Reporting Bayesian Analysis

Reporting Bayesian Analyses in Research Manuscripts

Bayesian analysis is rapidly gaining popularity across a wide array of research fields, and offers a flexible, intuitive, approachable framework that quantifies the incorporation of prior knowledge into statistical modeling

Reporting Bayesian Analyses

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Introduction

Bayesian analysis is rapidly gaining popularity across a wide array of research fields, and offers a flexible, intuitive, approachable framework that quantifies the incorporation of prior knowledge into statistical modeling and provides researchers with methods of interpretation that are more suited to uncertainty and prior data. Reporting Bayesian analyses appropriately is imperative to provide transparency, reproducibility, and intellectual rigor. The guidelines below are intended to help researchers to report Bayesian analyses in a way that clearly communicates and is thorough in helping readers discern methods, thereby enabling interpretable and robust results.

1. Specify Prior Probabilities

  • The first step in any Bayesian analysis is to provide prior probabilities. Priors can sometimes elicit the underlying beliefs and information researchers possess regarding the parameters before observing the data. The selection of priors can create large differences for the analysis.
  • For example, in clinical research, is the prior based on earlier studies, or expert opinion (Goligher et al., 2024). In contrast, more non-informative priors can be used if there is a lack of prior information as well if the intention is to let the data drive the conclusions.
  • Priors can take the context of informative priors and uninformative priors. Informative priors are beliefs that researchers base on significant prior knowledge or substantial belief significant characteristics about the parameters. With uninformative priors, the researcher selects the prior with the intent of having minimal effects on the analysis and scientifically provides a justification when little or unproductive prior information is available (Frühwirth-Schnatter et al., 2025).
  • The clear specification of priors is a prerequisite for establishing that the analysis is based on correct prior assumptions and is necessary for attributes like reproducibility with Bayesian methods.

    • 2. Justify the Selection of Priors

  • It’s not enough to simply describe the priors considered (a) so that research can illuminate novel hypotheses; and (b) set criteria for replication. Researchers are encouraged to justify why they selected those priors. Priors can be classified and selected based on different criteria (Reddin et al., 2023).
  • These include using prior published studies, engaging subject matter experts, or utilizing empirical data. There are also instances where priors can be based on theoretical underpinnings or the shape of the model.
  • In clinical trials, for example, informative prior distributions will be selected. Informative priors can provide the necessary basis for the analysis because they are based on knowledge from previous studies that are similar (Muehlemann et al., 2023). In alternative cases, researchers may select uninformative prior distributions to avoid imposing significant weight on the prior to make moderate assumptions.
  • Regardless of the approach selected to select priors, the author(s) should make it clear how they selected the priors, and if feasible, conduct a sensitivity analysis to determine how the study’s results might differ under different prior distributions (Liu et al., 2023).

    • 3. Describe the Statistical Model

  • Once priors have been established, it is necessary for the researcher to also clarify the statistical model being invoked in the Bayesian analysis, which consists of the likelihood function specifying how the data have assumed to have been generated and the way in which the data are related to the parameters (Cox et al., 2023).
  • The posterior distribution obtained from a Bayesian analysis is an important outcome and is obtained by combining the prior with the likelihood, producing an updated belief about the parameters after having viewed the data. The posterior distribution provides a full suite of potential values of the parameters rather than a point estimate so that researchers can be much richer in their conclusions.
  • As an example from behavioural research, Bayesian network meta-analysis can be conducted to synthesize the results of multiple studies (Liu et al., 2023). By also explaining the model assumptions and model structure, the reader is made aware of the foundations of the analysis and is better able to assess the veracity of the analysis – even in prior and posterior values only, some understanding of the analysis is genuine and possible for them to understand even when they do not analyse the totally expected posterior.

    • 4. Detail the Analytical Techniques

  • Bayesian analysis typically involves special procedures to derive the posterior distributions. There are a few different models (e.g., Markov Chain Monte Carlo (MCMC), Gibbs sampling, Hamiltonian Monte Carlo (HMC)) (Pfadt et al., 2023), and different procedures, such as simulating values, and sampling from a posterior distribution to approximate estimators of the parameters of interest.
  • The researcher needs to state the number of iterations used for the simulation, the burn-in time (if there is one), and any convergence diagnostics (e.g., trace plots, R-hat), because these are all important for establishing that the analyses converged to a given solution and that the results of the analyses are trustworthy (Jiang et al., 2023).
  • The analytical software and tools must be clearly identified. There are standard software for Bayesian analysis purposes; they are R, and all its packages (e.g. rstan, brms), and the dedicated tools, WinBUGS, JAGS, and PyMC. It is important for researchers to provide the software exact version they applied for reproducibility (Huth et al., 2023).
  • For example, in R, it is important to indicate the package you used, and the version, as well as R code that was directly related to any customization or scripts that may have been developed to customize the standard tools for the next researcher to replicate the analysis.

    • 6. Summarize the Posterior Distribution

  • Once the posterior distribution has been obtained, researchers should report the important summaries, focus on the measure of central tendency such as the posterior mean, median or mode. In contrast to frequentist methods that would use confidence intervals, credible intervals are the most common summary of a Bayesian analysis. Although most of the time people use the same 95% credible interval, it is important to understand that a 95% credible interval says that given the data and prior beliefs, the parameter is likely to fall somewhere in that interval.
  • For example, the researchers would say: “the posterior mean of the parameter is 1.85, with a 95% credible interval of 1.32–2.48” (Reddin et al., 2023). This is much more informative way of expressing uncertainty, especially to report about a probability that the parameter lies in the interval.

    • 7. Conduct Sensitivity Analysis

  • Sensitivity analysis is an important part of Bayesian analysis to assess if results change if different priors are used. Sensitivity analysis is important to assess the robustness of conclusions and make sure conclusions are not overly dependent on the priors (Muehlemann et al., 2023).
  • Sensitivity analysis is also a way to assess posterior estimates with alternative prior distributions when the analysis is rerun with different assumptions. This is an important aspect to begin to restore transparency in the process with information about how the conclusions hold up against differences in conditions.

    • Summary Checklist for Reporting Bayesian Analyses

    Reporting Element Required?
    Priors specified and justified Yes
    Statistical model clearly described Yes
    Computational techniques explained Yes
    Software/tools identified Yes
    Posterior summaries with credibility intervals Yes
    Sensitivity to priors assessed Recommended

    Conclusion

    In summary, Bayesian scientific knowledge provides a versatile framework for including knowledge of what has come before in a statistical model. Following the above recommendations will allow researchers reporting Bayesian analyses to make their findings more transparent, reproducible, and methodologically rigorous. Clearly reporting when priors are used, stipulating and justifying the priors, describing the statistical model, providing details of the computational and statistical routines, and performing sensitivity analyses will have a tremendous impact on the credibility of the analysis. By reporting in such elaborate detail, researchers also evolve knowledge and expand the ideas of Bayesian knowledge.
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    • References

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    2. Frühwirth-Schnatter, S., Hosszejni, D., & Lopes, H. F. (2025). Sparse Bayesian Factor Analysis when the number of factors is unknown (with Discussion). Bayesian Analysis, 20(1), 213-344.

    3. Goligher, E. C., Heath, A., & Harhay, M. O. (2024). Bayesian statistics for clinical research. The Lancet, 404(10457), 1067-1076.

    4. Huth, K. B., de Ron, J., Goudriaan, A. E., Luigjes, J., Mohammadi, R., van Holst, R. J., … & Marsman, M. (2023). Bayesian analysis of cross-sectional networks: A tutorial in R and JASP. Advances in Methods and Practices in Psychological Science, 6(4), 25152459231193334.

    5. Jiang, J. L., Ecker, C., & Rezzolla, L. (2023). Bayesian analysis of neutron-star properties with parameterized equations of state: The role of the likelihood functions. The Astrophysical Journal, 949(1), 11.

    6. Liu, Y., Béliveau, A., Wei, Y., Chen, M. Y., Record-Lemon, R., Kuo, P. L., … & Chen, G. (2023). A gentle introduction to Bayesian network meta-analysis using an automated R package. Multivariate Behavioral Research, 58(4), 706-722.

    7. Muehlemann, N., Zhou, T., Mukherjee, R., Hossain, M. I., Roychoudhury, S., & Russek-Cohen, E. (2023). A tutorial on modern Bayesian methods in clinical trials. Therapeutic Innovation & Regulatory Science, 57(3), 402-416.

    8. Pfadt, J. M., Bergh, D. V. D., Sijtsma, K., & Wagenmakers, E. J. (2023). A tutorial on Bayesian single-test reliability analysis with JASP. Behavior Research Methods, 55(3), 1069-1078.

    9. Reddin, C., Ferguson, J., Murphy, R., Clarke, A., Judge, C., Griffith, V., … & O’Donnell, M. J. (2023). Global mean potassium intake: a systematic review and Bayesian meta-analysis. European Journal of Nutrition, 62(5), 2027-2037.

    10. Sun, X., Hu, Y., Qin, Y., & Zhang, Y. (2024). Risk assessment of unmanned aerial vehicle accidents based on data-driven Bayesian networks. Reliability Engineering & System Safety, 248, 110185.