Update hiddenmarkovnormal.md

Author: RyoheiO

bayesml/hiddenmarkovnormal/hiddenmarkovnormal.md

@@ -7,10 +7,11 @@ The Hidden Markov model with the Gauss-Wishart prior distribution and the Dirich
 The stochastic data generative model is as follows:
 
 * $K \in \mathbb{N}$: number of latent classes
-* $\boldsymbol{z}_{n} \in \{ 0, 1 \}^K$: a one-hot vector representing the $n$-th latent class (latent variable)
+* $\boldsymbol{z} \in \{ 0, 1 \}^K$: a one-hot vector representing the latent class (latent variable)
 * $\boldsymbol{\pi} \in [0, 1]^K$: a parameter for latent classes, ($\sum_{k=1}^K \pi_k=1$)
+* $\boldsymbol{A}=(a_{k'k})_{0\leq k',k\leq K} \in [0, 1]^{K\times K}$: a parameter for latent classes, ($\sum_{k=1}^K a_{k'k}=1$)
 * $D \in \mathbb{N}$: a dimension of data
-* $\boldsymbol{x}_{n} \in \mathbb{R}^D$: the $n$-th data point
+* $\boldsymbol{x} \in \mathbb{R}^D$: a data point
 * $\boldsymbol{\mu}_k \in \mathbb{R}^D$: a parameter
 * $\boldsymbol{\mu} = \{ \boldsymbol{\mu}_k \}_{k=1}^K$
 * $\boldsymbol{\Lambda}_k \in \mathbb{R}^{D\times D}$ : a parameter (a positive definite matrix)
@@ -20,7 +21,7 @@ The stochastic data generative model is as follows:
 $$
 \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_{k'=1}^K a_{k'k}^{z_{n-1,k}z_{n,k}},\\
+    p(\boldsymbol{z}_{n} |\boldsymbol{z}_{n-1} ,\boldsymbol{A}) &= \prod_{k=1}^K \prod_{k'=1}^K a_{k'k}^{z_{n-1,k'}z_{n,k}},\\
     p(\boldsymbol{x}_{n} | \boldsymbol{\mu}, \boldsymbol{\Lambda}, \boldsymbol{z}_{n}) &= \prod_{k=1}^K \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_k,\boldsymbol{\Lambda}_k^{-1})^{z_{n,k}} \\
     &= \prod_{k=1}^K \left( \frac{| \boldsymbol{\Lambda}_{k} |^{1/2}}{(2\pi)^{D/2}} \exp \left\{ -\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu}_{k})^\top \boldsymbol{\Lambda}_{k} (\boldsymbol{x}-\boldsymbol{\mu}_{k}) \right\} \right)^{z_{n,k}},
 \end{align}
@@ -32,25 +33,28 @@ The prior distribution is as follows:
 * $\kappa_0 \in \mathbb{R}_{>0}$: a hyperparameter
 * $\nu_0 \in \mathbb{R}$: a hyperparameter ($\nu_0 > D-1$)
 * $\boldsymbol{W}_0 \in \mathbb{R}^{D\times D}$: a hyperparameter (a positive definite matrix)
-* $\boldsymbol{\alpha}_0 \in \mathbb{R}_{> 0}^K$: a hyperparameter
+* $\boldsymbol{\eta}_0 \in \mathbb{R}_{> 0}^K$: a hyperparameter
+* $\boldsymbol{\zeta}_{0,k'} \in \mathbb{R}_{> 0}^K$: a hyperparameter
 * $\mathrm{Tr} \{ \cdot \}$: a trace of a matrix
 * $\Gamma (\cdot)$: the gamma function
 
 $$
 \begin{align}
-    p(\boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_0,(\kappa_0 \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_0, \nu_0) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\alpha}_0) \\
+    p(\boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_0,(\kappa_0 \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_0, \nu_0) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\eta}_0) \prod_{k'=1}^{K}\mathrm{Dir}(\boldsymbol{a}_{k'}|\boldsymbol{\zeta}_{0,k'}), \\
     &= \Biggl[ \prod_{k=1}^K \left( \frac{\kappa_0}{2\pi} \right)^{D/2} |\boldsymbol{\Lambda}_k|^{1/2} \exp \left\{ -\frac{\kappa_0}{2}(\boldsymbol{\mu}_k -\boldsymbol{m}_0)^\top \boldsymbol{\Lambda}_k (\boldsymbol{\mu}_k - \boldsymbol{m}_0) \right\} \\
     &\qquad \times B(\boldsymbol{W}_0, \nu_0) | \boldsymbol{\Lambda}_k |^{(\nu_0 - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ \boldsymbol{W}_0^{-1} \boldsymbol{\Lambda}_k \} \right\} \Biggr] \\
-    &\qquad \times C(\boldsymbol{\alpha}_0)\prod_{k=1}^K \pi_k^{\alpha_{0,k}-1},\\
+    &\qquad \times C(\boldsymbol{\eta}_0)\prod_{k=1}^K \pi_k^{\eta_{0,k}-1}\\
+    &\qquad \times \prod_{k'=1}^KC(\boldsymbol{\zeta}_{0,k'})\prod_{k=1}^K a_{k'k}^{\zeta_{0,k',k}-1},\\
 \end{align}
 $$
 
-where $B(\boldsymbol{W}_0, \nu_0)$ and $C(\boldsymbol{\alpha}_0)$ are defined as follows:
+where $B(\boldsymbol{W}_0, \nu_0)$ and $C(\boldsymbol{\eta}_0)$ are defined as follows:
 
 $$
 \begin{align}
     B(\boldsymbol{W}_0, \nu_0) &= | \boldsymbol{W}_0 |^{-\nu_0 / 2} \left( 2^{\nu_0 D / 2} \pi^{D(D-1)/4} \prod_{i=1}^D \Gamma \left( \frac{\nu_0 + 1 - i}{2} \right) \right)^{-1}, \\
-    C(\boldsymbol{\alpha}_0) &= \frac{\Gamma(\sum_{k=1}^K \alpha_{0,k})}{\Gamma(\alpha_{0,1})\cdots\Gamma(\alpha_{0,K})}.
+    C(\boldsymbol{\eta}_0) &= \frac{\Gamma(\sum_{k=1}^K \eta_{0,k})}{\Gamma(\eta_{0,1})\cdots\Gamma(\eta_{0,K})},\\
+    C(\boldsymbol{\zeta}_{0,k'}) &= \frac{\Gamma(\sum_{k=1}^K \zeta_{0,k',k})}{\Gamma(\zeta_{0,k',1})\cdots\Gamma(\zeta_{0,k',K})}. 
 \end{align}
 $$
 
@@ -63,14 +67,14 @@ The apporoximate posterior distribution in the $t$-th iteration of a variational
 * $\kappa_{n,k}^{(t)} \in \mathbb{R}_{>0}$: a hyperparameter
 * $\nu_{n,k}^{(t)} \in \mathbb{R}$: a hyperparameter $(\nu_n > D-1)$
 * $\boldsymbol{W}_{n,k}^{(t)} \in \mathbb{R}^{D\times D}$: a hyperparameter (a positive definite matrix)
-* $\boldsymbol{\alpha}_n^{(t)} \in \mathbb{R}_{> 0}^K$: a hyperparameter
+* $\boldsymbol{\eta}_n^{(t)} \in \mathbb{R}_{> 0}^K$: a hyperparameter
 
 $$
 \begin{align}
-    q(\boldsymbol{z}^n, \boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{i=1}^n \mathrm{Cat} (\boldsymbol{z}_i | \boldsymbol{r}_i^{(t)}) \right\} \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_{n,k}^{(t)},(\kappa_{n,k}^{(t)} \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_{n,k}^{(t)}, \nu_{n,k}^{(t)}) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\alpha}_n^{(t)}) \\
+    q(\boldsymbol{z}^n, \boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{i=1}^n \mathrm{Cat} (\boldsymbol{z}_i | \boldsymbol{r}_i^{(t)}) \right\} \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_{n,k}^{(t)},(\kappa_{n,k}^{(t)} \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_{n,k}^{(t)}, \nu_{n,k}^{(t)}) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\eta}_n^{(t)}) \\
     &= \Biggl[ \prod_{i=1}^n \prod_{k=1}^K (r_{i,k}^{(t)})^{z_{i,k}} \Biggr] \Biggl[ \prod_{k=1}^K \left( \frac{\kappa_{n,k}^{(t)}}{2\pi} \right)^{D/2} |\boldsymbol{\Lambda}_k|^{1/2} \exp \left\{ -\frac{\kappa_{n,k}^{(t)}}{2}(\boldsymbol{\mu}_k -\boldsymbol{m}_{n,k}^{(t)})^\top \boldsymbol{\Lambda}_k (\boldsymbol{\mu}_k - \boldsymbol{m}_{n,k}^{(t)}) \right\} \\
     &\qquad \times B(\boldsymbol{W}_{n,k}^{(t)}, \nu_{n,k}^{(t)}) | \boldsymbol{\Lambda}_k |^{(\nu_{n,k}^{(t)} - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ ( \boldsymbol{W}_{n,k}^{(t)} )^{-1} \boldsymbol{\Lambda}_k \} \right\} \Biggr] \\
-    &\qquad \times C(\boldsymbol{\alpha}_n^{(t)})\prod_{k=1}^K \pi_k^{\alpha_{n,k}^{(t)}-1},\\
+    &\qquad \times C(\boldsymbol{\eta}_n^{(t)})\prod_{k=1}^K \pi_k^{\eta_{n,k}^{(t)}-1},\\
 \end{align}
 $$
 
@@ -84,8 +88,8 @@ $$
     \kappa_{n,k}^{(t+1)} &= \kappa_0 + N_k^{(t)}, \\
     (\boldsymbol{W}_{n,k}^{(t+1)})^{-1} &= \boldsymbol{W}_0^{-1} + \sum_{i=1}^{n} r_{i,k}^{(t)} (\boldsymbol{x}_i-\bar{\boldsymbol{x}}_k^{(t)})(\boldsymbol{x}_i-\bar{\boldsymbol{x}}_k^{(t)})^\top + \frac{\kappa_0 N_k^{(t)}}{\kappa_0 + N_k^{(t)}}(\bar{\boldsymbol{x}}_k^{(t)}-\boldsymbol{\mu}_0)(\bar{\boldsymbol{x}}_k^{(t)}-\boldsymbol{\mu}_0)^\top, \\
     \nu_{n,k}^{(t+1)} &= \nu_0 + N_k^{(t)},\\
-    \alpha_{n,k}^{(t+1)} &= \alpha_{0,k} + N_k^{(t)} \\
-    \ln \rho_{i,k}^{(t+1)} &= \psi (\alpha_{n,k}^{(t+1)}) - \psi ( {\textstyle \sum_{k=1}^K \alpha_{n,k}^{(t+1)}} ) \nonumber \\
+    \eta_{n,k}^{(t+1)} &= \eta_{0,k} + N_k^{(t)} \\
+    \ln \rho_{i,k}^{(t+1)} &= \psi (\eta_{n,k}^{(t+1)}) - \psi ( {\textstyle \sum_{k=1}^K \eta_{n,k}^{(t+1)}} ) \nonumber \\
     &\qquad + \frac{1}{2} \Biggl[ \sum_{d=1}^D \psi \left( \frac{\nu_{n,k}^{(t+1)} + 1 - d}{2} \right) + D \ln 2 + \ln | \boldsymbol{W}_{n,k}^{(t+1)} | \nonumber \\
     &\qquad - D \ln (2 \pi ) - \frac{D}{\kappa_{n,k}^{(t+1)}} - \nu_{n,k}^{(t+1)} (\boldsymbol{x}_i - \boldsymbol{m}_{n,k}^{(t+1)})^\top \boldsymbol{W}_{n,k}^{(t+1)} (\boldsymbol{x}_i - \boldsymbol{m}_{n,k}^{(t+1)}) \Biggr] \\
     r_{i,k}^{(t+1)} &= \frac{\rho_{i,k}^{(t+1)}}{\sum_{k=1}^K \rho_{i,k}^{(t+1)}}
@@ -102,8 +106,8 @@ The approximate predictive distribution is as follows:
 $$
 \begin{align}
     &p(x_{n+1}|x^n) \\
-    &= \frac{1}{\sum_{k=1}^K \alpha_{n,k}^{(t)}} \sum_{k=1}^K \alpha_{n,k}^{(t)} \mathrm{St}(x_{n+1}|\boldsymbol{\mu}_{\mathrm{p},k},\boldsymbol{\Lambda}_{\mathrm{p},k}, \nu_{\mathrm{p},k}) \\
-    &= \frac{1}{\sum_{k=1}^K \alpha_{n,k}^{(t)}} \sum_{k=1}^K \alpha_{n,k}^{(t)} \Biggl[ \frac{\Gamma (\nu_{\mathrm{p},k} / 2 + D / 2)}{\Gamma (\nu_{\mathrm{p},k} / 2)} \frac{|\boldsymbol{\Lambda}_{\mathrm{p},k}|^{1/2}}{(\nu_{\mathrm{p},k} \pi)^{D/2}} \nonumber \\
+    &= \frac{1}{\sum_{k=1}^K \eta_{n,k}^{(t)}} \sum_{k=1}^K \eta_{n,k}^{(t)} \mathrm{St}(x_{n+1}|\boldsymbol{\mu}_{\mathrm{p},k},\boldsymbol{\Lambda}_{\mathrm{p},k}, \nu_{\mathrm{p},k}) \\
+    &= \frac{1}{\sum_{k=1}^K \eta_{n,k}^{(t)}} \sum_{k=1}^K \eta_{n,k}^{(t)} \Biggl[ \frac{\Gamma (\nu_{\mathrm{p},k} / 2 + D / 2)}{\Gamma (\nu_{\mathrm{p},k} / 2)} \frac{|\boldsymbol{\Lambda}_{\mathrm{p},k}|^{1/2}}{(\nu_{\mathrm{p},k} \pi)^{D/2}} \nonumber \\
     &\qquad \qquad \qquad \qquad \qquad \times \left( 1 + \frac{1}{\nu_{\mathrm{p},k}} (\boldsymbol{x}_{n+1} - \boldsymbol{\mu}_{\mathrm{p},k})^\top \boldsymbol{\Lambda}_{\mathrm{p},k} (\boldsymbol{x}_{n+1} - \boldsymbol{\mu}_{\mathrm{p},k}) \right)^{-\nu_{\mathrm{p},k}/2 - D/2} \Biggr],
 \end{align}
 $$