[Paper Review] Calibrating Model-Based Inferences and Decisions

Michael Betancourt|arXiv (Cornell University)|Mar 22, 2018

Markov Chains and Monte Carlo Methods1 references22 citations

TL;DR

This paper presents a formal framework for calibrating model-based statistical inferences and decisions using frequentist and Bayesian methodologies, emphasizing sensitivity analysis and loss-based calibration. It demonstrates how to systematically assess uncertainty and decision reliability across model configurations, with applications to discovery claims and limit setting in complex experiments.

ABSTRACT

As the frontiers of applied statistics progress through increasingly complex experiments we must exploit increasingly sophisticated inferential models to analyze the observations we make. In order to avoid misleading or outright erroneous inferences we then have to be increasingly diligent in scrutinizing the consequences of those modeling assumptions. Fortunately model-based methods of statistical inference naturally define procedures for quantifying the scope of inferential outcomes and calibrating corresponding decision making processes. In this paper I review the construction and implementation of the particular procedures that arise within frequentist and Bayesian methodologies.

Motivation & Objective

• To address the growing challenge of ensuring reliable inferences in increasingly complex statistical models used in advanced experiments.
• To formalize procedures for quantifying the sensitivity of inferential outcomes to model assumptions and measurement variability.
• To develop practical calibration methods that link statistical models to decision-making under uncertainty, particularly in high-stakes scientific contexts.
• To bridge the gap between theoretical model-based inference and real-world experimental design by providing computationally feasible calibration techniques.
• To enable robust discovery claims and limit setting by rigorously assessing the reliability of p-values, posterior probabilities, and predictive scores.

Proposed method

• Uses mathematical formalism to define data generating processes as probability distributions over measurement spaces, with observations modeled as samples from a true process $\pi^*$.
• Applies loss functions to quantify inferential consequences, with frequentist calibration based on expected loss over all data generating processes in a model configuration space.
• Employs Bayesian calibration via posterior expectations of loss functions under the joint distribution of model parameters and data, incorporating prior uncertainty.
• Introduces predictive scores and posterior probabilities of the region of practical equivalence (ROPE) to assess model performance and practical significance.
• Utilizes Monte Carlo and Markov chain Monte Carlo methods for numerical computation of expected losses and posterior distributions in high-dimensional spaces.
• Proposes using automatic differentiation to estimate gradients of expected loss with respect to experimental design parameters, enabling optimization of experimental setups.

Experimental results

Research questions

• RQ1 How can inferential outcomes be systematically calibrated to reflect uncertainty across different model configurations in complex experiments?
• RQ2 What are the key differences and trade-offs between frequentist and Bayesian approaches to model calibration in terms of sensitivity and decision reliability?
• RQ3 How can calibration procedures be implemented in practice when analytic solutions are intractable, especially for high-dimensional models?
• RQ4 What role do loss functions and predictive scores play in evaluating the robustness of discovery claims and limit-setting procedures?
• RQ5 Can computational methods such as MCMC and automatic differentiation be leveraged to optimize experimental design based on calibrated performance metrics?

Key findings

• Frequentist calibration requires bounding the expected loss over all data generating processes in a model, which is analytically tractable only under simplifying assumptions about the model configuration space.
• Bayesian calibration involves computing posterior expectations of loss functions, which are often more amenable to numerical approximation but remain computationally intensive for complex models.
• The use of posterior quantiles in Bayesian limit setting naturally incorporates nuisance parameter uncertainty, providing robust bounds on phenomenological parameters.
• Predictive scores and ROPE probabilities offer practical alternatives to p-values for assessing practical significance and model performance in discovery contexts.
• Monte Carlo methods enable calibration of rare-event thresholds (e.g., $\mathcal{O}(10^{-7})$) but suffer from slow convergence, limiting their accuracy for extreme significance levels.
• Automatic differentiation of expected losses with respect to experimental design parameters offers a promising path toward optimizing experimental setups, though implementation remains challenging.