# Bayesian Methods in Nuclear Physics

A workshop on Bayesian Methods in Nuclar Physics was held at the Institute for Nuclear Theory at the University of Washington in Seattle from June 13 to July 8, 2016. These pages continue the discussion initiated at this program.

Goal: For statisticians and nuclear practitioners to jointly explore how Bayesian inference can enable progress on the frontiers of nuclear physics and open up new directions for the field.

## Questions and Discussion Topics

The following is a (partial) list of questions that were considered during the INT program. We strongly encourage participants and others to add their own questions (and answers) as well.

(Participants are also welcome to join the github project. To do so, please send us your github username.)

General questions:

• What do Bayesian techniques offer that frequentist statistics do not?
• Also, what kinds of problems ill-suited for Bayesian or frequentist approaches?
• What is the modern view of the conflict (if any) between Bayesian and frequentist statistics?
• What are the best references (e.g., texts or pedagogical reviews) for introductory Bayesian statistics and for advanced topics?
• As we compile lists: What are we missing? Are there more modern versions?
• What are the common or subtle pitfalls that novices to Bayesian methods fall into?
• What are we likely unaware of on the frontier of (Bayesian) statistical methods?
• D. Furnstahl: In interacting with applied mathematicians I've found that physicists are often using the Numerical Recipes version of numerical methods, while the state-of-the-art is one or two generations more advanced. What are the analogs for statistics?
• A Steiner: I'm currently using Goodman and Weare (2010)'s affine-invariant MCMC. Is there any way to do better? I'd like to get more accurate results with fewer samples. Will Metropolis-Hastings methods be superior if I have a sufficiently accurate proposal distribution?

Parameter estimation, model calibration, and model selection

• What is the difference between model calibration and parameter estimation?
• How should one do basic regression analysis?
• The old-school theoretical physics way is to do a least-squares fit with adding penalty terms for theoretical errors (which could be from the model or from the numerical method used to calculate the model) in quadrature to the data errors.
• When the theoretical systematic uncertainty is not known, one often determines the overall scale by requiring $\chi^2/\mathrm{dof} = 1$ (Birge factor). How is this done in Bayesian statistics?
• When should a nuclear model with systematic theory errors have a statistical distribution of residuals?
• What are appropriate Bayesian priors?
• A. Steiner: How does one deal with the ambiguity created by heteroscedasticity? E.g. if we have two types of data points in a $\chi^2$ fit, how do we decide the relative theoretical uncertainty between the two types?
• What approximations or techniques are useful for reducing computational cost?
• What is Approximate Bayesian Computation?
• What method should I use for calculating the evidence or odds ratios?
• e.g., simulated annealing, nested sampling, analytic approximations, ...
• What are the pros and cons?
• How do we propagate theoretical uncertainties (e.g., from truncations of an expansion or limitations of a physics model) to calculations of physics observables?

Priors:

• What is Bayesian model checking and how can it be used to minimize or validate the influence of priors?
• What are other ways to validate priors?
• How does empirical Bayes work and when is it useful (or dangerous)?
• How do we choose priors for systematic errors in physics?
• E.g., what general guidance is there?
• What range of priors should I consider?
• How does one choose a "non-informative" prior?

Software:

• What should we know about MCMC sampling algorithms and software?
• MCMC programs are often a black box to physicists.
• What are recommended implementations for different types of physics applications?
• Are there parallelized versions?
• What are the pitfalls or "tricks" in using MCMC?
• Should one use more than one algorithm?
• Autocorrelations in MCMC
• A. Steiner: I'm using the method outlined here similar to the acor program used in emcee.
• What are good programs for visualization (e.g., of projected posteriors)?
• What are the best software options for Python, C++, R, ...

Other topics:

• Inconsistent data (or model)
• Outliers
• Model and uncertainty extrapolation
• Empirical Bayes
• Emulation
• A Steiner: In nuclear astrophysics, in order to perform a proper uncertainty quantification, we need two things: (i) the correlations between masses in the Atomic Mass Evaluation, and (ii) the correlations between parameters in popular mass models (e.g. FRDM). How do we get those?
• A. Steiner: What can be understood from the analogy between a particle propagator and a conditional probability distribution? Or does the fact that the former is defined over complex numers spoil the analogy?