Update HMM resume

Author: RyoheiO

bayesml/hiddenmarkovnormal/hiddenmarkovnormal.md

@@ -9,7 +9,8 @@ The stochastic data generative model is as follows:
 * $K \in \mathbb{N}$: number of latent classes
 * $\boldsymbol{z} \in \{ 0, 1 \}^K$: 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, ($\sum_{k=1}^K a_{k'k}=1$)
+* $a_{jk}$ : 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, ($\sum_{k=1}^K a_{jk}=1$)
 * $D \in \mathbb{N}$: a dimension of data
 * $\boldsymbol{x} \in \mathbb{R}^D$: a data point
 * $\boldsymbol{\mu}_k \in \mathbb{R}^D$: a parameter
@@ -21,7 +22,7 @@ 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_{j=1}^K a_{jk}^{z_{n-1,j}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}
@@ -34,17 +35,17 @@ The prior distribution is as follows:
 * $\nu_0 \in \mathbb{R}$: a hyperparameter ($\nu_0 > D-1$)
 * $\boldsymbol{W}_0 \in \mathbb{R}^{D\times D}$: a hyperparameter (a positive definite matrix)
 * $\boldsymbol{\eta}_0 \in \mathbb{R}_{> 0}^K$: a hyperparameter
-* $\boldsymbol{\zeta}_{0,k'} \in \mathbb{R}_{> 0}^K$: a hyperparameter
+* $\boldsymbol{\zeta}_{0,j} \in \mathbb{R}_{> 0}^K$: a hyperparameter
 * $\mathrm{Tr} \{ \cdot \}$: a trace of a matrix
 * $\Gamma (\cdot)$: the gamma function
 
 $$
 \begin{align}
-    p(\boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_0,(\kappa_0 \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_0, \nu_0) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\eta}_0) \prod_{k'=1}^{K}\mathrm{Dir}(\boldsymbol{a}_{k'}|\boldsymbol{\zeta}_{0,k'}), \\
+    p(\boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi},\boldsymbol{A}) &= \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_0,(\kappa_0 \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_0, \nu_0) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\eta}_0) \prod_{j=1}^{K}\mathrm{Dir}(\boldsymbol{a}_{j}|\boldsymbol{\zeta}_{0,j}), \\
     &= \Biggl[ \prod_{k=1}^K \left( \frac{\kappa_0}{2\pi} \right)^{D/2} |\boldsymbol{\Lambda}_k|^{1/2} \exp \left\{ -\frac{\kappa_0}{2}(\boldsymbol{\mu}_k -\boldsymbol{m}_0)^\top \boldsymbol{\Lambda}_k (\boldsymbol{\mu}_k - \boldsymbol{m}_0) \right\} \\
-    &\qquad \times B(\boldsymbol{W}_0, \nu_0) | \boldsymbol{\Lambda}_k |^{(\nu_0 - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ \boldsymbol{W}_0^{-1} \boldsymbol{\Lambda}_k \} \right\} \Biggr] \\
-    &\qquad \times C(\boldsymbol{\eta}_0)\prod_{k=1}^K \pi_k^{\eta_{0,k}-1}\\
-    &\qquad \times \prod_{k'=1}^KC(\boldsymbol{\zeta}_{0,k'})\prod_{k=1}^K a_{k'k}^{\zeta_{0,k',k}-1},\\
+    &\qquad \times B(\boldsymbol{W}_0, \nu_0) | \boldsymbol{\Lambda}_k |^{(\nu_0 - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ \boldsymbol{W}_0^{-1} \boldsymbol{\Lambda}_k \} \right\} \\
+    &\qquad \times C(\boldsymbol{\eta}_0)\pi_k^{\eta_{0,k}-1}\\
+    &\qquad \times \prod_{j=1}^KC(\boldsymbol{\zeta}_{0,j}) a_{jk}^{\zeta_{0,j,k}-1}\Biggr],\\
 \end{align}
 $$
 
@@ -54,7 +55,7 @@ $$
 \begin{align}
     B(\boldsymbol{W}_0, \nu_0) &= | \boldsymbol{W}_0 |^{-\nu_0 / 2} \left( 2^{\nu_0 D / 2} \pi^{D(D-1)/4} \prod_{i=1}^D \Gamma \left( \frac{\nu_0 + 1 - i}{2} \right) \right)^{-1}, \\
     C(\boldsymbol{\eta}_0) &= \frac{\Gamma(\sum_{k=1}^K \eta_{0,k})}{\Gamma(\eta_{0,1})\cdots\Gamma(\eta_{0,K})},\\
-    C(\boldsymbol{\zeta}_{0,k'}) &= \frac{\Gamma(\sum_{k=1}^K \zeta_{0,k',k})}{\Gamma(\zeta_{0,k',1})\cdots\Gamma(\zeta_{0,k',K})}. 
+    C(\boldsymbol{\zeta}_{0,j}) &= \frac{\Gamma(\sum_{k=1}^K \zeta_{0,j,k})}{\Gamma(\zeta_{0,j,1})\cdots\Gamma(\zeta_{0,j,K})}. 
 \end{align}
 $$
 <!--