### Abstract

We describe a C++ framework that uses graphical processing units (GPUs) to accelerate basic hierarchical Bayesian computation, and use it to compute the posterior distribution for the parameters of a galaxy luminosity function based on data with photometric noise and selection effects arising from a flux-limited detection threshold. In addition to estimating the population-level luminosity function parameters, the framework also provides estimates of the absolute magnitude of each object, which automatically correct for Eddington bias. To sample the posterior, we implement a Metropolis–Hastings-within-Gibbs algorithm that alternates between exploring the population-level and member-level parameters. The algorithm exploits conditional independence in hierarchical Bayesian models, which makes member-level exploration readily parallelizable on a GPU. The framework uses adaptive MCMC, automatically tuning Metropolis–Hastings proposal distributions on-the-fly. We present a simulation study demonstrating the accuracy and computational scaling of the algorithm. In addition, we compare hierarchical Bayesian estimation with maximum likelihood estimation (known to provide inconsistent estimates in this setting), providing a concrete demonstration of the benefits of using hierarchical models for luminosity function inference.

Original language | English |
---|---|

Pages (from-to) | 247-256 |

Number of pages | 10 |

Journal | Astronomy and Computing |

Volume | 25 |

DOIs | |

Publication status | Published - Oct 1 2018 |

### Fingerprint

### Keywords

- Astrostatistics
- Graphical processing units (GPUs)
- Hierarchical Bayesian models
- Markov processes
- Metropolis-within-Gibbs sampling
- Parallel computing

### ASJC Scopus subject areas

- Astronomy and Astrophysics
- Computer Science Applications
- Space and Planetary Science

### Cite this

*Astronomy and Computing*,

*25*, 247-256. https://doi.org/10.1016/j.ascom.2018.10.004

**GPU-accelerated hierarchical Bayesian estimation of luminosity functions using flux-limited observations with photometric noise.** / Szalai-Gindl, J. M.; Loredo, T. J.; Kelly, B. C.; Csabai, I.; Budavári, T.; Dobos, L.

Research output: Contribution to journal › Article

*Astronomy and Computing*, vol. 25, pp. 247-256. https://doi.org/10.1016/j.ascom.2018.10.004

}

TY - JOUR

T1 - GPU-accelerated hierarchical Bayesian estimation of luminosity functions using flux-limited observations with photometric noise

AU - Szalai-Gindl, J. M.

AU - Loredo, T. J.

AU - Kelly, B. C.

AU - Csabai, I.

AU - Budavári, T.

AU - Dobos, L.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - We describe a C++ framework that uses graphical processing units (GPUs) to accelerate basic hierarchical Bayesian computation, and use it to compute the posterior distribution for the parameters of a galaxy luminosity function based on data with photometric noise and selection effects arising from a flux-limited detection threshold. In addition to estimating the population-level luminosity function parameters, the framework also provides estimates of the absolute magnitude of each object, which automatically correct for Eddington bias. To sample the posterior, we implement a Metropolis–Hastings-within-Gibbs algorithm that alternates between exploring the population-level and member-level parameters. The algorithm exploits conditional independence in hierarchical Bayesian models, which makes member-level exploration readily parallelizable on a GPU. The framework uses adaptive MCMC, automatically tuning Metropolis–Hastings proposal distributions on-the-fly. We present a simulation study demonstrating the accuracy and computational scaling of the algorithm. In addition, we compare hierarchical Bayesian estimation with maximum likelihood estimation (known to provide inconsistent estimates in this setting), providing a concrete demonstration of the benefits of using hierarchical models for luminosity function inference.

AB - We describe a C++ framework that uses graphical processing units (GPUs) to accelerate basic hierarchical Bayesian computation, and use it to compute the posterior distribution for the parameters of a galaxy luminosity function based on data with photometric noise and selection effects arising from a flux-limited detection threshold. In addition to estimating the population-level luminosity function parameters, the framework also provides estimates of the absolute magnitude of each object, which automatically correct for Eddington bias. To sample the posterior, we implement a Metropolis–Hastings-within-Gibbs algorithm that alternates between exploring the population-level and member-level parameters. The algorithm exploits conditional independence in hierarchical Bayesian models, which makes member-level exploration readily parallelizable on a GPU. The framework uses adaptive MCMC, automatically tuning Metropolis–Hastings proposal distributions on-the-fly. We present a simulation study demonstrating the accuracy and computational scaling of the algorithm. In addition, we compare hierarchical Bayesian estimation with maximum likelihood estimation (known to provide inconsistent estimates in this setting), providing a concrete demonstration of the benefits of using hierarchical models for luminosity function inference.

KW - Astrostatistics

KW - Graphical processing units (GPUs)

KW - Hierarchical Bayesian models

KW - Markov processes

KW - Metropolis-within-Gibbs sampling

KW - Parallel computing

UR - http://www.scopus.com/inward/record.url?scp=85056187465&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056187465&partnerID=8YFLogxK

U2 - 10.1016/j.ascom.2018.10.004

DO - 10.1016/j.ascom.2018.10.004

M3 - Article

VL - 25

SP - 247

EP - 256

JO - Astronomy and Computing

JF - Astronomy and Computing

SN - 2213-1337

ER -