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The typicality principle and its implications for statistics and data science

Yiran Jiang, Zeyu Zhang, Ryan Martin, Chuanhai Liu

TL;DR

This work introduces the typicality principle, a falsification-based lens for statistics and data science that judges theory adequacy by how typical data are under a given model. It couples this principle with a data-dependent regularization, ρ_λ^typ(x,θ) = ℓ_x(θ) − λ r_x^typ(θ) where r_x^typ(θ) = −\log pval_x(θ), to counteract non-regularities that plague maximum likelihood estimation. Through Le Cam's mixture, the Neyman–Scott problem, and Stein's mean-vector-length problem, the approach demonstrates improved point estimation and supports reliable uncertainty quantification within an inferential-model framework. The paper also elaborates practical construction of typicality contours τ_x(θ), discusses connections to existing statistical principles, and outlines paths for broader application to data science challenges and AI systems. Overall, the typicality principle offers a principled, regularization-based route to robust inference and calibrated uncertainty without requiring informative priors.

Abstract

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in machine learning and artificial intelligence in years to come. Along these lines, the present paper advances a novel {\em typicality principle} which states, roughly, that if the observed data is sufficiently ``atypical'' in a certain sense relative to a posited theory, then that theory is unwarranted. This emphasis on typicality brings familiar but often overlooked background notions like model-checking to the inferential foreground. One instantiation of the typicality principle is in the context of parameter estimation, where we propose a new typicality-based regularization strategy that leans heavily on goodness-of-fit testing. The effectiveness of this new regularization strategy is illustrated in three non-trivial examples where ordinary maximum likelihood estimation fails miserably. We also demonstrate how the typicality principle fits within a bigger picture of reliable and efficient uncertainty quantification.

The typicality principle and its implications for statistics and data science

TL;DR

This work introduces the typicality principle, a falsification-based lens for statistics and data science that judges theory adequacy by how typical data are under a given model. It couples this principle with a data-dependent regularization, ρ_λ^typ(x,θ) = ℓ_x(θ) − λ r_x^typ(θ) where r_x^typ(θ) = −\log pval_x(θ), to counteract non-regularities that plague maximum likelihood estimation. Through Le Cam's mixture, the Neyman–Scott problem, and Stein's mean-vector-length problem, the approach demonstrates improved point estimation and supports reliable uncertainty quantification within an inferential-model framework. The paper also elaborates practical construction of typicality contours τ_x(θ), discusses connections to existing statistical principles, and outlines paths for broader application to data science challenges and AI systems. Overall, the typicality principle offers a principled, regularization-based route to robust inference and calibrated uncertainty without requiring informative priors.

Abstract

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in machine learning and artificial intelligence in years to come. Along these lines, the present paper advances a novel {\em typicality principle} which states, roughly, that if the observed data is sufficiently ``atypical'' in a certain sense relative to a posited theory, then that theory is unwarranted. This emphasis on typicality brings familiar but often overlooked background notions like model-checking to the inferential foreground. One instantiation of the typicality principle is in the context of parameter estimation, where we propose a new typicality-based regularization strategy that leans heavily on goodness-of-fit testing. The effectiveness of this new regularization strategy is illustrated in three non-trivial examples where ordinary maximum likelihood estimation fails miserably. We also demonstrate how the typicality principle fits within a bigger picture of reliable and efficient uncertainty quantification.
Paper Structure (13 sections, 2 theorems, 22 equations, 7 figures)

This paper contains 13 sections, 2 theorems, 22 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Typicality principle
  4. Examples: non-regular estimation
  5. Le Cam's mixture
  6. Neyman--Scott problem
  7. Stein's mean vector length
  8. Reliable uncertainty quantification
  9. Typicality principle, revisited
  10. Putting the principle into practice
  11. Relation to other statistical principles
  12. Stein's mean vector length, again
  13. Conclusion

Key Result

Proposition 1

A test procedure that rejects the hypothesis $H$ if and only if $x$ is sufficiently atypical relative to $H$ at level $\alpha$, i.e., rejects if and only if $\tau_x(H) \leq \alpha$, controls the frequentist Type I error in the sense that

Figures (7)

  • Figure 1: Log-likelihood surfaces from three viewing angles.
  • Figure 2: Log p-value surfaces for the Kolmogorov--Smirnov test.
  • Figure 3: Profiled log p-values for the Kolmogorov--Smirnov test.
  • Figure 4: Profiles of the objective function $\rho_\lambda^\text{typ}(\mu,\sigma^2)$ for different $\lambda$.
  • Figure 5: The objective function $\sigma^2 \mapsto \rho_\lambda^\text{typ}(x,\sigma^2)$ with varying choices of $\lambda$. The true $\Sigma^2$ value is denoted as the vertical dashed line.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2