A preferential attachment model for the stellar initial mass function

Abstract

Accurate specification of a likelihood function is becoming increasingly difficult in many inference problems in astronomy. As sample sizes resulting from astronomical surveys continue to grow, deficiencies in the likelihood function lead to larger biases in key parameter estimates. These deficiencies result from the oversimplification of the physical processes that generated the data, and from the failure to account for observational limitations. Unfortunately, realistic models often do not yield an analytical form for the likelihood. The estimation of a stellar initial mass function (IMF) is an important example. The stellar IMF is the mass distribution of stars initially formed in a given cluster of stars, a population which is not directly observable due to stellar evolution and other disruptions and observational limitations of the cluster. There are several difficulties with specifying a likelihood in this setting since the physical processes and observational challenges result in measurable masses that cannot legitimately be considered independent draws from an IMF. This work improves inference of the IMF by using an approximate Bayesian computation approach that both accounts for observational and astrophysical effects and incorporates a physically-motivated model for star cluster formation. The methodology is illustrated via a simulation study, demonstrating that the proposed approach can recover the true posterior in realistic situations, and applied to observations from astrophysical simulation data.

Approximate Bayesian computation for finite mixture models

“…First, to select the sequence 1:T we implement an adaptive approach based on the accepted distances from the previous time step. That is, t is set to a particular quantile of the accepted distances from time step t−1 [32,[44][45][46][47][48]. Selecting too high of a quantile at which to shrink the tolerance could result in a decrease in computational efficiency because more iterations would be needed to shrink the tolerance to a small enough value.…”

“…The decreasing tolerance sequence's impact on the achievement of the global maximum is addressed in two ways in the proposed algorithm. First, we initialize the tolerance sequence by oversampling in the first iteration of the ABC-PMC algorithm [47]. Let N be the desired number of particles used to approximate the posterior distribution, then the initial tolerance, 1 , can be adaptively selected by sampling N init = lN draws from the prior, for some l ∈ Z + .…”

Adaptive Approximate Bayesian Computation Tolerance Selection

“…J=1 , the distances of the accepted particles from iteration t − 1 (Cisewski- Kehe et al, 2019;Ishida et al, 2015;Lenormand et al, 2013;Simola et al, 2019;Weyant et al, 2013), or (iii) adaptively selecting t based on some quantile of the effective sample size (ESS) values (Del Moral et al, 2012;Numminen et al, 2013). These approaches can lead to inefficient sampling as discussed below and demonstrated in the simulation study in Section 3.…”

“…Approximate Bayesian Computation (ABC) provides a framework for inference in situations where the relationship between the data and the parameters does not lead to a tractable likelihood function, but where forward simulation of the data-generating process is possible. ABC has been used in many areas of science such as biology (Thornton and Andolfatto, 2006), epidemiology (McKinley et al, 2009;Numminen et al, 2013), ecology (Beaumont, 2010), population modeling (Toni et al, 2009), modeling the population effects of a vaccine (Corander et al, 2017), dark matter direct detection (Simola et al, 2019), and astronomy (Cameron and Pettitt, 2012;Cisewski-Kehe et al, 2019;Ishida et al, 2015;Schafer and Freeman, 2012;Weyant et al, 2013). The basic ABC algorithm (Pritchard et al, 1999;Rubin, 1984;Tavaré et al, 1997) can be explained in four steps.…”

“…ABC has been successfully used in numerous situations where the likelihood function is intractable and the number of parameters varies from 2 to 5 (e.g. Beaumont et al 2009;Cisewski-Kehe et al 2019;Csilléry et al 2010;Cornuet et al 2008;Del Moral et al 2012;Gutmann and Corander 2016;Järvenpää et al 2016;Jennings and Madigan 2016;Numminen et al 2013;Silk et al 2013;Simola et al 2019;Sisson et al 2007;Toni et al 2009). Our algorithm is designed to significantly improve upon the original ABC-PMC method under similar circumstances.…”