Principled priors for Bayesian inference of circular models

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.

💡 Research Summary

This paper addresses a long‑standing gap in Bayesian circular statistics: the principled selection of priors for the concentration (or dispersion) parameters of common circular distributions. While many circular models such as the von Mises, Cardioid, and Wrapped Cauchy have been widely used, existing priors (Gamma, Beta, Minimum Message Length‑based, or ad‑hoc conjugate forms) are either heuristic, computationally burdensome, or prone to over‑fitting when data are scarce.

The authors propose to adopt the Penalized Complexity (PC) prior framework, originally introduced by Simpson et al. (2017), as a systematic way to construct default priors that favour simpler models. The PC approach rests on four principles: (i) Occam’s razor – prefer the simplest model that explains the data; (ii) a quantitative measure of model complexity defined by the Kullback‑Leibler (KL) distance d between the flexible model and a base (simpler) model; (iii) a constant‑rate penalisation that forces the prior density to decay exponentially with d; and (iv) a user‑defined scaling parameter that translates prior knowledge into a probability statement about the distance.

In the circular context the base model is the circular uniform distribution, obtained when the concentration parameter equals zero (κ = 0 for von Mises, ℓ = 0 for Cardioid, ρ = 0 for Wrapped Cauchy). The authors derive explicit PC priors for each distribution:

• von Mises (κ) – Using an analytic or approximated KL distance d(κ), the PC prior takes the form
   p(κ) = λ exp(−λ d(κ)) |∂d/∂κ|,
  where λ is chosen by specifying a probability such as “κ ≤ κ₀ with probability p₀”.

• Cardioid (ℓ) – The KL distance simplifies to d(ℓ) = −log(1 − 2ℓ), yielding
   p(ℓ) = λ exp(−λ d(ℓ)) · 2/(1 − 2ℓ).

• Wrapped Cauchy (ρ) – With d(ρ) = −log(1 − ρ) the prior becomes
   p(ρ) = λ exp(−λ d(ρ)) · 1/(1 − ρ).

These expressions guarantee that a substantial prior mass is placed near the uniform case, thereby automatically discouraging unnecessarily concentrated models. The scaling parameter λ provides a transparent way for practitioners to encode domain knowledge without resorting to arbitrary hyper‑parameters.

A comprehensive simulation study evaluates the PC priors against traditional Gamma, Beta, and MML‑based priors across a range of sample sizes (n = 20, 50, 200) and concentration levels (low, medium, high). Performance metrics include mean absolute error, 95 % credible‑interval width, and leave‑one‑out predictive information criterion (LOO‑IC). The PC priors consistently yield lower estimation error and tighter credible intervals, especially in small‑sample regimes where over‑fitting is most severe.

Two real‑world applications illustrate practical benefits. First, wind‑direction data from a coastal observatory are modeled with von Mises and Wrapped Cauchy likelihoods; second, customer arrival times recorded on a 24‑hour clock are analyzed similarly. In both cases, models equipped with PC priors achieve better predictive scores (5–12 % improvement in LOO‑IC) and produce posterior concentration estimates that are more plausible and easier to interpret than those obtained with conventional priors.

The discussion acknowledges limitations: the KL distance for von Mises requires numerical approximation, and the choice of λ, while intuitive, remains somewhat subjective. The authors suggest future work on multivariate circular models, extensions to non‑standard circular families, and data‑driven methods for calibrating λ.

In summary, the paper delivers the first unified, principled framework for prior selection in Bayesian circular models. By embedding the notion of model simplicity directly into the prior through the PC construction, it mitigates over‑fitting, simplifies prior elicitation, and enhances both inferential robustness and interpretability for a wide array of directional data problems.