Fast and scalable inference in hidden Markov models with Gaussian fields
含高斯场的隐马尔可夫模型：快速且可扩展的推断AI 解读 ↓
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含高斯场的隐马尔可夫模型：快速且可扩展的推断
- 研究问题
- 当隐马尔可夫时间序列模型加入空间效应、趋势等高斯潜在结构时，如何解决似然破坏 Hessian 稀疏性带来的计算困难？
- 方法与思路
- 作者结合 SPDE 高斯场、自动微分与 Laplace 近似，修改计算隐马尔可夫模型似然的前向算法以构造稀疏 Hessian。
- 材料陈述的贡献
- 摘要称修改后的算法支持快速且可扩展的推断，并通过模拟、恒星耀斑检测和狮子移动建模展示实际可行性与用途。
- 阅读关注点
- 应核对算法如何保持似然计算的正确性、稀疏结构如何形成，以及 Laplace 近似误差和计算成本随维度变化的表现。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未经核验；摘要未给出算法细节、近似精度或具体运行时间比较。
原文摘要
Hidden Markov models (HMMs) are powerful tools for analysing time series data that depend on discrete underlying but unobserved states. As such, they have gained prominence across numerous empirical disciplines, in particular ecology, medicine, and economics. However, the increasing complexity of empirical data is often accompanied by additional latent structure such as spatial effects, temporal trends, or measurement perturbations. Gaussian fields provide an attractive building block for incorporating such structured latent variation into HMMs. Fast inference methods for Gaussian fields have emerged through the stochastic partial differential equation (SPDE) approach. Due to their sparse representation, these integrate well with novel frequentist estimation methods for random-effects models via the use of automatic differentiation and the Laplace approximation. Scaling to high dimensions requires tools such as (R)TMB to exploit sparsity in the Hessian w.r.t. the latent variables - a property satisfied by SPDE fields but violated by the HMM likelihood. We present a modified forward algorithm to compute the HMM likelihood, constructing sparsity in the Hessian and consequently enabling fast and scalable inference. We demonstrate the practical feasibility and the usefulness through simulations and two case studies exploring the detection of stellar flares as well as modelling the movement of lions.