Bayesian inference for Poisson–hidden Markov models

细致地讨论一下用贝叶斯方法估计Poisson-HMM模型, 该问题来自 W. Zcchini 的 Hidden Markov Models for Time Series 一书第7章.

1. 问题概述

说是有一个 Poisson–HMM { X t } \{X_t\} { Xt​} on $m$ states, with underlying Markov chain { C t } \{C_t\} { Ct​}. We denote the state-dependent means by $\lambda =({\lambda }_1,\cdots ,{\lambda }_m)$, and the transition probability matrix of the Markov chain by $\Gamma$.

假设 $m$ 是已知的, { X t } \{X_t\} { Xt​} 是一元的, 观测值为 X ( T ) = { X 1 , X 2 , ⋯   , X T } X^{(T)}=\{X_1, X_2, \cdots, X_T\} X(T)={ X1​,X2​,⋯,XT​}.

2. 简要分析

要用贝叶斯方法估计该模型, 就是要获得从 Θ = { λ \Theta=\{\lambda Θ={ λ 和 $\Gamma \}$ 的后验分布 $P(\Theta \mid X^{(T)})$ 里的大量抽样. 按贝叶斯方法套路, $$P(\Theta \mid X^{(T)})\propto P(X^{(T)}\mid \Theta )P(\Theta )$$

$P(X^{(T)}\mid \Theta )$ 需要 summing over all possible sequences of latent states $S^{(T)}$. If the time series has length $T$ and there are $m$ possible states, this sum is over $m^T$ sequences, making direct computation intractable for long series. $$P(X^{(T)}\mid \Theta )=\sum _{S^T}P(X^{(T)},S^{(T)}\mid \Theta )$$

直接干不行, 需要退而从 $P(\Theta ,S^{(T)}\mid X^{(T)})$ 这儿抽样, 理论推导见 Tanner and Wong (1987), 结论就是下面算法经过足够多次(burn-in)迭代后, { Θ g , Θ g + 1 , ⋯   } \{\Theta^{g}, \Theta^{g+1}, \cdots\} { Θg,Θg+1,⋯} 就是来自 $p(\Theta \mid X^{(T)})$ 的样本.

这个算法就是Gibbs sampling, 是个循环迭代过程:

• In order to obtain the posterior distribution $p(\Theta \mid X^{(T)})$, we have to initialize $\Theta$ to ${\Theta }^1$ and get a sample of latent states $S^{(T),1}$ from $P(S^{(T)}\mid X^{(T)},{\Theta }^1)$.
• Then draw ${\Theta }^2$ (不是平方, 2是序号) from $P(\Theta \mid X^{(T)},S^{(T),1})$ which utilizes the factorization of the complete-data likelihood and conjugate relationship of the prior and the posterior distribution of $\Theta$.
  • $P(\Theta \mid X^{(T)},S^{(T)})\propto P(X^{(T)},S^{(T)}\mid \Theta )P(\Theta )$
  • The complete-data likelihood $P(X$