Posterior inference for Dirichlet process mixture models is analytically intractable and typically relies on Markov chain Monte Carlo methods, which can become computationally prohibitive at moderate to large sample sizes. In this work, we investigate the performance of Laplace and skew-Laplace posterior approximations for density estimation in this setting. Through an extensive numerical study covering four simulation scenarios with sample sizes ranging from n = 20 to n = 2,000 and four standard real datasets, we compare the standard Laplace approximation, its skew-corrected extension, and a slice sampling benchmark, assessing accuracy through total variation distance and computational efficiency through runtime. Our results show that the Gaussian Laplace approximation is more effective in this setting than might be anticipated, and that the skew-Laplace approximation consistently improves posterior recovery while remaining substantially faster than state-of-the-art Markov chain Monte Carlo samplers across all settings considered. In particular, the use of skew-Laplace in place of the standard Laplace approximation is especially beneficial in more complex density structures, where we observe error reductions typically on the order of 30%.

翻译：狄利克雷过程混合模型的后验推断在解析上难以处理，通常依赖马尔可夫链蒙特卡洛方法，但该方法在中到大样本量下可能产生高昂的计算成本。本研究探讨了在该场景下用于密度估计的拉普拉斯近似与偏态拉普拉斯近似的性能。通过涵盖四种模拟场景（样本量从n=20到n=2,000）及四个标准真实数据集的广泛数值研究，我们比较了标准拉普拉斯近似、其偏态修正扩展以及切片采样基准方法，基于全变差距离评估准确性，并根据运行时间评估计算效率。结果表明，高斯拉普拉斯近似在该场景下的有效性超出预期，而偏态拉普拉斯近似在持续提升后验恢复性能的同时，在所有测试设置下仍显著快于最先进的马尔可夫链蒙特卡洛采样器。特别地，在更复杂的密度结构中，使用偏态拉普拉斯替代标准拉普拉斯近似尤为优势，观察到的误差降低幅度通常达30%左右。