The INT Program INT-16-2a on "Bayesian Methods in Nuclear Physics" was held from June 13 through July 8, 2016. It had 51 registered participants and 35 scheduled talks with focused afternoon discussions. Bayesian statistics is a well-developed field, although it has not been part of the traditional education of nuclear theorists. In schematic form, Bayesian statistics treats the parameters or the model/theory as genuine random variables. It then uses Bayes theorem of probabilities to provide a recipe to compute their probability distribution (the "posterior") in terms of prior information (e.g., about the data) and a likelihood function. For applications to tting ("parameter estimation"), the posterior lets us infer, given the data we have measured, the most probable values of the parameters and predict values of observables with confidence intervals. Other applications involve deciding between alternative explanations or parameterizations ("model selection"). In practice, there are pitfalls in the implementation of this formalism and it is often a computationally hard problem.

Interest in Bayesian statistics has increased significantly in the past 10 years. The wide availability of large-scale computing resources has made the computation of the integrals needed for Bayesian inference easier. Modern experimental and observational facilities generate large amounts of data, often best analyzed and characterized through Bayesian methods. Bayesian methods are often preferred for under-constrained fits and inverse convolutions. In nuclear science, Bayesian methods have found their way into such areas as nuclear data, lattice QCD, dense matter, effective field theory, nuclear reactions, and parton distribution functions. These sub-fields have generally turned to Bayesian inference methods independently and in some cases without access to expert advice and guidance from professional statisticians.

This program brought statisticians and nuclear practitioners together to explore how Bayesian inference could enable progress on the frontiers of nuclear physics and open up new directions for the field. The program was able to

• facilitate cross communication and collaboration on Bayesian applications among the nuclear sub-fields;
• provide the opportunity for nuclear physicists who were unfamiliar with Bayesian methods to start applying them to new problems;
• learn from the experts about innovative and advanced uses of Bayesian statistics, and best practices in applying them;
• learn about advanced computational tools and methods;
• critically examine the application of Bayesian methods to particular physics problems in the various subfields.

Broad communication was achieved through two morning talks by participants on particular applications of Bayesian methods, supplemented by informal afternoon meetings. In addition, statisticians gave an overview lecture/tutorial on Bayesian inference and computational methods each Monday morning, and Friday were entirely devoted to discussions on specific topics that arose during the week, with impromptu presentations and white board sessions. Among the various topics discussed repeatedly in the program were:

• Parameter estimation, model calibration, and model selection;
• Gaussian processes: what, why, how;
• Prior distributions: definition, validation, verification;
• Markov Chain Monte Carlo (MCMC) software;
• How to quantify systematic errors with Bayesian techniques;
• Bayesian model calibration and Approximate Bayesian Computation (ABC).