Posterior-Based Metrics (αₚₒₛₜ, βₚₒₛₜ)

• Posterior-based metrics (αₚₒₛₜ, βₚₒₛₜ) are quantitative measures derived from Bayesian posteriors that support model calibration, robustness analysis, and clustering.
• They have dual formulations: one using BGMM for cluster assignment and another as an Integral Probability Metric to assess divergence between distributions.
• These metrics provide actionable insights for anomaly detection and stability analysis by quantifying posterior concentration, separation, and calibration.

Posterior-based metrics, commonly denoted as $\alpha_{\text{post}}$ and $\beta_{\text{post}}$, are quantitative tools for interrogating Bayesian posterior distributions, particularly in model calibration, robustness analysis, and probabilistic clustering applications. Their formal definitions, operational meaning, and empirical roles vary contextually across domains such as Bayesian inverse problems, probabilistic cluster detection for side-channel security, and calibrated Bayesian inference. The notation is not universal but is consistently used to denote interpretable posterior-derived statistics or distances, either between distributions or between cluster-bound assignments. The sections below synthesize key definitions, methodologies, theoretical foundations, and representative empirical uses, referencing established research and state-of-the-art applications.

1. Definitions and Formalism

The precise interpretation of $\alpha_{\text{post}}$ and $\beta_{\text{post}}$ depends on context:

• Posterior Mean Assignment Metrics: In the context of probabilistic clustering (e.g., for EM side-channel analysis), $\alpha_{\text{post}}$ and $\beta_{\text{post}}$ represent sample-averaged posterior probabilities of "normal" and "anomalous" cluster membership, respectively, under a Bayesian Gaussian Mixture Model (BGMM) (Tahghigh et al., 3 Feb 2026).
• Integral Probability Metrics (IPMs) Between Posteriors: In Bayesian inverse problems, $\alpha_{\text{post}}$ and $\beta_{\text{post}}$ are dual formulations of IPMs that quantify the divergence between posteriors induced by different priors or likelihoods, with test-function (dual) and coupling (primal) representations (Garbuno-Inigo et al., 2023).

Notation	Definition/Role	Reference
$\alpha_{\text{post}}$	Mean normal-cluster BGMM posterior, or IPM dual (test-function) form	(Tahghigh et al., 3 Feb 2026, Garbuno-Inigo et al., 2023)
$\beta_{\text{post}}$	Mean anomaly-cluster BGMM posterior, or IPM primal (coupling/wasserstein) form	(Tahghigh et al., 3 Feb 2026, Garbuno-Inigo et al., 2023)

In both contexts, these metrics serve to quantify posterior concentration, separation, or divergence, supporting interpretable statistical decision-making.

2. Methodological Foundations and Derivation

A. EM Side-Channel Detection (Tahghigh et al., 3 Feb 2026)

1. BGMM Computation: Given $\beta_{\text{post}}$0 one-dimensional feature vectors $\beta_{\text{post}}$1, a BGMM with $\beta_{\text{post}}$2 components is fit, producing weights $\beta_{\text{post}}$3 and component densities $\beta_{\text{post}}$4. For each trace $\beta_{\text{post}}$5, the posterior responsibility is:

$\beta_{\text{post}}$6

1. Cluster Assignment: The dominant component $\beta_{\text{post}}$7 is labeled "normal," others above a threshold as "anomalous."
2. Per-Trace Confidences: $\beta_{\text{post}}$8, $\beta_{\text{post}}$9.
3. Aggregate Metrics:

$\alpha_{\text{post}}$0

By construction, $\alpha_{\text{post}}$1.

1. Decision Support: $\alpha_{\text{post}}$2 and $\alpha_{\text{post}}$3 are integrated with the Bayesian Information Criterion ($\alpha_{\text{post}}$4BIC) and Mahalanobis separation $\alpha_{\text{post}}$5 for anomaly detection, producing smooth and interpretable anomaly-confidence scores.

B. Posterior Stability via IPM (Garbuno-Inigo et al., 2023)

Given two posteriors $\alpha_{\text{post}}$6:

• Dual (Test-function) Form:

$\alpha_{\text{post}}$7

where $\alpha_{\text{post}}$8 is the set of $\alpha_{\text{post}}$9-Lipschitz functions.

• Primal (Coupling) Form:

$\beta_{\text{post}}$0

where $\beta_{\text{post}}$1 is a cost function and $\beta_{\text{post}}$2 is the set of all couplings.

• When $\beta_{\text{post}}$3 is a metric, the two coincide by Kantorovich-Rubinstein duality:

$\beta_{\text{post}}$4

where $\beta_{\text{post}}$5 is the $\beta_{\text{post}}$6-Wasserstein distance.

3. Interpretation and Theoretical Properties

EM and BGMM Posterior Aggregates

• $\beta_{\text{post}}$7: Nearly all data classified as normal; low false-positive rate.
• $\beta_{\text{post}}$8: Significant anomalous behavior detected.
• $\beta_{\text{post}}$9 ensures a mutual-exclusivity interpretation under the two-state model.
• Posterior-based metric values support explicit detection rules, e.g., high-confidence anomalous decisions for $\alpha_{\text{post}}$0 and $\alpha_{\text{post}}$1 (Tahghigh et al., 3 Feb 2026).

IPM-based Posterior Divergences

• $\alpha_{\text{post}}$2, ensuring that the divergence vanishes only for identical posteriors.
• Triangle inequality and Lipschitz continuity in likelihood/prior perturbations provide robustness quantification under model/data uncertainty (Garbuno-Inigo et al., 2023).
• The flexibility of the cost function $\alpha_{\text{post}}$3 enables adaptation to problem-specific regularity properties.

4. Calibration and Posterior Region Coverage

A related class of posterior-based metrics, exemplified by the spread-control parameter $\alpha_{\text{post}}$4 in general posterior calibration, aims to ensure that nominal posterior credible regions achieve desired frequentist coverage (Syring et al., 2015). Although $\alpha_{\text{post}}$5 is not labeled $\alpha_{\text{post}}$6 or $\alpha_{\text{post}}$7, it acts as a posterior-based tuning metric for adjusting the scale of the posterior distribution:

• The calibration algorithm iterates $\alpha_{\text{post}}$8 using bootstrap-based empirical coverage estimates to solve $\alpha_{\text{post}}$9.
• This produces HPD credible regions $\beta_{\text{post}}$0 with empirical frequentist coverage matching nominal Bayesian levels.

This suggests that posterior-based metrics more generally encompass not just distances or aggregate assignment probabilities, but also tuning parameters for posterior dispersion and frequentist calibration.

5. Practical Applications

EM-Based Reference-Free Hardware Trojan Detection

$\beta_{\text{post}}$1 and $\beta_{\text{post}}$2, together with $\beta_{\text{post}}$3BIC and Mahalanobis $\beta_{\text{post}}$4, enable a fully reference-free, statistically grounded anomaly (Trojan) detection rule. Empirical results on AES-128 designs show that $\beta_{\text{post}}$5 robustly distinguishes HT-free ($\beta_{\text{post}}$6) from Trojan-activated states ($\beta_{\text{post}}$7), correlating with high evidence for anomalies ($\beta_{\text{post}}$8BIC $\beta_{\text{post}}$9) and substantial cluster separation ($\alpha_{\text{post}}$0) (Tahghigh et al., 3 Feb 2026).

Robust Bayesian Inference and Model Confirmation

For Bayesian inverse problems and model robustness tasks, $\alpha_{\text{post}}$1/$\alpha_{\text{post}}$2 (via IPM formalism) quantify sensitivity of the posterior to perturbations, enabling principled stability analysis of data-driven priors, surrogate likelihoods (e.g., neural networks), and hyperparameter selection (Garbuno-Inigo et al., 2023). Variation in these metrics yields a direct diagnostic of epistemic uncertainty propagation and model misspecification effects.

Calibration of Bayesian Credible Regions

Posterior-based spread metrics (such as $\alpha_{\text{post}}$3) support interval and region calibration, ensuring that inferred uncertainties have coverage properties justified by the data and model (Syring et al., 2015). This is especially important in misspecified or risk-based posteriors (e.g., Gibbs posteriors for median estimation).

6. Limitations and Considerations

• In clustering, only two-state (normal/anomaly) settings guarantee $\alpha_{\text{post}}$4; more complex cluster topologies require different metrics or generalization.
• The IPM-based metrics are only as informative as the choice of the cost function $\alpha_{\text{post}}$5 and the regularity of the posterior family.
• Calibration algorithms based on posterior-based metrics (e.g., $\alpha_{\text{post}}$6 adjustment) may incur substantial computational costs due to nested bootstrapping and posterior sampling (Syring et al., 2015).
• For severely misshapen or misaligned posteriors, calibration via spread metrics may not remedy geometric bias or multimodal distortions. This affects both interpretability of $\alpha_{\text{post}}$7, $\alpha_{\text{post}}$8 and the accuracy of calibrated regions.

7. Summary and Unifying Perspective

Posterior-based metrics such as $\alpha_{\text{post}}$9 and $\beta_{\text{post}}$0 provide interpretable, mathematically grounded quantitative summaries of posterior assignment, divergence, or dispersion, with direct applications in anomaly detection, model calibration, and robustness analysis across a spectrum of Bayesian inference tasks. In clustering, they quantify assignment confidence; in IPM analysis, they measure posterior differences due to data/model perturbation; and in posterior calibration, they tune coverage-adjustment. The choice of formulation and operational meaning is tailored by application, but the underlying principle is consistent: statistical conclusions are grounded in explicit properties of the Bayesian posterior, enabling rigorous, reference-free analysis and robust decision support (Tahghigh et al., 3 Feb 2026, Garbuno-Inigo et al., 2023, Syring et al., 2015).