Pathologies in statistics

by W. B. Meitei, PhD

In statistics, the term "pathological" or "pathologies" refers to examples, cases, or behaviours that are unusual, counterintuitive, or problematic within the typical theoretical framework. These pathologies highlight instances where standard assumptions or commonly accepted rules fail, leading to results that may appear strange, misleading, or mathematically "monstrous." They often expose limitations or gaps in existing theories and motivate the development of improved methods or new theoretical insights.

Definition: Pathologies in statistics denote problematic or extreme examples where statistical methods, assumptions, or models break down or produce misleading results. Such cases may defy intuition or expected behaviours and often require special treatment.

Purpose: These "pathological" cases serve as important counterexamples that sharpen understanding by demonstrating where existing approaches fail or need refinement. They can reveal hidden pitfalls in data analysis or hypothesis testing.

Examples of Statistical Pathologies

• Bartlett’s paradox: When using overly diffuse (improper or very broad) priors in Bayesian hypothesis testing, the Bayes factor may perversely favour the null hypothesis regardless of the data. This counterintuitive outcome is a classic pathological behaviour.
• Jeffreys-Lindley paradox: With large samples, Bayesian and frequentist methods yield conflicting conclusions due to how evidence scales with sample size and prior specification, a fundamental pathology in hypothesis testing frameworks.
• Pathological convergence: Certain estimators may fail usual consistency or convergence properties under specific data or design conditions, leading to irregular or misleading inference.
• Non-identifiability or ill-posed problems: Situations where parameters cannot be estimated uniquely, causing standard estimators to behave erratically.
• Pathological functions and distributions: In a broader mathematical context, examples like nowhere-differentiable functions (e.g., Weierstrass function) demonstrate pathological behaviours that challenge classical assumptions, indirectly affecting statistical modelling.

Why Are Pathologies Important?

• They reveal limitations of standard statistical tools and assumptions.
• Prompt the creation of robust methods and more careful model specifications.
• Highlight the need for informed prior choices in Bayesian analysis and for caution in interpreting frequentist measures like p-values.
• Stimulate research and theoretical development to address these challenges, such as cake priors or adaptive testing methods developed to overcome specific paradoxes.

Suggested Readings:

1. Sudre, C. H., Cardoso, M. J., Bouvy, W., Biessels, G. J., Barnes, J., & Ourselin, S. (2014). Bayesian model selection for pathological data. International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 323-330). Cham: Springer International Publishing.
2. Pathological. Wolfram Mathworld.

Suggested Citation: Meitei, W. B. (2025). Pathologies in statistics. WBM STATS.