Pairwise-MFV Approach

• Pairwise-MFV is a robust statistical framework that employs point-to-point ratios and the most frequent value (MFV) statistic to decouple nuisance parameters.
• The approach enhances sample efficiency and outlier resistance, enabling accurate lifetime estimation in nuclear experiments and joint PMF recovery via coupled matrix factorization.
• Its algorithm combines iterative MFV estimation with nonnegative matrix factorization, achieving superior performance in sparse and high-dimensional data scenarios.

The pairwise-MFV approach encompasses statistical methodologies that leverage point-to-point ratios and robust "most frequent value" (MFV) statistics for high-precision parameter estimation and joint probability recovery in scenarios where classical regression or tensor-based methods are challenged by outliers, sample-complexity, or parameter-correlations. The term spans robust lifetime estimation from time series (notably for HPGe gamma-ray detector data in nuclear experiments) and latent-factor inference for joint PMFs from pairwise marginals using coupled nonnegative matrix factorization. Across these domains, pairwise constructions serve to decorrelate nuisance parameters, eliminate dependence on amplitude or global normalization, summarize heavy-tailed estimator distributions, and maximize sample-efficiency in high dimensions.

1. Foundations of the Pairwise-MFV Framework

The pairwise-MFV methodology centers on algebraic constructions of estimators and statistical summaries that are inherently robust against correlated noise and outlying measurements. In exponential decay analysis, one observes background-adjusted net peak areas $N_i = y_i$ with statistical uncertainty $\sigma_i$ at times $t_i$ and assumes a decay law $A(t)=A_0\,e^{-t/\tau}$ (Golovko, 30 Dec 2025). Forming the point-to-point ratio between measurements $(i,j)$,

$$R_{ij} = \frac{N_i}{N_j} = \exp\left[-\frac{t_i-t_j}{\tau}\right],$$

cancels dependence on initial amplitude $A_0$, yielding a two-point, amplitude-independent lifetime estimator:

$\tau_{ij} = \frac{t_i-t_j}{\ln(N_j/N_i)}.$

Similarly, in high-dimensional joint PMF inference from pairwise marginals, the coupled-NMF approach models $P(i_1,\dots,i_N)$ via latent factors. Each pairwise marginal is factorized as $P_{jk}=A_j\,\mathrm{Diag}(\lambda)\,A_k^\top$, and recovery proceeds by minimizing divergence (e.g., KL) across all available pairs (Ibrahim et al., 2020).

Pairwise-MFV is thus characterized by:

• Algebraic pairwise ratio constructions for parameter estimation or probabilistic modeling.
• Decorrelation of key parameters (e.g., amplitude and lifetime, or global normalizations).
• Down-weighting or insensitivity to outliers by utilizing the MFV statistic or robust matrix factorization.
• Avoidance of non-linear amplitude fitting or high-order tensor decompositions.

2. Steiner’s Most Frequent Value (MFV) Statistic

Central to robust parameter summarization, MFV is a density-mode estimator suitable for heavy-tailed Cauchy-like distributions arising from ratios of Gaussian variables (Golovko, 30 Dec 2025). MFV fits the collection $\{\tau_{ij}\}$ to

$$f(x;\,M,\varepsilon) = \frac{1}{\pi}\frac{\varepsilon}{(x-M)^2+\varepsilon^2},$$

where $M$ roughly represents the mode ("most frequent value") and $\varepsilon$ the "dihesion" (scale). Initialization proceeds with $M^{(0)}=\text{median}(\{\tau_{ij}\})$, $$\varepsilon^{(0)}={\sqrt{3}/2} (\max\tau_{ij}-\min\tau_{ij})$$, followed by iterative weighted averaging:

$$w_k^{(n)} = \frac{\varepsilon^{(n)2}}{(\tau_k-M^{(n)})^2+\varepsilon^{(n)2}},$$

$$M^{(n+1)} = \frac{\sum_k w_k^{(n)}\tau_k}{\sum_k w_k^{(n)}}, \quad \varepsilon^{(n+1)2} = 3\frac{\sum_k w_k^{(n)2}(\tau_k-M^{(n)})^2}{\sum_k w_k^{(n)2}}.$$

Convergence yields $M$ as the MFV estimate and $\varepsilon$ as the scale, with an internal $1\sigma$ uncertainty

$$\sigma_M = \frac{\varepsilon}{\sqrt{n_{\text{eff}}}}, \quad n_{\text{eff}} = \sum_k w_k.$$

MFV differentiates itself from the median by continuous weighting of tail outliers, resulting in median-like outlier resistance but reduced estimator variance.

3. Statistical Robustness and Sample Complexity

Statistical properties of pairwise-MFV include amplitude–lifetime decorrelation and effective handling of heavy tails (Golovko, 30 Dec 2025):

• The estimator $\tau_{ij}$ is insensitive to amplitude $A_0$; no joint fit is needed, and the anti-correlation present in standard regression is eliminated.
• The distribution of $\tau_{ij}$ is heavy-tailed (Cauchy-like), with rare statistical fluctuations generating extreme outliers.
• MFV’s weighting scheme concentrates inference on dense clusters, discarding tails efficiently.
• Asymptotically, uncertainty $\sigma_M$ reflects the Gaussian-like core, not global heavy tails.

In joint PMF reconstruction from pairwise marginals, sample complexity is notably reduced compared to tensor approaches. For $I$ categories, pairwise marginals induce an error of $O(\sqrt{I^2/(Sp^2)})$, whereas third-order marginals scale as $O(\sqrt{I^3/(Sp^3)})$ (Ibrahim et al., 2020). Therefore, pairwise-MFV offers practical gains in sparse sampling regimes, enabling accurate recovery where tensor decomposition fails.

4. Algorithmic Implementation

Pairwise-MFV in lifetime estimation proceeds through multiple steps (Golovko, 30 Dec 2025):

• Input background-subtracted net counts $y_k=N_k$ and times $t_k$.
• Compute $\tau_{ij}$ for all $i<j$ with $N_i,N_j>0$.
• Estimate MFV $M$ and $\varepsilon$ via iterative scheme above.
• Quantify statistical uncertainty using bootstrap, constructing confidence intervals over resampled $\{t_k,y_k,\sigma_k\}$.
• Assess systematic errors by shifting $y_k$, $t_k$, or scaling $\sigma_k$ and propagating impact on MFV.

For joint PMF reconstruction, the CNMF-SPA-EM suite entails:

• Stacking observed pairwise marginals into a block matrix $\widetilde X$, choosing partitions $S_1,S_2$.
• Applying the Successive Projection Algorithm (SPA) to select well-separated columns.
• Extracting factor estimates $A_j$, $A_k$ and latent vector $\lambda$ through normalization and nonnegative least-squares.
• Refining with the economical EM algorithm, which guarantees convergence and monotonic likelihood increase.
• Selecting matrix rank $F$ by cross-validation or error-elbow technique; determining convergence via relative likelihood changes.

Recommended implementation strategies include initializing with CNMF-SPA, running 5–20 EM iterations, employing KL-divergence for PMF fitting, and ensuring partition/tall-matrix separability (Ibrahim et al., 2020).

5. Comparative Evaluation and Applications

Pairwise-MFV exhibits distinct empirical advantages. In HPGe detector experiments, the pairwise-MFV estimate $$\tau_{\mathrm{MFV}} = 4.0959 \pm 0.0007_{\mathrm{stat}} \pm 0.0110_{\mathrm{syst}}\ \mathrm{d}$$ and half-life $$T_{1/2} = 2.8391 \pm 0.0005_{\mathrm{stat}} \pm 0.0076_{\mathrm{syst}}\ \mathrm{d}$$ match conventional regression within statistical errors but offer superior reproducibility, amplitude decorrelation, and resistance to gain drifts or local background fluctuations (Golovko, 30 Dec 2025).

In joint PMF recovery for classification and recommender systems, CNMF-SPA-EM attains competitive or superior accuracy versus tensor-based methods at substantial computational savings. Real-data cases include UCI classification (>$92\%$ accuracy on "Votes" data) and MovieLens recommendations (RMSE $0.78$ on "Action"), matching specialist matrix factorization approaches (Ibrahim et al., 2020).

6. Methodological Extensions and Significance

Methodological extensions of pairwise-MFV cover robust regression, context-aware and cross-domain recommender systems, and high-dimensional signal reconstruction. By decoupling nuisance global parameters, resisting outliers, and maximizing sample efficiency, pairwise-MFV approaches establish templates for reliable inference in heavily correlated, noisy, or sample-starved environments.

In nuclear physics and astrophysics, pairwise-MFV enables stable decay parameter extraction over long time scales. In machine learning, CNMF-SPA-EM empowers practitioners to infer complex latent structures from limited, incomplete, or pairwise-only data, broadening applicability to domains previously restricted by the curse of dimensionality. This suggests continued relevance for pairwise-MFV in scalable, interpretable, and robust inference frameworks across physical and data sciences.