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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), \\

bayesml/multivariate_normal/__init__.py

@@ -12,7 +12,7 @@
 * :math:`\boldsymbol{x} \in \mathbb{R}^D`: a data point
 * :math:`\boldsymbol{\mu} \in \mathbb{R}^D`: a parameter
 * :math:`\boldsymbol{\Lambda} \in \mathbb{R}^{D\times D}` : a parameter (a positive definite matrix)
-* :math:`| \boldsymbol{\Lambda} | \in \mathbb{R}`: the determinant of :math:`\boldsymbol{\Lambda}_0`
+* :math:`| \boldsymbol{\Lambda} | \in \mathbb{R}`: the determinant of :math:`\boldsymbol{\Lambda}`
 
 .. math::
     p(\boldsymbol{x} | \boldsymbol{\mu}, \boldsymbol{\Lambda}) &= \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu},\boldsymbol{\Lambda}^{-1}) \\
@@ -51,7 +51,7 @@
 * :math:`\boldsymbol{x}^n = (\boldsymbol{x}_1, \boldsymbol{x}_2, \dots , \boldsymbol{x}_n) \in \mathbb{R}^{D\times n}`: given data
 * :math:`\boldsymbol{m}_n \in \mathbb{R}^{D}`: a hyperparameter
 * :math:`\kappa_n \in \mathbb{R}_{>0}`: a hyperparameter
-* :math:`\nu_n \in \mathbb{R}`: a hyperparameter :math:`(\nu_0 > D-1)`
+* :math:`\nu_n \in \mathbb{R}`: a hyperparameter :math:`(\nu_n > D-1)`
 * :math:`\boldsymbol{W}_n \in \mathbb{R}^{D\times D}`: a hyperparameter (a positive definite matrix)
 
 .. math::

bayesml/normal/__init__.py

@@ -57,7 +57,7 @@
     \mathbb{E}[\mu | x^n] &= m_n & \left( \alpha_n > \frac{1}{2} \right), \\
     \mathbb{V}[\mu | x^n] &= \frac{\beta_n \alpha_n}{\alpha_n (\alpha_n - 1)} & (\alpha_n > 1), \\
     \mathbb{E}[\tau | x^n] &= \frac{\alpha_n}{\beta_n}, \\
-    \mathbb{V}[\tau | x^n] &= \frac{\alpha_n}{\beta_n^2}.
+    \mathbb{V}[\tau | x^n] &= \frac{\alpha_n}{\beta_n^2},
 
 where the updating rule of the hyperparameters is
 
@@ -66,7 +66,7 @@
     m_n &= \frac{\kappa_0 m_0 + n \bar{x}}{\kappa_0 + n}, \\
     \kappa_n &= \kappa_0 + n, \\
     \alpha_n &= \alpha_0 + \frac{n}{2}, \\
-    \beta_n &=  \beta_0 + \frac{1}{2} \left( \sum_{i=0}^n (x_i - \bar{x})^2 + \frac{\kappa_0 n}{\kappa_n + n} (\bar{x} - m_0)^2 \right).
+    \beta_n &=  \beta_0 + \frac{1}{2} \left( \sum_{i=1}^n (x_i - \bar{x})^2 + \frac{\kappa_0 n}{\kappa_n + n} (\bar{x} - m_0)^2 \right).
 
 The predictive distribution is as follows:
 
@@ -77,7 +77,7 @@
 
 .. math::
     p(x_{n+1} | x^{n} ) &= \mathrm{St}(x_{n+1} | \mu_\mathrm{p}, \lambda_\mathrm{p}, \nu_\mathrm{p}) \\
-    &= \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} (x_{n+1} - \mu_\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} (x_{n+1} - \mu_\mathrm{p})^2}{\nu_\mathrm{p}} \right)^{-\nu_\mathrm{p}/2 - 1/2},
 
 .. math::
     \mathbb{E}[x_{n+1} | x^n] &= \mu_\mathrm{p} & (\nu_\mathrm{p} > 1), \\

bayesml/poisson/__init__.py

@@ -12,7 +12,7 @@
 * :math:`\lambda \in \mathbb{R}_{>0}`: a parameter
 
 .. math::
-    p(x | \lambda) = \mathrm{Po}(x|\lambda) = \frac{ \lambda^{x} }{x!}\exp \{ -\lambda \}
+    p(x | \lambda) = \mathrm{Po}(x|\lambda) = \frac{ \lambda^{x} }{x!}\exp \{ -\lambda \}.
 
 The prior distribution is as follows:
 
@@ -38,13 +38,13 @@
 
 .. 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 + \sum_{i=1}^n x_i\\
-    \beta_n &= \beta_0 + n
+    \alpha_n &= \alpha_0 + \sum_{i=1}^n x_i,\\
+    \beta_n &= \beta_0 + n.
 
 The predictive distribution is as follows:
 

docs/bayesml.autoregressive.html

@@ -336,14 +336,14 @@ <h1>bayesml.autoregressive package<a class="headerlink" href="#bayesml-autoregre
 <div class="math notranslate nohighlight">
 \[\mathrm{St}(x_{n+1}|m_\mathrm{p}, \lambda_\mathrm{p}, \nu_\mathrm{p})
 = \frac{\Gamma (\nu_\mathrm{p}/2 + 1/2)}{\Gamma (\nu_\mathrm{p}/2)}
-\left( \frac{m_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2}
+\left( \frac{\lambda_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2}
 \left[ 1 + \frac{\lambda_\mathrm{p}(x_{n+1}-m_\mathrm{p})^2}{\nu_\mathrm{p}} \right]^{-\nu_\mathrm{p}/2 - 1/2}.\]</div>
 <div class="math notranslate nohighlight">
 \[\begin{split}\mathbb{E}[x_{n+1} | x^n] &amp;= m_\mathrm{p} &amp; (\nu_\mathrm{p} &gt; 1), \\
 \mathbb{V}[x_{n+1} | x^n] &amp;= \frac{1}{\lambda_\mathrm{p}} \frac{\nu_\mathrm{p}}{\nu_\mathrm{p}-2} &amp; (\nu_\mathrm{p} &gt; 2),\end{split}\]</div>
 <p>where the parameters are obtained from the hyperparameters of the posterior distribution as follows.</p>
 <div class="math notranslate nohighlight">
-\[\begin{split}m_\mathrm{p} &amp;= \mu_n^\top \boldsymbol{x}'_n,\\
+\[\begin{split}m_\mathrm{p} &amp;= \boldsymbol{\mu}_n^\top \boldsymbol{x}'_n,\\
 \lambda_\mathrm{p} &amp;= \frac{\alpha_n}{\beta_n} (1 + (\boldsymbol{x}'_n)^\top \boldsymbol{\Lambda}_n^{-1} \boldsymbol{x}'_n)^{-1},\\
 \nu_\mathrm{p} &amp;= 2 \alpha_n.\end{split}\]</div>
 <dl class="py class">

docs/bayesml.bernoulli.html

@@ -281,7 +281,7 @@ <h1>bayesml.bernoulli package<a class="headerlink" href="#bayesml-bernoulli-pack
 <li><p><span class="math notranslate nohighlight">\(B(\cdot,\cdot): \mathbb{R}_{&gt;0} \times \mathbb{R}_{&gt;0} \to \mathbb{R}_{&gt;0}\)</span>: the Beta function</p></li>
 </ul>
 <div class="math notranslate nohighlight">
-\[p(\theta) = \mathrm{Beta}(\theta|\alpha_0,\beta_0) = \frac{1}{B(\alpha_0, \beta_0)} \theta^{\alpha_0} (1-\theta)^{\beta_0}.\]</div>
+\[p(\theta) = \mathrm{Beta}(\theta|\alpha_0,\beta_0) = \frac{1}{B(\alpha_0, \beta_0)} \theta^{\alpha_0 - 1} (1-\theta)^{\beta_0 - 1}.\]</div>
 <div class="math notranslate nohighlight">
 \[\begin{split}\mathbb{E}[\theta] &amp;= \frac{\alpha_0}{\alpha_0 + \beta_0}, \\
 \mathbb{V}[\theta] &amp;= \frac{\alpha_0 \beta_0}{(\alpha_0 + \beta_0)^2 (\alpha_0 + \beta_0 + 1)}.\end{split}\]</div>
@@ -292,10 +292,10 @@ <h1>bayesml.bernoulli package<a class="headerlink" href="#bayesml-bernoulli-pack
 <li><p><span class="math notranslate nohighlight">\(\beta_n \in \mathbb{R}_{&gt;0}\)</span>: a hyperparameter</p></li>
 </ul>
 <div class="math notranslate nohighlight">
-\[p(\theta | x^n) = \mathrm{Beta}(\theta|\alpha_n,\beta_n) = \frac{1}{B(\alpha_n, \beta_n)} \theta^{\alpha_n} (1-\theta)^{\beta_n},\]</div>
+\[p(\theta | x^n) = \mathrm{Beta}(\theta|\alpha_n,\beta_n) = \frac{1}{B(\alpha_n, \beta_n)} \theta^{\alpha_n - 1} (1-\theta)^{\beta_n - 1},\]</div>
 <div class="math notranslate nohighlight">
 \[\begin{split}\mathbb{E}[\theta | x^n] &amp;= \frac{\alpha_n}{\alpha_n + \beta_n}, \\
-\mathbb{V}[\theta | x^n] &amp;= \frac{\alpha_n \beta_n}{(\alpha_n + \beta_n)^2 (\alpha_n + \beta_n + 1)}.\end{split}\]</div>
+\mathbb{V}[\theta | x^n] &amp;= \frac{\alpha_n \beta_n}{(\alpha_n + \beta_n)^2 (\alpha_n + \beta_n + 1)},\end{split}\]</div>
 <p>where the updating rule of the hyperparameters is</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}\alpha_n = \alpha_0 + \sum_{i=1}^n I \{ x_i = 1 \},\\
@@ -308,13 +308,13 @@ <h1>bayesml.bernoulli package<a class="headerlink" href="#bayesml-bernoulli-pack
 <li><p><span class="math notranslate nohighlight">\(\theta_\mathrm{p} \in [0,1]\)</span>: a parameter</p></li>
 </ul>
 <div class="math notranslate nohighlight">
-\[p(x_{n+1} | x^n) = \mathrm{Bern}(x_{n+1}|\theta_\mathrm{p}) =\theta_\mathrm{p}^{x_{n+1}}(1-\theta_\mathrm{p})^{1-x_{n+1}}\]</div>
+\[p(x_{n+1} | x^n) = \mathrm{Bern}(x_{n+1}|\theta_\mathrm{p}) =\theta_\mathrm{p}^{x_{n+1}}(1-\theta_\mathrm{p})^{1-x_{n+1}},\]</div>
 <div class="math notranslate nohighlight">
 \[\begin{split}\mathbb{E}[x_{n+1} | x^n] &amp;= \theta_\mathrm{p}, \\
-\mathbb{V}[x_{n+1} | x^n] &amp;= \theta_\mathrm{p} (1 - \theta_\mathrm{p}).\end{split}\]</div>
+\mathbb{V}[x_{n+1} | x^n] &amp;= \theta_\mathrm{p} (1 - \theta_\mathrm{p}),\end{split}\]</div>
 <p>where the parameters are obtained from the hyperparameters of the posterior distribution as follows.</p>
 <div class="math notranslate nohighlight">
-\[\theta_\mathrm{p} = \frac{\alpha_n}{\alpha_n + \beta_n}\]</div>
+\[\theta_\mathrm{p} = \frac{\alpha_n}{\alpha_n + \beta_n}.\]</div>
 <dl class="py class">
 <dt class="sig sig-object py" id="bayesml.bernoulli.GenModel">
 <em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">bayesml.bernoulli.</span></span><span class="sig-name descname"><span class="pre">GenModel</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">theta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">h_alpha</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">h_beta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#bayesml.bernoulli.GenModel" title="Permalink to this definition">¶</a></dt>

docs/bayesml.categorical.html

@@ -278,7 +278,7 @@ <h1>bayesml.categorical package<a class="headerlink" href="#bayesml-categorical-
 \mathrm{Cov}[x_k, x_{k'} | \boldsymbol{\theta}] &amp;= -\theta_k \theta_{k'}.\end{split}\]</div>
 <p>The prior distribution is as follows:</p>
 <ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(\boldsymbol{\alpha}_0 \in \mathbb{R}_{&gt;0}\)</span>: a hyperparameter</p></li>
+<li><p><span class="math notranslate nohighlight">\(\boldsymbol{\alpha}_0 \in \mathbb{R}_{&gt;0}^d\)</span>: a hyperparameter</p></li>
 <li><p><span class="math notranslate nohighlight">\(\Gamma (\cdot)\)</span>: the gamma function</p></li>
 <li><p><span class="math notranslate nohighlight">\(\tilde{\alpha}_0 = \sum_{k=1}^d \alpha_{0,k}\)</span></p></li>
 <li><p><span class="math notranslate nohighlight">\(C(\boldsymbol{\alpha}_0)=\frac{\Gamma(\tilde{\alpha}_0)}{\Gamma(\alpha_{0,1})\cdots\Gamma(\alpha_{0,d})}\)</span></p></li>
@@ -307,7 +307,7 @@ <h1>bayesml.categorical package<a class="headerlink" href="#bayesml-categorical-
 \[\alpha_{n,k} = \alpha_{0,k} + \sum_{i=1}^n x_{i,k}, \quad (k \in \{ 1, 2, \dots , d \}).\]</div>
 <p>The predictive distribution is as follows:</p>
 <ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(x_{n+1} \in \{ 0, 1\}^d\)</span>: a new data point</p></li>
+<li><p><span class="math notranslate nohighlight">\(\boldsymbol{x}_{n+1} \in \{ 0, 1\}^d\)</span>: a new data point</p></li>
 <li><p><span class="math notranslate nohighlight">\(\boldsymbol{\theta}_\mathrm{p} \in [0, 1]^d\)</span>: the hyperparameter of the posterior (<span class="math notranslate nohighlight">\(\sum_{k=1}^d \theta_{\mathrm{p},k} = 1\)</span>)</p></li>
 </ul>
 <div class="math notranslate nohighlight">
@@ -318,7 +318,7 @@ <h1>bayesml.categorical package<a class="headerlink" href="#bayesml-categorical-
 \mathrm{Cov}[x_{n+1,k}, x_{n+1,k'} | \boldsymbol{x}^n] &amp;= -\theta_{\mathrm{p},k} \theta_{\mathrm{p},k'},\end{split}\]</div>
 <p>where the parameters are obtained from the hyperparameters of the posterior distribution as follows:</p>
 <div class="math notranslate nohighlight">
-\[\boldsymbol{\theta}_{\mathrm{p},k} = \frac{\alpha_{n,k}}{\sum_{k=1}^d \alpha_{n,k}}, \quad (k \in \{ 1, 2, \dots , d \}).\]</div>
+\[\theta_{\mathrm{p},k} = \frac{\alpha_{n,k}}{\sum_{k=1}^d \alpha_{n,k}}, \quad (k \in \{ 1, 2, \dots , d \}).\]</div>
 <dl class="py class">
 <dt class="sig sig-object py" id="bayesml.categorical.GenModel">
 <em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">bayesml.categorical.</span></span><span class="sig-name descname"><span class="pre">GenModel</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">degree</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">theta_vec</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">h_alpha_vec</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#bayesml.categorical.GenModel" title="Permalink to this definition">¶</a></dt>