A variational framework for modal estimation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912915208011776 |
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| author | LeMinh, Tâm Arbel, Julyan Forbes, Florence Nguyen, Hien Duy |
| author_facet | LeMinh, Tâm Arbel, Julyan Forbes, Florence Nguyen, Hien Duy |
| contents | We approach multivariate mode estimation through Gibbs distributions and introduce GERVE (Gibbs-measure Entropy-Regularised Variational Estimation), a likelihood-free framework that approximates Gibbs measures directly from samples by maximizing an entropy-regularised variational objective with natural-gradient updates. GERVE brings together kernel density estimation, mean-shift, variational inference, and annealing in a single platform for mode estimation. It fits Gaussian mixtures that concentrate on high-density regions and yields cluster assignments from responsibilities, with reduced sensitivity to the chosen number of components. We provide theory in two regimes: as the Gibbs temperature approaches zero, mixture components converge to population modes; at fixed temperature, maximisers of the empirical objective exist, are consistent, and are asymptotically normal. We also propose a bootstrap procedure for per-mode confidence ellipses and stability scores. Simulation and real-data studies show accurate mode recovery and emergent clustering, robust to mixture overspecification. GERVE is a practical likelihood-free approach when the number of modes or groups is unknown and full density estimation is impractical. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_17956 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A variational framework for modal estimation LeMinh, Tâm Arbel, Julyan Forbes, Florence Nguyen, Hien Duy Methodology Statistics Theory Computation We approach multivariate mode estimation through Gibbs distributions and introduce GERVE (Gibbs-measure Entropy-Regularised Variational Estimation), a likelihood-free framework that approximates Gibbs measures directly from samples by maximizing an entropy-regularised variational objective with natural-gradient updates. GERVE brings together kernel density estimation, mean-shift, variational inference, and annealing in a single platform for mode estimation. It fits Gaussian mixtures that concentrate on high-density regions and yields cluster assignments from responsibilities, with reduced sensitivity to the chosen number of components. We provide theory in two regimes: as the Gibbs temperature approaches zero, mixture components converge to population modes; at fixed temperature, maximisers of the empirical objective exist, are consistent, and are asymptotically normal. We also propose a bootstrap procedure for per-mode confidence ellipses and stability scores. Simulation and real-data studies show accurate mode recovery and emergent clustering, robust to mixture overspecification. GERVE is a practical likelihood-free approach when the number of modes or groups is unknown and full density estimation is impractical. |
| title | A variational framework for modal estimation |
| topic | Methodology Statistics Theory Computation |
| url | https://arxiv.org/abs/2602.17956 |