LIKELIHOOD-FREE COSMOLOGICAL INFERENCE WITH TYPE Ia SUPERNOVAE: APPROXIMATE BAYESIAN COMPUTATION FOR A COMPLETE TREATMENT OF UNCERTAINTY

Abstract

Cosmological inference becomes increasingly difficult when complex data-generating processes cannot be modeled by simple probability distributions. With the everincreasing size of data sets in cosmology, there is increasing burden placed on adequate modeling; systematic errors in the model will dominate where previously these were swamped by statistical errors. For example, Gaussian distributions are an insufficient representation for errors in quantities like photometric redshifts. Likewise, it can be difficult to quantify analytically the distribution of errors that are introduced in complex fitting codes. Without a simple form for these distributions, it becomes difficult to accurately construct a likelihood function for the data as a function of parameters of interest. Approximate Bayesian computation (ABC) provides a means of probing the posterior distribution when direct calculation of a sufficiently accurate likelihood is intractable. ABC allows one to bypass direct calculation of the likelihood but instead relies upon the ability to simulate the forward process that generated the data. These simulations can naturally incorporate priors placed on nuisance parameters, and hence these can be marginalized in a natural way. We present and discuss ABC methods in the context of supernova cosmology using data from the SDSS-II Supernova Survey. Assuming a flat cosmology and constant dark energy equation of state we demonstrate that ABC can recover an accurate posterior distribution. Finally we show that ABC can still produce an accurate posterior distribution when we contaminate the sample with Type IIP supernovae.

Evidence of a Non-universal Stellar Initial Mass Function. Insights from HST Optical Imaging of Six Ultra-faint Dwarf Milky Way Satellites∗

“…ABC is an approach to Bayesian inference designed for problems where evaluating the actual posterior probability distribution is computationally infeasible or actually impossible. While astronomers have invented many ABC-like techniques, the conscious use of ABC in astronomy started with Cameron & Pettitt (2012) and Weyant et al (2013). Knowing that a technique is an example of ABC is useful because there is a large technical literature on the statistical properties of many types of ABC.…”

A new model to predict weak-lensing peak counts

“…This allows us to compare observations with our model without the need to define a likelihood function or to assume any Gaussian distribution. For example, model discrimination can be carried out using the false discovery rate method (FDR, Benjamini & Hochberg 1995, an application can be found in Pires et al 2009), approximate Bayesian computation (ABC, see for example Cameron & Pettitt 2012;Weyant et al 2013), or other statistical techniques. Another powerful advantage of our model is its flexibility.…”

The potential of likelihood-free inference of cosmological parameters with weak lensing data

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