per grid element. Effectively \(\bar{N}\) will absorb the details of the normalization of \(R_p\).