Wright-Fisher隐马尔可夫模型中预测与平滑的渐近性及有限样本界
Asymptotics and finite sample bounds for prediction and smoothing in Wright-Fisher hidden Markov models
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- Duke University（杜克大学）
- University of Torino（都灵大学）
- New York University Abu Dhabi（阿布扎比纽约大学）
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中文总结 AI 辅助
本研究针对Wright-Fisher隐马尔可夫模型，证明预测与平滑分布在重复抽样下收敛于群体转移律和桥律，并给出有限样本界及渐近分布，适用于古DNA等时间序列数据。
中文摘要 AI 辅助
我们研究了隐马尔可夫模型中的预测与平滑问题，其中潜在信号由多类型Wright-Fisher扩散给出，观测为离散时间的分类数据，其动机来自重复抽样时间序列设置，包括时间分箱的古DNA数据，在这些数据中，噪声频率计数在有限多个时间点被记录。我们的重点是在亲本独立突变下可得的精确贝叶斯预测分布和平滑分布，及其在时间内部重复抽样下的大样本目标。对于固定的采集时间网格和发散的时间内部样本量，我们证明了精确的Wright-Fisher预测分布和平滑分布在总变差距离下收敛到极限相邻频率处的相应群体转移律和桥律。然后，我们推导了预测律的显式有限样本控制以及边际平滑器的相应有限样本界。最后，在检查时刻，我们证明了联合条件分布集中在目标频率上，其活跃坐标渐近服从高斯分布，而真实频率为零的坐标以更快的速度收敛到Gamma极限。我们的框架对检查时刻之间的依赖性不施加除时间内部抽样之外的任何限制。分析依赖于Kingman聚结块计数过程的固定区间尾界，该尾界本身具有独立的意义。
英文摘要
We study prediction and smoothing in hidden Markov models with a latent signal given by a multi-type Wright-Fisher diffusion and discrete-time categorical observations, motivated by repeated-sampling time-series settings, including temporally binned ancient-DNA data, in which noisy frequency counts are recorded at finitely many times. Our focus is on the exact Bayesian predictive and smoothing distributions available under parent-independent mutation, in relation to their large-sample targets under repeated within-time sampling. For a fixed collection-time grid and diverging within-time sample sizes, we show that the exact Wright-Fisher predictive and smoothing distributions converge in total variation to the corresponding population transition and bridge laws at the limiting neighboring frequencies. We then derive explicit finite-sample control for the predictive law and a corresponding finite-sample bound for the marginal smoother. Finally, at the inspection times, we show that the joint conditional law concentrates at the target frequencies and that its active coordinates are asymptotically Gaussian, while coordinates with zero true frequencies converge to Gamma limits at faster rates. Our regime imposes no restriction on dependence across inspection times beyond within-time sampling. The analysis rests on a fixed-interval tail bound for Kingman's coalescent block-counting process, which is of independent interest.
