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arxiv: 2502.18223 · v4 · submitted 2025-02-25 · 📊 stat.ME

Penalizing complexity priors for Bayesian inference of circular models

Xiang Ye , Janet Van Niekerk , H\r{a}vard Rue This is my paper

Pith reviewed 2026-05-23 02:49 UTC · model grok-4.3

classification 📊 stat.ME
keywords Bayesian inferencecircular statisticspenalized complexity priorshyperpriorsdirectional dataprior selectionvon Mises distribution
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The pith

Penalized complexity priors contract to simpler models for Bayesian circular inference when information is limited.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework for choosing hyperpriors in Bayesian models of circular data such as wind directions or arrival times. It constructs penalized complexity priors for standard circular distributions so that the priors automatically favor simpler models when data provide little guidance on parameter values. Simulation studies and a case study compare these priors against existing hyperprior choices. The work addresses the underdeveloped state of prior selection for circular statistical models in Bayesian settings.

Core claim

We propose a framework for selecting hyperpriors that contracts to a simpler model 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 hyperpriors in the literature are conducted through simulation studies and a practical case study.

What carries the argument

Penalized Complexity (PC) priors, which penalize deviation from a base model to induce automatic contraction toward simplicity in the hyperprior.

If this is right

  • Bayesian models for directional data gain a default hyperprior choice that shrinks toward the simpler base model.
  • Prior elicitation for circular parameters becomes less dependent on strong historical information.
  • Circular statistical models can be built with greater consistency across different applications.
  • The approach supplies a template for extending similar prior constructions to other circular families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contraction property could reduce overfitting in small-sample circular analyses common in environmental monitoring.
  • Extensions might test whether the same PC construction improves inference for other periodic data types such as seasonal time series.
  • If the priors perform well in the reported case study, practitioners analyzing 24-hour cycles could adopt them as a standard default.

Load-bearing premise

That the PC priors will demonstrably outperform or improve upon existing hyperpriors for circular models under limited information, as shown after the simulation studies and case study.

What would settle it

A simulation in which PC priors for a circular distribution such as the von Mises fail to contract toward the base model or produce worse predictive performance than standard hyperpriors when sample size or information is low.

Figures

Figures reproduced from arXiv: 2502.18223 by H\r{a}vard Rue, Janet Van Niekerk, Xiang Ye.

Figure 1
Figure 1. Figure 1: Density of von Mises distributions with low (left) and high (right) concentration. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Density of cardioid (left) and wrapped Cauchy (right) distributions. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relationship between popular circular distributions. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: κ0 = 0 base model PC prior density (π (κ), left) and log-log density (log π (log κ), right) [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: κ0 → ∞ base model PC prior density (π (κ), left) and log-log density (log π (log κ), right). 4.1.1 Interpretable Parameter Transformation Since κ ∈ [0,∞), following the discussion on interpretable transformation in Section 3.4, we suggest the transformation Q (κ) = 2π 1+κ ∈ [0, 2π], which represents the radian of a circle. There￾fore, the scaling parameter λ for the PC prior for the concentration parameter… view at source ↗
Figure 6
Figure 6. Figure 6: ℓ0 = 0 base model PC prior density (π (ℓ), left) and log-log density (log π (log ℓ), right) [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: ℓ0 → 0.5 base model PC prior density (π (ℓ), left) and log-log density (log π (log ℓ), right). 4.2.1 Interpretable Parameter Transformation The parameter ℓ serves as a shape-concentration parameter, controlling the “cardiodity” of the density. Unlike the von Mises (vM) and wrapped Cauchy (WC) distributions, it is not appro￾priate to define Q (ℓ) using similar ideas due to the unique role of ℓ in determinin… view at source ↗
Figure 8
Figure 8. Figure 8: PC prior density (π (ρ), left) and log-log density (log π (log ρ), right). Although the point mass boundary for WC distribution ρ = 1 is not well-defined, ρ still controls the concentration behavior of the distribution from uniform to approximate point mass. Therefore, to compute the scaling parameter of the PC prior, the same idea of interpretable trans￾formation for a parameter could be employed. For ρ ∈… view at source ↗
Figure 9
Figure 9. Figure 9: The PC prior density in distance scale. To check the distance scale behavior of other priors, the transformation in Equation 8 is required. 5.1.1 Von Mises Distribution [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Priors for κ in parameter (left) and distance scale with the base model at κ0 = 0 (right). The solid line is the PC (λ = 0.92) prior, the dashed line is the Gamma (1, 0.34), the dotted line is the h2 prior and the dot-dash line is the h3 prior [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Priors for κ in parameter (left) and distance scale with the base model at κ0 → ∞(right). The solid line is the PC (λ = 1.26) prior, the dashed line is the Gamma (1, 0.34), the dotted line is the h2 prior and the dot-dash line is the h3 prior. When the base model is at point mass (κ0 → ∞), [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Priors for [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Beta prior and PC (λ = 1) prior density in parameter (π (ρ), left) and distance scale (π (d), right). Therefore, a Beta prior with a < 1 is considered acceptable. When users have sufficient information to set an informative prior, a Beta prior with a < 0 is a reasonable choice, as it maintains the ability to prevent “overconfidence”. However, selection of the hyperparameter b significantly influences the … view at source ↗
Figure 15
Figure 15. Figure 15: Simulation Study for the priors for concentration parameter [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Simulation Study for the priors for concentration parameter [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Simulation Study for the priors for concentration parameter [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Example code for choosing α values. After fitting the model, the posterior mean and the 95% credible interval for κ are given in [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 20
Figure 20. Figure 20: Posterior means and 95% credible intervals of [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
read the original 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 framework for selecting hyperpriors that contracts to a simpler model 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 hyperpriors 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework for hyperprior selection in Bayesian circular models based on Penalized Complexity (PC) priors. These priors are constructed to contract toward simpler base models (e.g., uniform or point-mass distributions) when data information is limited. The authors derive PC priors for standard circular distributions, compare them against existing hyperpriors via simulation studies and a real-data case study, and discuss implications for prior elicitation in directional statistics.

Significance. If the PC priors demonstrably improve inference under limited information while preserving the geometry of circular parameter spaces, the work would fill a documented gap in Bayesian circular statistics. The extension of the PC prior framework to this setting is a natural and potentially useful contribution, especially for applications such as wind-direction or periodic time-series modeling where informative priors are difficult to elicit.

major comments (2)
  1. [Simulation studies] Simulation studies section: the design does not isolate the limited-information regime with metrics that directly quantify contraction (e.g., posterior probability mass near the base model or Kullback-Leibler divergence to the simpler model). Without such metrics or explicit small-sample regimes, the central claim that PC priors outperform existing hyperpriors under insufficient information cannot be evaluated from the reported results.
  2. [§3] §3 (PC prior construction): the penalization term for circular parameters appears to rely on a Euclidean distance in the tangent space; it is unclear whether this distance respects the periodic topology or whether the resulting prior remains proper after marginalization over the concentration parameter.
minor comments (2)
  1. [Notation] Notation for the base model and the penalization parameter is introduced inconsistently between the abstract, §2, and the simulation tables.
  2. [Tables] Table captions for the simulation results do not state the number of Monte Carlo replications or the exact sample sizes used in the low-information scenarios.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments, which help clarify how to strengthen the presentation of our PC prior framework for circular models. We address each major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [Simulation studies] Simulation studies section: the design does not isolate the limited-information regime with metrics that directly quantify contraction (e.g., posterior probability mass near the base model or Kullback-Leibler divergence to the simpler model). Without such metrics or explicit small-sample regimes, the central claim that PC priors outperform existing hyperpriors under insufficient information cannot be evaluated from the reported results.

    Authors: We agree that direct quantitative metrics of contraction would make the evaluation of the limited-information regime more transparent. Although our simulations already include small sample sizes (n = 10, 20, 50) to represent data scarcity, we will revise the simulation studies section to report the posterior probability mass in a neighborhood of the base model and the Kullback-Leibler divergence between the posterior and the base model for both PC priors and the comparator hyperpriors. These additions will be presented alongside the existing results to allow direct assessment of the contraction property. revision: yes

  2. Referee: [§3] §3 (PC prior construction): the penalization term for circular parameters appears to rely on a Euclidean distance in the tangent space; it is unclear whether this distance respects the periodic topology or whether the resulting prior remains proper after marginalization over the concentration parameter.

    Authors: We thank the referee for raising this point of clarity. The distance employed in the PC prior penalization is the geodesic (arc-length) distance on the circle, which is invariant to the periodic identification and therefore respects the topology; it is not a Euclidean distance in the tangent space. Propriety of the marginal hyperprior follows from the standard PC construction (exponential decay of the prior density with respect to the distance), which remains integrable after integrating out the concentration parameter. We will revise §3 to state the distance explicitly, add a short derivation confirming topological invariance, and include a brief remark on marginal propriety. These clarifications will be incorporated in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proposes a new framework applying PC priors to circular distributions and validates performance via simulation studies and a case study with explicit comparisons to existing hyperpriors in the literature. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or unverified self-citation chain; the central claims rest on the proposed construction for circular cases plus external empirical benchmarks rather than tautological equivalence to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract.

pith-pipeline@v0.9.0 · 5698 in / 973 out tokens · 29593 ms · 2026-05-23T02:49:25.032073+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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