\form#0:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \]
\form#1:$[\theta r]$
\form#2:$\psi=\psi(y_{1:t},u_{1:t})$
\form#3:$u_t$
\form#4:$e_t$
\form#5:\[ e_t \sim \mathcal{N}(0,1). \]
\form#6:$ y_t $
\form#7:$\theta,r$
\form#8:$ dt = [y_t psi_t] $
\form#9:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \]
\form#10:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \]
\form#11:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\]
\form#12:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]
\form#13:$\psi$
\form#14:$w=[w_1,\ldots,w_n]$
\form#15:$\theta_i$
\form#16:$\Theta$
\form#17:$\Theta = [\theta_1,\ldots,\theta_n,w]$
\form#18:$A=Ch' Ch$
\form#19:$Ch$
\form#20:\[M = L'DL\]
\form#21:$L$
\form#22:$D$
\form#23:$V = V + w v v'$
\form#24:$C$
\form#25:$V = C*V*C'$
\form#26:$V = C'*V*C$
\form#27:$V$
\form#28:$x$
\form#29:$x= v'*V*v$
\form#30:$x= v'*inv(V)*v$
\form#31:$U$
\form#32:$A'D0 A$
\form#33:$L'DL$
\form#34:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$
\form#35:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \]
\form#36:$ f(rvc) = \int f(rv,rvc) d\ rv $
\form#37:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \]
\form#38:$f_i(x)$
\form#39:$f(x)$
\form#40:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \]
\form#41:$y_t$
\form#42:$ c_t $
\form#43:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \]
\form#44:$x=$
\form#45:$ x $
\form#46:$ f_x()$
\form#47:$ [x_1 , x_2 , \ldots \ $
\form#48:$ f_x(rv)$
\form#49:$x \sim epdf(rv|cond)$
\form#50:$ t $
\form#51:$ t+1 $
\form#52:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $
\form#53:$t$
\form#54:$[y_{t} y_{t-1} ...]$
\form#55:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$
\form#56:$ f(x_t|x_{t-1}) $
\form#57:$ f(d_t|x_t) $
\form#58:$p$
\form#59:$p\times$
\form#60:$n$
\form#61:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \]
\form#62:$\gamma=\sum_i \beta_i$
\form#63:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \]
\form#64:$\beta$
\form#65:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \]
\form#66:$mu=A*rvc+mu_0$
\form#67:$\mu$
\form#68:$k$
\form#69:$\alpha=k$
\form#70:$\beta=k/\mu$
\form#71:$\mu/\sqrt(k)$
\form#72:$ \mu $
\form#73:$ k $
\form#74:$ \alpha=\mu/k^2+2 $
\form#75:$ \beta=\mu(\alpha-1)$
\form#76:$ \mu/\sqrt(k)$
\form#77:$l$
\form#78:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\]
\form#79:$\mathcal{I}$
\form#80:$\theta$
\form#81:$\alpha$
\form#82:$ \Psi $
\form#83:$ \nu $
\form#84:$ \nu-p-1 $
\form#85:$w$
\form#86:$x^{(i)}, i=1..n$
\form#87:$f(x) = a$
\form#88:$f(x) = Ax+B$
\form#89:$f(x,u)$
\form#90:$f(x,u) = Ax+Bu$
\form#91:$f(x0,u0)$
\form#92:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$
\form#93:$u$
\form#94:$A=\frac{d}{du}f(x,u)|_{x0,u0}$
\form#95:$ f(D) $
\form#96:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]
\form#97:$ f(a|b,c) $
\form#98:$ f(b) $
\form#99:$ f(c) $
\form#100:\begin{eqnarray} x_t &= &A x_{t-1} + B u_{t} + v_t,\\ y_t &= &C x_{t} + D u_{t} + w_t, \end{eqnarray}
\form#101:$ x_t $
\form#102:$ A, B, C, D$
\form#103:$v_t, w_t$
\form#104:$Q, R$
\form#105:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray}
\form#106:$ g(), h() $
\form#107:\[ y_t = \theta' \psi_t + \rho e_t \]
\form#108:$[\theta,\rho]$
\form#109:$\psi_t$
\form#110:$\mathcal{N}(0,1)$
\form#111:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \]
\form#112:\[ \nu_t = \sum_{i=0}^{n} 1 \]
\form#113:$ \theta_t , r_t $
\form#114:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \]
\form#115:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \]
\form#116:$ \phi $
\form#117:$ \phi \in [0,1]$
\form#118:\[ \mathrm{win_length} = \frac{1}{1-\phi}\]
\form#119:$ \phi=0.9 $
\form#120:$ V_0 , \nu_0 $
\form#121:$ V_t , \nu_t $
\form#122:$ \phi<1 $
\form#123:$ [d_1, d_2, \ldots d_t] $
\form#124:\[ f(x_i|y_i), i=1..n \]
\form#125:$ \cup [x_i,y_i] $
\form#126:\[ f(z_i|y_i,x_i) f(x_i|y_i) f(y_i) i=1..n \]
\form#127:$ z_i $
\form#128:$ y_i={}, z_i={}, \forall i $
\form#129:$ f(z_i|x_i,y_i) $
\form#130:$ f(d_{t+1} |d_{t+h-1}, \ldots d_{t}) $
\form#131:$ \log(x)\sim \mathcal{N}(\mu,\sigma^2) $
\form#132:\[ x \sim \frac{1}{x\sigma\sqrt{2\pi}}\exp{-\frac{1}{2\sigma^2}(\log(x)-\mu)} \]