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Principled priors for Bayesian inference of circular models

Xiang Ye, Janet Van Niekerk, Håvard Rue

TL;DR

This work tackles the challenge of priors in Bayesian circular statistics by introducing Penalized Complexity (PC) priors that penalize model complexity via a distance based on the Kullback–Leibler divergence from a simpler base circular model. The framework is instantiated for three foundational circular distributions—von Mises, cardioid, and wrapped Cauchy—providing explicit PC priors and practical calibration via a mean resultant length or equivalent interpretables. Through extensive simulations and a wind-direction case study, the authors demonstrate that PC priors offer robust performance, favorable boundary behavior, and improved protection against overfitting compared to traditional priors, with the von Mises base model (circular uniform base) and the cardioid base model yielding particularly stable results. The methodology supports a principled, user-tunable balance between prior informativeness and regularization, and the authors outline future work to extend the approach to circular covariates, joint priors, and multi-parameter circular models, enhancing Bayesian circular inference in diverse applications.

Abstract

Advancements in computational power and methodologies have enabled research on massive datasets. However, tools for analyzing data with directional or periodic characteristics, such as wind directions and customers' arrival time in 24-hour clock, remain underdeveloped. While statisticians have proposed circular distributions for such analyses, significant challenges persist in constructing circular statistical models, particularly in the context of Bayesian methods. These challenges stem from limited theoretical development and a lack of historical studies on prior selection for circular distribution parameters. In this article, we propose a principled, practical and systematic framework for selecting priors that effectively prevents overfitting in circular scenarios, especially when there is insufficient information to guide prior selection. We introduce well-examined Penalized Complexity (PC) priors for the most widely used circular distributions. Comprehensive comparisons with existing priors in the literature are conducted through simulation studies and a practical case study. Finally, we discuss the contributions and implications of our work, providing a foundation for further advancements in constructing Bayesian circular statistical models.

Principled priors for Bayesian inference of circular models

TL;DR

This work tackles the challenge of priors in Bayesian circular statistics by introducing Penalized Complexity (PC) priors that penalize model complexity via a distance based on the Kullback–Leibler divergence from a simpler base circular model. The framework is instantiated for three foundational circular distributions—von Mises, cardioid, and wrapped Cauchy—providing explicit PC priors and practical calibration via a mean resultant length or equivalent interpretables. Through extensive simulations and a wind-direction case study, the authors demonstrate that PC priors offer robust performance, favorable boundary behavior, and improved protection against overfitting compared to traditional priors, with the von Mises base model (circular uniform base) and the cardioid base model yielding particularly stable results. The methodology supports a principled, user-tunable balance between prior informativeness and regularization, and the authors outline future work to extend the approach to circular covariates, joint priors, and multi-parameter circular models, enhancing Bayesian circular inference in diverse applications.

Abstract

Advancements in computational power and methodologies have enabled research on massive datasets. However, tools for analyzing data with directional or periodic characteristics, such as wind directions and customers' arrival time in 24-hour clock, remain underdeveloped. While statisticians have proposed circular distributions for such analyses, significant challenges persist in constructing circular statistical models, particularly in the context of Bayesian methods. These challenges stem from limited theoretical development and a lack of historical studies on prior selection for circular distribution parameters. In this article, we propose a principled, practical and systematic framework for selecting priors that effectively prevents overfitting in circular scenarios, especially when there is insufficient information to guide prior selection. We introduce well-examined Penalized Complexity (PC) priors for the most widely used circular distributions. Comprehensive comparisons with existing priors in the literature are conducted through simulation studies and a practical case study. Finally, we discuss the contributions and implications of our work, providing a foundation for further advancements in constructing Bayesian circular statistical models.

Paper Structure

This paper contains 30 sections, 5 theorems, 61 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Von Mises Distribution
  4. Cardioid Distribution
  5. Wrapped Cauchy Distribution
  6. Methodology
  7. Penalizing complexity prior for circular models
  8. Specific cases
  9. Von Mises Distribution
  10. Cardioid Distribution
  11. Wrapped Cauchy Distribution
  12. Simulation Study and Comparisons
  13. Properties of PC and other common priors
  14. Von Mises Distribution
  15. Cardioid Distribution
  16. ...and 15 more sections

Key Result

Proposition 3.1

The PC prior for concentration parameter $\kappa$ of the von Mises distribution with the base model at $\kappa_{0}=0$ has density where $\lambda > 0$, and the corresponding CDF is

Figures (21)

  • Figure 1: Relationship between popular circular distributions. $\kappa$, $\ell$ and $\rho$ are the concentration parameters for von Mises, cardioid and wrapped Cauchy distributions.
  • Figure 2: vM density for small (left) and large (right) $\kappa$ values with $\mu = \pi$ .
  • Figure 3: Cardioid density (left) and wrapped Cauchy density (right) with $\mu = \pi$ for different concentration parameter values.
  • Figure 4: Distance ($d\left(\kappa\right)$) between vM distribution and circular uniform distribution (left) and the PC prior density in distance scale for different $\lambda$ values (right).
  • Figure 5: $\kappa_{0}=0$ base model PC prior density (left) and log-log density (right).
  • ...and 16 more figures

Theorems & Definitions (5)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5