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.

翻译：我们通过吉布斯分布研究多元模态估计问题，并提出了GERVE（吉布斯测度熵正则化变分估计）——一种无需似然函数的框架，通过最大化具有自然梯度更新的熵正则化变分目标，直接从样本中近似吉布斯测度。GERVE将核密度估计、均值漂移、变分推断和退火技术集成于统一的模态估计平台。该方法拟合集中于高密度区域的高斯混合模型，并通过责任度产生聚类分配，对所选分量数量具有较低的敏感性。我们在两种机制下提供理论分析：当吉布斯温度趋近于零时，混合分量收敛于总体模态；在固定温度下，经验目标的最大化估计量存在、具有一致性且渐近正态。我们还提出了一种用于计算各模态置信椭圆和稳定性评分的自助法程序。仿真与真实数据研究表明，该方法能准确恢复模态并实现涌现聚类，对混合模型过参数化具有鲁棒性。当模态或群组数量未知且全密度估计不可行时，GERVE是一种实用的无需似然函数的方法。