Develop metatree resume

bayesml/autoregressive/__init__.py

@@ -85,7 +85,7 @@
 .. math::
     \mathrm{St}(x_{n+1}|m_\mathrm{p}, \lambda_\mathrm{p}, \nu_\mathrm{p})
     = \frac{\Gamma (\nu_\mathrm{p}/2 + 1/2)}{\Gamma (\nu_\mathrm{p}/2)}
-    \left( \frac{m_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2}
+    \left( \frac{\lambda_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2}
     \left[ 1 + \frac{\lambda_\mathrm{p}(x_{n+1}-m_\mathrm{p})^2}{\nu_\mathrm{p}} \right]^{-\nu_\mathrm{p}/2 - 1/2}.
 
 .. math::
@@ -95,7 +95,7 @@
 where the parameters are obtained from the hyperparameters of the posterior distribution as follows.
 
 .. math::
-    m_\mathrm{p} &= \mu_n^\top \boldsymbol{x}'_n,\\
+    m_\mathrm{p} &= \boldsymbol{\mu}_n^\top \boldsymbol{x}'_n,\\
     \lambda_\mathrm{p} &= \frac{\alpha_n}{\beta_n} (1 + (\boldsymbol{x}'_n)^\top \boldsymbol{\Lambda}_n^{-1} \boldsymbol{x}'_n)^{-1},\\
     \nu_\mathrm{p} &= 2 \alpha_n.
 """

bayesml/bernoulli/__init__.py

@@ -23,7 +23,7 @@
 * :math:`B(\cdot,\cdot): \mathbb{R}_{>0} \times \mathbb{R}_{>0} \to \mathbb{R}_{>0}`: the Beta function
 
 .. math::
-    p(\theta) = \mathrm{Beta}(\theta|\alpha_0,\beta_0) = \frac{1}{B(\alpha_0, \beta_0)} \theta^{\alpha_0} (1-\theta)^{\beta_0}.
+    p(\theta) = \mathrm{Beta}(\theta|\alpha_0,\beta_0) = \frac{1}{B(\alpha_0, \beta_0)} \theta^{\alpha_0 - 1} (1-\theta)^{\beta_0 - 1}.
 
 .. math::
     \mathbb{E}[\theta] &= \frac{\alpha_0}{\alpha_0 + \beta_0}, \\
@@ -36,11 +36,11 @@
 * :math:`\beta_n \in \mathbb{R}_{>0}`: a hyperparameter
 
 .. math::
-    p(\theta | x^n) = \mathrm{Beta}(\theta|\alpha_n,\beta_n) = \frac{1}{B(\alpha_n, \beta_n)} \theta^{\alpha_n} (1-\theta)^{\beta_n},
+    p(\theta | x^n) = \mathrm{Beta}(\theta|\alpha_n,\beta_n) = \frac{1}{B(\alpha_n, \beta_n)} \theta^{\alpha_n - 1} (1-\theta)^{\beta_n - 1},
 
 .. math::
     \mathbb{E}[\theta | x^n] &= \frac{\alpha_n}{\alpha_n + \beta_n}, \\
-    \mathbb{V}[\theta | x^n] &= \frac{\alpha_n \beta_n}{(\alpha_n + \beta_n)^2 (\alpha_n + \beta_n + 1)}.
+    \mathbb{V}[\theta | x^n] &= \frac{\alpha_n \beta_n}{(\alpha_n + \beta_n)^2 (\alpha_n + \beta_n + 1)},
 
 where the updating rule of the hyperparameters is
 
@@ -56,16 +56,16 @@
 * :math:`\theta_\mathrm{p} \in [0,1]`: a parameter
 
 .. math::
-    p(x_{n+1} | x^n) = \mathrm{Bern}(x_{n+1}|\theta_\mathrm{p}) =\theta_\mathrm{p}^{x_{n+1}}(1-\theta_\mathrm{p})^{1-x_{n+1}}
+    p(x_{n+1} | x^n) = \mathrm{Bern}(x_{n+1}|\theta_\mathrm{p}) =\theta_\mathrm{p}^{x_{n+1}}(1-\theta_\mathrm{p})^{1-x_{n+1}},
 
 .. math::
     \mathbb{E}[x_{n+1} | x^n] &= \theta_\mathrm{p}, \\
-    \mathbb{V}[x_{n+1} | x^n] &= \theta_\mathrm{p} (1 - \theta_\mathrm{p}).
+    \mathbb{V}[x_{n+1} | x^n] &= \theta_\mathrm{p} (1 - \theta_\mathrm{p}),
 
 where the parameters are obtained from the hyperparameters of the posterior distribution as follows.
 
 .. math::
-    \theta_\mathrm{p} = \frac{\alpha_n}{\alpha_n + \beta_n}
+    \theta_\mathrm{p} = \frac{\alpha_n}{\alpha_n + \beta_n}.
 """
 
 from ._bernoulli import GenModel

bayesml/categorical/__init__.py

@@ -23,7 +23,7 @@
 
 The prior distribution is as follows:
 
-* :math:`\boldsymbol{\alpha}_0 \in \mathbb{R}_{>0}`: a hyperparameter
+* :math:`\boldsymbol{\alpha}_0 \in \mathbb{R}_{>0}^d`: a hyperparameter
 * :math:`\Gamma (\cdot)`: the gamma function
 * :math:`\tilde{\alpha}_0 = \sum_{k=1}^d \alpha_{0,k}`
 * :math:`C(\boldsymbol{\alpha}_0)=\frac{\Gamma(\tilde{\alpha}_0)}{\Gamma(\alpha_{0,1})\cdots\Gamma(\alpha_{0,d})}`
@@ -58,7 +58,7 @@
 
 The predictive distribution is as follows:
 
-* :math:`x_{n+1} \in \{ 0, 1\}^d`: a new data point
+* :math:`\boldsymbol{x}_{n+1} \in \{ 0, 1\}^d`: a new data point
 * :math:`\boldsymbol{\theta}_\mathrm{p} \in [0, 1]^d`: the hyperparameter of the posterior (:math:`\sum_{k=1}^d \theta_{\mathrm{p},k} = 1`)
 
 .. math::
@@ -72,7 +72,7 @@
 where the parameters are obtained from the hyperparameters of the posterior distribution as follows:
 
 .. math::
-    \boldsymbol{\theta}_{\mathrm{p},k} = \frac{\alpha_{n,k}}{\sum_{k=1}^d \alpha_{n,k}}, \quad (k \in \{ 1, 2, \dots , d \}).
+    \theta_{\mathrm{p},k} = \frac{\alpha_{n,k}}{\sum_{k=1}^d \alpha_{n,k}}, \quad (k \in \{ 1, 2, \dots , d \}).
 """
 
 from ._categorical import GenModel

bayesml/exponential/__init__.py

@@ -42,13 +42,13 @@
 
 .. math::
     \mathbb{E}[\lambda | x^n] &= \frac{\alpha_n}{\beta_n}, \\
-    \mathbb{V}[\lambda | x^n] &= \frac{\alpha_n}{\beta_n^2}.
+    \mathbb{V}[\lambda | x^n] &= \frac{\alpha_n}{\beta_n^2},
 
 where the updating rule of the hyperparameters is
 
 .. math::
-    \alpha_n &= \alpha_0 + n\\
-    \beta_n &= \beta_0 + \sum_{i=1}^n x_i
+    \alpha_n &= \alpha_0 + n,\\
+    \beta_n &= \beta_0 + \sum_{i=1}^n x_i.
 
 
 The predictive distribution is as follows:
@@ -58,7 +58,7 @@
 * :math:`\eta_\mathrm{p} \in \mathbb{R}_{>0}`: the hyperparameter of the posterior
 
 .. math::
-    p(x_{n+1}|x^n)=\mathrm{Lomax}(x_{n+1}|\alpha_\mathrm{p},\eta_\mathrm{p}) = \frac{\alpha_\mathrm{p}}{\eta_\mathrm{p}}\left(1+\frac{x}{\eta_\mathrm{p}}\right)^{-(\alpha_\mathrm{p}+1)},
+    p(x_{n+1}|x^n)=\mathrm{Lomax}(x_{n+1}|\alpha_\mathrm{p},\eta_\mathrm{p}) = \frac{\alpha_\mathrm{p}}{\eta_\mathrm{p}}\left(1+\frac{x_{n+1}}{\eta_\mathrm{p}}\right)^{-(\alpha_\mathrm{p}+1)},
 
 .. math::
     \mathbb{E}[x_{n+1} | x^n] &=

bayesml/linearregression/__init__.py

@@ -10,7 +10,7 @@
 * :math:`d \in \mathbb N`: a dimension
 * :math:`\boldsymbol{x} = [x_1, x_2, \dots , x_d] \in \mathbb{R}^d`: an explanatory variable. If you consider an intercept term, it should be included as one of the elements of :math:`\boldsymbol{x}`.
 * :math:`y\in\mathbb{R}`: an objective variable
-* :math:`\tau \in\mathbb{R}`: a parameter
+* :math:`\tau \in\mathbb{R}_{>0}`: a parameter
 * :math:`\boldsymbol{\theta}\in\mathbb{R}^{d}`: a parameter
 
 .. math::
@@ -25,7 +25,7 @@
 The prior distribution is as follows:
 
 * :math:`\boldsymbol{\mu_0} \in \mathbb{R}^d`: a hyperparameter
-* :math:`\boldsymbol{\Lambda_0} \in \mathbb{R}^{d\times d}`: a hyperparameter
+* :math:`\boldsymbol{\Lambda_0} \in \mathbb{R}^{d\times d}`: a hyperparameter (a positive definite matrix)
 * :math:`\alpha_0\in \mathbb{R}_{>0}`: a hyperparameter
 * :math:`\beta_0\in \mathbb{R}_{>0}`: a hyperparameter
 
@@ -78,7 +78,7 @@
 
 .. math::
     p(y_{n+1} | \boldsymbol{X}, \boldsymbol{y}, \boldsymbol{x}_{n+1} ) &= \mathrm{St}\left(y_{n+1} \mid m_\mathrm{p}, \lambda_\mathrm{p}, \nu_\mathrm{p}\right) \\
-    &= \frac{\Gamma (\nu_\mathrm{p} / 2) + 1/2}{\Gamma (\nu_\mathrm{p} / 2)} \left( \frac{\lambda_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2} \left( 1 + \frac{\lambda_\mathrm{p} (y_{n+1} - m_\mathrm{p})^2}{\nu_\mathrm{p}} \right)^{-\nu_\mathrm{p}/2 - 1/2},
+    &= \frac{\Gamma (\nu_\mathrm{p} / 2 + 1/2 )}{\Gamma (\nu_\mathrm{p} / 2)} \left( \frac{\lambda_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2} \left( 1 + \frac{\lambda_\mathrm{p} (y_{n+1} - m_\mathrm{p})^2}{\nu_\mathrm{p}} \right)^{-\nu_\mathrm{p}/2 - 1/2},
 
 .. math::
     \mathbb{E}[y_{n+1} | \boldsymbol{X}, \boldsymbol{y}, \boldsymbol{x}_{n+1}] &= m_\mathrm{p} & (\nu_\mathrm{p} > 1), \\