贝叶斯推断与深度神经网络在死亡率预测中的应用与对比研究

Application and Comparative Study of Bayesian Inference and Deep Neural Networks in Mortality Rate Prediction

摘要: 随着社会的不断发展和死亡率的下降，死亡率数据分析及未来预测一直是精算师、人口学家和统计学家的研究重点。经典的单因子Lee-Carter模型通过对数转换将死亡率分解为年龄和时间项，尽管形式简单，但在处理复杂死亡率模式时表现有限。为此，我们采用了神经网络DNN模型，通过多层神经网络捕捉数据中的非线性关系，期望有所改善。同时，在经典Heligman-Pollard模型的基础上，引入了动态分量和随机游走结构，放宽了参数平稳性假设，从而能更精准地描述不同年龄段死亡率的变化。我们还使用GMRF模型，通过高效捕捉跨年龄组和跨年份的死亡率相关性，并通过基于梯度的MCMC采样器提高采样效率，以期减少误差。最后，结合均方根误差(RMSE)、平均绝对误差(MAE)和平均绝对百分比误差(MAPE)等评价指标，对不同模型的结果进行了比较分析。结果表明，GMRF模型在预测精度方面表现最佳，Heligman-Pollard模型适用于描述死亡率的动态变化，而DNN模型则适合于大规模数据且需捕捉非线性关系的场景。

Abstract: With society’s development and declining mortality rates, mortality data analysis and forecasting have been key research areas for actuaries, demographers, and statisticians. The Lee-Carter model decomposes mortality into age and time components using a logarithmic transformation. While simple, it has limitations in handling complex mortality patterns. To improve, we used a neural network (DNN) model to capture nonlinear relationships in the data. Additionally, we enhanced the Heligman-Pollard model with dynamic components and random walk structure, allowing more accurate description of age-specific mortality changes. We also use GMRF model to capture the mortality correlation across age groups and years efficiently, and improve the sampling efficiency through gradient-based MCMC sampler in order to reduce errors. Finally, we compared the model’s using metrics like RMSE, MAE, and MAPE. The results show that the GMRF model performs best in accuracy, the Heligman-Pollard model excels at describing dynamic mortality changes, and the DNN model is best for large datasets with nonlinear relationships.