### Abstract

The well-known Bayes theorem assumes that a posterior distribution is a probability distribution. However, the posterior distribution may no longer be a probability distribution if an improper prior distribution (non-probability measure) such as an unbounded uniform prior is used. Improper priors are often used in the astronomical literature to reflect a lack of prior knowledge, but checking whether the resulting posterior is a probability distribution is sometimes neglected. It turns out that 23 out of 75 articles (30.7 per cent) published online in two renowned astronomy journals (ApJ and MNRAS) between 2017 Jan 1 and Oct 15 make use of Bayesian analyses without rigorously establishing posterior propriety. A disturbing aspect is that a Gibbs-type Markov chain Monte Carlo (MCMC) method can produce a seemingly reasonable posterior sample even when the posterior is not a probability distribution (Hobert & Casella 1996). In such cases, researchers may erroneously make probabilistic inferences without noticing that the MCMC sample is from a non-existing probability distribution. We review why checking posterior propriety is fundamental in Bayesian analyses, and discuss how to set up scientifically motivated proper priors.

Original language | English (US) |
---|---|

Pages (from-to) | 277-285 |

Number of pages | 9 |

Journal | Monthly Notices of the Royal Astronomical Society |

Volume | 481 |

Issue number | 1 |

DOIs | |

State | Published - Nov 21 2018 |

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### All Science Journal Classification (ASJC) codes

- Astronomy and Astrophysics
- Space and Planetary Science

### Cite this

*Monthly Notices of the Royal Astronomical Society*,

*481*(1), 277-285. https://doi.org/10.1093/mnras/sty2326

}

*Monthly Notices of the Royal Astronomical Society*, vol. 481, no. 1, pp. 277-285. https://doi.org/10.1093/mnras/sty2326

**How proper are Bayesian models in the astronomical literature?** / Tak, Hyungsuk; Ghosh, Sujit K.; Ellis, Justin A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - How proper are Bayesian models in the astronomical literature?

AU - Tak, Hyungsuk

AU - Ghosh, Sujit K.

AU - Ellis, Justin A.

PY - 2018/11/21

Y1 - 2018/11/21

N2 - The well-known Bayes theorem assumes that a posterior distribution is a probability distribution. However, the posterior distribution may no longer be a probability distribution if an improper prior distribution (non-probability measure) such as an unbounded uniform prior is used. Improper priors are often used in the astronomical literature to reflect a lack of prior knowledge, but checking whether the resulting posterior is a probability distribution is sometimes neglected. It turns out that 23 out of 75 articles (30.7 per cent) published online in two renowned astronomy journals (ApJ and MNRAS) between 2017 Jan 1 and Oct 15 make use of Bayesian analyses without rigorously establishing posterior propriety. A disturbing aspect is that a Gibbs-type Markov chain Monte Carlo (MCMC) method can produce a seemingly reasonable posterior sample even when the posterior is not a probability distribution (Hobert & Casella 1996). In such cases, researchers may erroneously make probabilistic inferences without noticing that the MCMC sample is from a non-existing probability distribution. We review why checking posterior propriety is fundamental in Bayesian analyses, and discuss how to set up scientifically motivated proper priors.

AB - The well-known Bayes theorem assumes that a posterior distribution is a probability distribution. However, the posterior distribution may no longer be a probability distribution if an improper prior distribution (non-probability measure) such as an unbounded uniform prior is used. Improper priors are often used in the astronomical literature to reflect a lack of prior knowledge, but checking whether the resulting posterior is a probability distribution is sometimes neglected. It turns out that 23 out of 75 articles (30.7 per cent) published online in two renowned astronomy journals (ApJ and MNRAS) between 2017 Jan 1 and Oct 15 make use of Bayesian analyses without rigorously establishing posterior propriety. A disturbing aspect is that a Gibbs-type Markov chain Monte Carlo (MCMC) method can produce a seemingly reasonable posterior sample even when the posterior is not a probability distribution (Hobert & Casella 1996). In such cases, researchers may erroneously make probabilistic inferences without noticing that the MCMC sample is from a non-existing probability distribution. We review why checking posterior propriety is fundamental in Bayesian analyses, and discuss how to set up scientifically motivated proper priors.

UR - http://www.scopus.com/inward/record.url?scp=85054084560&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054084560&partnerID=8YFLogxK

U2 - 10.1093/mnras/sty2326

DO - 10.1093/mnras/sty2326

M3 - Article

AN - SCOPUS:85054084560

VL - 481

SP - 277

EP - 285

JO - Monthly Notices of the Royal Astronomical Society

JF - Monthly Notices of the Royal Astronomical Society

SN - 0035-8711

IS - 1

ER -