Multimodality of the likelihood in Gaussian mixtures is a well-known problem. The choice of the initial parameter vector for the numerical optimizer may affect whether the optimizer finds the global maximum, or gets trapped in a local maximum of the likelihood. We propose to use Hamiltonian Monte Carlo (HMC) to explore the part of the parameter space which has a high likelihood. Each sampled parameter vector is used as the initial value for quasi-Newton optimizer, and the resulting sample of (maximum) likelihood values is used to determine if the likelihood is multimodal. We use a single simulated data set from a three component bivariate mixture to develop and test the method. We use state-of-the-art HCM software, but experience difficulties when trying to directly apply HMC to the full model with 15 parameters. To improve the mixing of the Markov Chain we explore various tricks, and conclude that for the dataset at hand we have found the global maximum likelihood estimate.

翻译：高斯混合模型中似然的多模态性是一个众所周知的问题。数值优化器初始参数向量的选择会直接影响其能否找到全局最大值，或陷入似然函数的局部极大值。我们提出使用哈密顿蒙特卡洛（HMC）方法探索具有高似然值的参数空间区域。每个采样得到的参数向量被用作拟牛顿优化器的初始值，通过所得（最大）似然值的样本分布判断似然函数是否具有多模态性。我们采用一个包含三个分量的二元混合模拟数据集来开发和测试该方法。虽然使用了最先进的HCM软件，但在直接对具有15个参数的完整模型应用HMC时遇到困难。为改进马尔可夫链的混合效率，我们尝试了多种技巧，最终确认对于当前数据集已找到全局最大似然估计。