Unconditional Density Estimation ¶ Data ¶ D = { y n } , y n ∈ R D y \begin{aligned}
\mathcal{D} &=\left\{y_n \right\}, && &&
y_n\in\mathbb{R}^{D_y}
\end{aligned} D = { y n } , y n ∈ R D y Parametric ¶ Joint Distribution : p ( y , θ ) = p ( y ∣ θ ) p ( θ ) \begin{aligned}
\text{Joint Distribution}: && &&
p(y,\theta) &= p(y|\theta)p(\theta)
\end{aligned} Joint Distribution : p ( y , θ ) = p ( y ∣ θ ) p ( θ ) Generative ¶ Joint Distribution : p ( y , z , θ ) = p ( y ∣ z , θ ) p ( z ∣ θ ) p ( θ ) \begin{aligned}
\text{Joint Distribution}: && &&
p(y,z,\theta) &= p(y|z,\theta)p(z|\theta)p(\theta)
\end{aligned} Joint Distribution : p ( y , z , θ ) = p ( y ∣ z , θ ) p ( z ∣ θ ) p ( θ ) Conditional Density Estimation ¶ Data ¶ D = { y n , x n } , y n ∈ R D y x n ∈ R D x \begin{aligned}
\mathcal{D} &=\left\{y_n, x_n \right\}, && &&
y_n\in\mathbb{R}^{D_y} &&
x_n\in\mathbb{R}^{D_x}
\end{aligned} D = { y n , x n } , y n ∈ R D y x n ∈ R D x Parametric ¶ Joint Distribution : p ( y , x , θ ) = p ( y ∣ x , θ ) p ( θ ) \begin{aligned}
\text{Joint Distribution}: && &&
p(y,x,\theta) &= p(y|x,\theta)p(\theta)
\end{aligned} Joint Distribution : p ( y , x , θ ) = p ( y ∣ x , θ ) p ( θ ) Generative ¶ Joint Distribution I : p ( y , x , z , θ ) = p ( y ∣ z , x , θ ) p ( z ∣ θ ) p ( θ ) Joint Distribution II : p ( y , x , z , θ ) = p ( y ∣ z , θ ) p ( z ∣ x , θ ) p ( θ ) Joint Distribution III : p ( y , x , z , θ ) = p ( y ∣ x , z , θ ) p ( z ∣ x , θ ) p ( θ ) \begin{aligned}
\text{Joint Distribution I}: && &&
p(y,x,z,\theta) &= p(y|z,x,\theta)p(z|\theta)p(\theta) \\
\text{Joint Distribution II}: && &&
p(y,x,z,\theta) &= p(y|z,\theta)p(z|x,\theta)p(\theta) \\
\text{Joint Distribution III}: && &&
p(y,x,z,\theta) &= p(y|x,z,\theta)p(z|x,\theta)p(\theta) \\
\end{aligned} Joint Distribution I : Joint Distribution II : Joint Distribution III : p ( y , x , z , θ ) p ( y , x , z , θ ) p ( y , x , z , θ ) = p ( y ∣ z , x , θ ) p ( z ∣ θ ) p ( θ ) = p ( y ∣ z , θ ) p ( z ∣ x , θ ) p ( θ ) = p ( y ∣ x , z , θ ) p ( z ∣ x , θ ) p ( θ ) Dynamical Models ¶ Observations ¶ D = { y t } t = 1 T , y t ∈ R D y \begin{aligned}
\mathcal{D} &=\left\{y_t \right\}_{t=1}^T, && &&
y_t\in\mathbb{R}^{D_y}
\end{aligned} D = { y t } t = 1 T , y t ∈ R D y Parametric (Global, IID ) ¶ p ( y 1 : T , θ ) = p ( θ ) ∏ t = 1 T p ( y t ∣ θ ) \begin{aligned}
p(y_{1:T},\theta) &= p(\theta)\prod_{t=1}^T p(y_t|\theta)
\end{aligned} p ( y 1 : T , θ ) = p ( θ ) t = 1 ∏ T p ( y t ∣ θ ) Parametric (Local) ¶ p ( y 1 : T , θ 0 : T ) = p ( θ 0 ) ∏ t = 1 T p ( y t ∣ θ t ) \begin{aligned}
p(y_{1:T},\theta_{0:T}) &= p(\theta_0)\prod_{t=1}^T p(y_t|\theta_t)
\end{aligned} p ( y 1 : T , θ 0 : T ) = p ( θ 0 ) t = 1 ∏ T p ( y t ∣ θ t ) Generative ¶ p ( y 1 : T , z 1 : T , θ ) = p ( θ ) p ( z 0 ) ∏ t = 1 T p ( y t ∣ z t , θ ) p ( z t ∣ z t − 1 , θ ) \begin{aligned}
p(y_{1:T},z_{1:T},\theta) &= p(\theta)p(z_0)\prod_{t=1}^T p(y_t|z_t,\theta)p(z_t|z_{t-1},\theta)
\end{aligned} p ( y 1 : T , z 1 : T , θ ) = p ( θ ) p ( z 0 ) t = 1 ∏ T p ( y t ∣ z t , θ ) p ( z t ∣ z t − 1 , θ ) Conditional Generative ¶ p ( y 1 : T , x 1 : T , z 1 : T , θ ) = p ( θ ) p ( z 0 ) ∏ t = 1 T p ( y t ∣ z t , θ ) p ( z t ∣ z t − 1 , x t , θ ) \begin{aligned}
p(y_{1:T},x_{1:T},z_{1:T},\theta) &= p(\theta)p(z_0)\prod_{t=1}^T p(y_t|z_t,\theta)p(z_t|z_{t-1}, x_{t},\theta)
\end{aligned} p ( y 1 : T , x 1 : T , z 1 : T , θ ) = p ( θ ) p ( z 0 ) t = 1 ∏ T p ( y t ∣ z t , θ ) p ( z t ∣ z t − 1 , x t , θ ) Dynamical ¶ p ( y 1 : T , u 0 : T , θ ) = p ( θ ) p ( u 0 ) ∏ t = 1 T p ( y t ∣ u t , θ ) p ( u t ∣ u t − 1 , θ ) \begin{aligned}
p(y_{1:T},u_{0:T},\theta) &= p(\theta)p(u_0)\prod_{t=1}^T p(y_t|u_t,\theta)p(u_t|u_{t-1},\theta)
\end{aligned} p ( y 1 : T , u 0 : T , θ ) = p ( θ ) p ( u 0 ) t = 1 ∏ T p ( y t ∣ u t , θ ) p ( u t ∣ u t − 1 , θ ) Conditional Dynamical ¶ p ( y 1 : T , x 1 : T , u 0 : T , θ ) = p ( θ ) p ( u 0 ) ∏ t = 1 T p ( y t ∣ u t , x t , θ ) p ( u t ∣ u t − 1 , x t , θ ) \begin{aligned}
p(y_{1:T},x_{1:T}, u_{0:T},\theta) &= p(\theta)p(u_0)\prod_{t=1}^T p(y_t|u_t,x_t,\theta)p(u_t|u_{t-1}, x_{t},\theta)
\end{aligned} p ( y 1 : T , x 1 : T , u 0 : T , θ ) = p ( θ ) p ( u 0 ) t = 1 ∏ T p ( y t ∣ u t , x t , θ ) p ( u t ∣ u t − 1 , x t , θ ) Generative Dynamical ¶ p ( y 1 : T , u 1 : T , z 0 : T , θ ) = p ( θ ) p ( z 0 ) ∏ t = 1 T p ( y t ∣ u t , θ ) p ( u t ∣ z t , θ ) p ( z t ∣ z t − 1 , θ ) \begin{aligned}
p(y_{1:T},u_{1:T}, z_{0:T},\theta) &= p(\theta)p(z_0)\prod_{t=1}^T p(y_t|u_t,\theta)p(u_t|z_{t}, \theta)p(z_t|z_{t-1},\theta)
\end{aligned} p ( y 1 : T , u 1 : T , z 0 : T , θ ) = p ( θ ) p ( z 0 ) t = 1 ∏ T p ( y t ∣ u t , θ ) p ( u t ∣ z t , θ ) p ( z t ∣ z t − 1 , θ ) Conditional Generative Dynamical ¶ p ( y 1 : T , u 1 : T , x 1 : T , z 0 : T , θ ) = p ( θ ) p ( z 0 ) ∏ t = 1 T p ( y t ∣ u t , x t , θ ) p ( u t ∣ z t , x t , θ ) p ( z t ∣ z t − 1 , θ ) \begin{aligned}
p(y_{1:T},u_{1:T}, x_{1:T},z_{0:T},\theta) &= p(\theta)p(z_0)\prod_{t=1}^T p(y_t|u_t,x_t,\theta)p(u_t|z_{t}, x_{t},\theta)p(z_t|z_{t-1},\theta)
\end{aligned} p ( y 1 : T , u 1 : T , x 1 : T , z 0 : T , θ ) = p ( θ ) p ( z 0 ) t = 1 ∏ T p ( y t ∣ u t , x t , θ ) p ( u t ∣ z t , x t , θ ) p ( z t ∣ z t − 1 , θ )