隐马尔可夫模型参数的变分推断方法研究

The Study of Variational Inference Methods for HMMs’ Parameters

摘要: 隐马尔可夫模型因其功能强大且拥有坚实的数理基础而被广泛应用于众多领域。然而，其传统参数估计方法通常基于最大似然估计，难以有效刻画复杂分布结构及模型不确定性，尤其在存在异常值或厚尾特征的金融数据中表现受限。故本文引入变分推断框架，构造了一种基于多元t分布的变分隐马尔可夫模型，并给出了详细推导过程。然后，通过数值模拟实验测试了其参数估计与状态预测的正确性与有效性；最后，使用S&P500与CSI300两套真实股指数据进行了实证分析，结果表明VBtHMM模型参数估计更加高效稳健，在应对含异常值或非高斯噪声数据建模中，仍能保持较强的参数估计与状态预测能力。

Abstract: Hidden Markov Models (HMMs) have been widely used in various fields due to their strong modeling capabilities and sound mathematical foundation. However, traditional parameter estimation of HMMs mainly relies on Maximum Likelihood Estimation (MLE), which fails to effectively capture complex distributional structures and quantify model uncertainty—this limitation is more obvious when modeling financial data with outliers or heavy-tailed distributions. To address this problem, this paper proposes a variational HMM based on multivariate t-distribution under the Variational Inference (VI) framework, with detailed mathematical derivations provided. Numerical simulations are carried out to verify the validity and effectiveness of the proposed model in parameter estimation and latent state prediction. Finally, empirical analyses using real S&P 500 and CSI 300 data show that the VBtHMM achieves more efficient and robust parameter estimation, and maintains strong performance in parameter estimation and state prediction even for data with outliers or non-Gaussian noise.