Practical Guidance for Bayesian Inference in Astronomy

The paper addresses the increasing adoption of Bayesian inference in astronomy and the accompanying challenges due to inconsistent notation, terminology, and methodological choices across sub-fields and between astronomy and statistics. While Bayesian methods offer advantages such as full posterior uncertainty quantification, interpretability of credible intervals, natural marginalization over nuisance parameters, and incorporation of measurement error and incomplete data, training in the field has lagged behind usage. The study’s purpose is to translate and harmonize concepts and notation between disciplines and to provide practical guidance for specifying Bayesian models, choosing likelihoods and priors, making posterior-based inferences, and performing posterior predictive checks. The importance lies in improving clarity, reproducibility, and robustness of Bayesian analyses in astronomy, thereby facilitating scientific discovery.

The article synthesizes relevant work from both astronomy and statistics. It references common Bayesian software and algorithms used in astronomy (e.g., emcee, Stan, PyMC3, BUGS, NIMBLE, JAGS) and foundational and applied texts in Bayesian data analysis (e.g., Gelman et al., Carlin & Louis). It surveys literature on likelihood principles and modeling (e.g., Schervish; Berger & Wolpert), prior specification and their impacts (e.g., Jeffreys’ priors, empirical Bayes, effective prior sample size methods), and computational methods for posterior approximation (e.g., MCMC, INLA, variational Bayes, ABC). In astronomy, it cites applications such as Gaia parallax inference, exoplanet detection, and Hubble constant estimation, and prior work on distance inference from parallax (Bailer-Jones and collaborators). The review is integrated throughout sections rather than as a standalone review, providing context for best practices and pitfalls.

The paper provides methodological guidance rather than a single empirical protocol, structured around a running example of distance inference from parallax measurements. - Notation and Bayes’ theorem: Define data vectors and parameters; the posterior p(θ|y) ∝ p(y|θ)p(θ) with p(y) the normalizing constant. Emphasize the distinction between the likelihood function viewed as a function of parameters given fixed data and probability density functions, and provide translation of notation used across disciplines. - Likelihood specification: Stress careful choice of the sampling distribution reflecting the data-generating process and measurement pipeline. Warn against default Gaussian assumptions when Poisson or t-distributions may be more appropriate, or when temporal/spatial correlation violates exchangeability. Encourage explicit mathematical specification including truncation, censoring, measurement errors, correlated errors, biases, sampling bias, and missing data. - Running parallax example (single star): Observed parallax ϖ with known standard error σϖ is modeled as ϖ|π ∼ Normal(π, σϖ^2), with distance r related by r = 1/π. Likelihood in terms of r becomes Normal(ϖ | 1/r, σϖ^2). - Prior specification: Contrast improper “flat” priors with proper, weakly-informative, and scientifically motivated priors. For parallax/distance, evaluate (i) truncated uniform prior on parallax, (ii) truncated uniform prior on distance, and (iii) the exponentially decreasing space density prior r^2 exp(−r/L) truncated to [r_min, r_max], motivated by spatial volume and a disk scale length L (Bailer-Jones et al.). Emphasize that uniformity is not invariant to reparameterization and that priors must respect physical constraints and be fully specified for reproducibility. - Posterior inference: Compute posteriors p(r|ϖ) ∝ likelihood × prior for each prior choice; closed forms are unavailable but can be approximated via MCMC or other methods. Discuss posterior summaries (means, medians, modes, credible intervals), visualization via corner plots and density plots, and propagation of uncertainty to derived quantities via posterior samples. - Extended example (star cluster): With N stars assumed at common distance r_cluster, combine independent Gaussian likelihoods to form a joint likelihood; the effective parallax ϖ_eff is a precision-weighted combination. The posterior for r_cluster (or π_cluster) is obtained using a prior for the cluster parallax; demonstrate how increasing data reduces prior influence. - Posterior predictive checking: Define posterior predictive distribution p(ỹ|y) and recommend simulating from it to compare observed and replicated data. Advocate Q–Q plots over overlaid densities for assessing discrepancies. In the cluster example, Q–Q plots reveal tail mismatches, prompting model reconsideration (e.g., allowing distance dispersion or uncertain measurement variances). - Sensitivity analyses and EPSS: Recommend sensitivity analyses over different priors and introduce effective prior sample size to quantify prior influence, with simple calculation in conjugate settings and references for generalizations beyond conjugacy.

- Harmonized terminology and notation: Clarifies differences across astronomy and statistics, particularly around the likelihood function, random variables, densities, and target distributions, facilitating cross-disciplinary understanding. - Likelihood choice matters: Demonstrates that default Gaussian assumptions can be inappropriate; proper modeling of truncation, censoring, measurement error, correlation, and biases is essential. - Prior influence is substantial: For parallax-to-distance inference, different weakly or physically motivated priors produce notably different posterior shapes and summaries, even with the same data. Uniformity depends on parameterization, and physically motivated priors (e.g., r^2 exp(−r/L)) often better reflect astrophysical realities. - Data aggregation reduces prior impact: In the star cluster example, as more stellar measurements are included, the data increasingly dominate the posterior, diminishing sensitivity to the prior. - Posterior predictive checking reveals model shortcomings: Q–Q plots of simulated vs. observed parallaxes show discrepancies in distribution tails, suggesting potential model refinements (e.g., allowing intra-cluster distance dispersion or modeling uncertainty in σϖ). - Practical best practices: Enumerates concrete recommendations—explicit model specification, careful likelihood construction, informative priors when justified, sensitivity analyses, posterior visualization and summaries, and posterior predictive checks. - Adoption context: Notes rapid growth of Bayesian usage in astronomy (over 2,377 refereed papers with Bayesian content on ADS; over 237 in 2021), underscoring the need for standardized guidance.

The guidance addresses the core challenges of inconsistent terminology, notation, and modeling choices by providing a cross-disciplinary translation and a set of best practices. Through the parallax distance example, the paper demonstrates how choices of likelihood and prior directly affect posterior inferences and derived scientific conclusions. Posterior predictive checks and sensitivity analyses are shown as practical tools to diagnose model fit and quantify the influence of prior assumptions. The relevance to astrostatistics is substantial: improved clarity and rigor in Bayesian workflows enhance reproducibility, facilitate collaboration between astronomers and statisticians, and lead to more reliable inference in contexts with complex noise, measurement processes, and non-Gaussian data common in astronomy.

The paper consolidates and clarifies Bayesian terminology and notation for astronomers and offers practical guidance for model specification, prior and likelihood selection, posterior inference, visualization, sensitivity analyses, and posterior predictive checking. Key recommendations include: explicitly specifying models and using correct terminology; constructing likelihoods as functions of parameters given the data and including all material contributors from the measurement process; preferring informative, physically motivated priors when justified and carefully evaluating non-informative priors under reparameterization; testing posterior sensitivity to priors; providing open-source code; and performing posterior predictive checks with appropriate diagnostics like Q–Q plots. The authors emphasize that Bayesian inference is broad and that many subtopics were not covered, encouraging future, more specialized contributions and continued interdisciplinary collaboration.

- The work is a practical guide rather than a comprehensive introduction to Bayesian analysis; many subtopics (e.g., detailed computational algorithms, hierarchical modeling variants, model comparison, rigorous theory of priors) are only briefly mentioned or omitted. - Illustrations rely on a simplified running example (parallax-based distance inference) with assumptions such as known Gaussian measurement errors; real-world complexities (e.g., distance dispersion within clusters, non-Gaussian errors, unknown error variances) may require more elaborate models. - Posterior predictive checks in the example reveal tail discrepancies, indicating potential model misspecification that is not fully resolved within the paper. - The paper does not present new empirical datasets or benchmarks beyond illustrative examples; conclusions are primarily methodological recommendations.