Develop hiddenmarkovautoregressive resume

bayesml/hiddenmarkovautoregressive/hiddenmarkovautoregressive.md

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+<!-- Document Author
+Koki Kazama <kokikazama@aoni.waseda.jp>
+-->
+The Hidden Markov model with the Gauss-Wishart prior distribution and the Dirichlet prior distribution.
+
+The stochastic data generative model is as follows:
+
+* $K \in \mathbb{N}$: number of latent classes
+* $n \in \mathbb{N}$: time index
+* $\boldsymbol{z}_{n} \in \{ 0, 1 \}^K$: a one-hot vector representing the latent class (latent variable)
+* $\boldsymbol{\pi}=[\pi_{1},\dots,\pi_{K}]^{\top} \in [0, 1]^K$: a parameter for latent classes, ($\sum_{k=1}^K \pi_k=1$)
+* $a_{jk}$ : transition probability to latent state k under latent state j
+* $\boldsymbol{A}=(a_{jk})_{0\leq j,k\leq K} \in [0, 1]^{K\times K}$: a parameter for latent classes, ($\sum_{k=1}^K a_{jk}=1$)
+* $d \in \mathbb{N}$: the degree of the model
+* $\boldsymbol{x}'_n := [1, x_{n-d}, x_{n-d+1}, \dots , x_{n-1}]^\top \in \mathbb{R}^{d+1}$. Here we assume $x_n$ for $n < 1$ is given as a initial value.
+* $\boldsymbol{\theta}_{k} \in \mathbb{R}^{d+1}$: a regression coefficient parameter
+* $\boldsymbol{\theta}:=\{\boldsymbol{\theta}_{k}\}_{k=1}^{K}$
+* $x_n \in \mathbb{R}$: a data point at $n$
+* $\tau_{k} \in \mathbb{R}_{>0}$: a precision parameter of noise
+* $\boldsymbol{\tau}:=\{\tau_{k}\}_{k=1}^{K}$.
+
+
+$$
+\begin{align}
+    p(\boldsymbol{z}_{1} | \boldsymbol{\pi}) &= \mathrm{Cat}(\boldsymbol{z}_{1}|\boldsymbol{\pi}) = \prod_{k=1}^K \pi_k^{z_{1,k}},\\
+    p(\boldsymbol{z}_{n} |\boldsymbol{z}_{n-1} ,\boldsymbol{A}) &= \prod_{k=1}^K \prod_{j=1}^K a_{jk}^{z_{n-1,j}z_{n,k}},\\
+    p(x_n | \boldsymbol{x}'_{n-1}, \boldsymbol{\theta}, \boldsymbol{\tau}) &= \prod_{k=1}^{K}\mathcal{N}(x_n|\boldsymbol{\theta}_{k}^\top \boldsymbol{x}'_{n-1}, \tau_{k}^{-1})^{z_{k}} \\
+    &= \prod_{k=1}^{K}\left(\sqrt{\frac{\tau_{k}}{2 \pi}} \exp \left\{ -\frac{\tau_{k}}{2} (x_n - \boldsymbol{\theta}_{k}^\top \boldsymbol{x}'_{n-1})^2 \right\}\right)^{z_{k}}
+\end{align}
+$$
+
+The prior distribution is as follows:
+
+* $\boldsymbol{\mu}_0 \in \mathbb{R}^{d+1}$: a hyperparameter for $\boldsymbol{\theta}$
+* $\boldsymbol{\Lambda}_0 \in \mathbb{R}^{(d+1) \times (d+1)}$: a hyperparameter for $\boldsymbol{\theta}$ (a positive definite matrix)
+* $| \boldsymbol{\Lambda}_0 | \in \mathbb{R}$: the determinant of $\boldsymbol{\Lambda}_0$
+* $\alpha_0 \in \mathbb{R}_{>0}$: a hyperparameter for $\tau$
+* $\beta_0 \in \mathbb{R}_{>0}$: a hyperparameter for $\tau$
+* $\Gamma(\cdot): \mathbb{R}_{>0} \to \mathbb{R}$: the Gamma function
+* $\boldsymbol{\eta}_0=(\eta_{0,1},\dots,\eta_{0,K}) \in \mathbb{R}_{> 0}^K$: a hyperparameter
+* $\boldsymbol{\zeta}_{0,j}=(\zeta_{0,j,1},\dots,\zeta_{0,j,K}) \in \mathbb{R}_{> 0}^K$: a hyperparameter for $j=1,\dots,K$
+* $\boldsymbol{a}_{j}=\{a_{j,k}\}_{k=1}^{K}$ for $j=1,\dots,K$
+* $\mathrm{Tr} \{ \cdot \}$: a trace of a matrix
+
+$$
+\begin{align}
+    &p(\boldsymbol{\theta}, \boldsymbol{\tau},\boldsymbol{\pi},\boldsymbol{A}) 
+    \\
+    &=\left[\prod_{k=1}^{K} \mathcal{N}(\boldsymbol{\theta}_{k}|\boldsymbol{\mu}_0, (\tau _{k}\boldsymbol{\Lambda}_0)^{-1}) \mathrm{Gam}(\tau_{k}|\alpha_0,\beta_0)\right]
+    \\
+    &\qquad\times\left[\mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\eta}_0) \prod_{k=1}^{K}\mathrm{Dir}(\boldsymbol{a}_{j}|\boldsymbol{\zeta}_{0,j})\right]\\
+    &=\left[\prod_{k=1}^{K}\frac{|\tau_{k}\boldsymbol{\Lambda}_0|^{1/2}}{(2 \pi)^{(d+1)/2}} 
+    \exp \left\{ -\frac{\tau_{k}}{2} (\boldsymbol{\theta}_{k} - \boldsymbol{\mu}_0)^\top 
+    \boldsymbol{\Lambda}_0 (\boldsymbol{\theta}_{k} - \boldsymbol{\mu}_0) \right\}
+    \frac{\beta_0^{\alpha_0}}{\Gamma (\alpha_0)} \tau_{k}^{\alpha_0 - 1} \exp \{ -\beta_0 \tau_{k} \}\right]
+    \\
+    &\qquad\times\left[ \prod_{k=1}^KC(\boldsymbol{\eta}_0)\pi_k^{\eta_{0,k}-1}\biggl]\times \biggl[\prod_{j=1}^KC(\boldsymbol{\zeta}_{0,j})\prod_{k=1}^K a_{jk}^{\zeta_{0,j,k}-1}\right],
+\end{align}
+$$
+​
+where $C(\boldsymbol{\eta}_0)$ is defined as follows:
+​
+$$
+\begin{align}
+    C(\boldsymbol{\eta}_0) &= \frac{\Gamma(\sum_{k=1}^K \eta_{0,k})}{\Gamma(\eta_{0,1})\cdots\Gamma(\eta_{0,K})},\\
+    C(\boldsymbol{\zeta}_{0,j}) &= \frac{\Gamma(\sum_{k=1}^K \zeta_{0,j,k})}{\Gamma(\zeta_{0,j,1})\cdots\Gamma(\zeta_{0,j,K})}. 
+\end{align}
+$$