\form#0:$f(x)$
\form#1:$x$
\form#2:$ f( x | y) $
\form#3:$ x $
\form#4:$ y $
\form#5:$ u_t $
\form#6:$ y_t $
\form#7:$ d_t=[y_t,u_t, \ldots ]$
\form#8:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \]
\form#9:$y_t$
\form#10:$ c_t $
\form#11:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \]
\form#12:$x=$
\form#13:$ f_x()$
\form#14:$ [x_1 , x_2 , \ldots \ $
\form#15:$ f_x(rv)$
\form#16:$x \sim epdf(rv|cond)$
\form#17:$[Up_{t-1},Up_{t-2}, \ldots]$
\form#18:$ t $
\form#19:$ t+1 $
\form#20:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $
\form#21:$ f(d_{t+1} |d_{t+h-1}, \ldots d_{t}) $
\form#22:$t$
\form#23:$[y_{t} y_{t-1} ...]$
\form#24:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$
\form#25:$ f(x_t|x_{t-1}) $
\form#26:$ f(d_t|x_t) $
\form#27:\[ L(y,u) = (y-y_{req})'Q_y (y-y_{req}) + (u-u_{req})' Q_u (u-u_{req}) \]
\form#28:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \]
\form#29:$[\theta r]$
\form#30:$\psi=\psi(y_{1:t},u_{1:t})$
\form#31:$u_t$
\form#32:$e_t$
\form#33:\[ e_t \sim \mathcal{N}(0,1). \]
\form#34:$\theta,r$
\form#35:$ dt = [y_t psi_t] $
\form#36:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \]
\form#37:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \]
\form#38:\[ x_{t+1} = Ax_t + B u_t + R^{1/2} e_t, y_t=Cx_t+Du_t + R^{1/2}w_t, \]
\form#39:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\]
\form#40:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]
\form#41:$\psi$
\form#42:$w=[w_1,\ldots,w_n]$
\form#43:$\theta_i$
\form#44:$\Theta$
\form#45:$\Theta = [\theta_1,\ldots,\theta_n,w]$
\form#46:$A=Ch' Ch$
\form#47:$Ch$
\form#48:$f(x) = a$
\form#49:$f(x) = Ax+B$
\form#50:$f(x,u)$
\form#51:$f(x,u) = Ax+Bu$
\form#52:$f(x0,u0)$
\form#53:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$
\form#54:$u$
\form#55:$A=\frac{d}{du}f(x,u)|_{x0,u0}$
\form#56:\[M = L'DL\]
\form#57:$L$
\form#58:$D$
\form#59:$V = V + w v v'$
\form#60:$C$
\form#61:$V = C*V*C'$
\form#62:$V = C'*V*C$
\form#63:$V$
\form#64:$x= v'*V*v$
\form#65:$x= v'*inv(V)*v$
\form#66:$U$
\form#67:$A'D0 A$
\form#68:$L'DL$
\form#69:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$
\form#70:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \]
\form#71:$ f(rvc) = \int f(rv,rvc) d\ rv $
\form#72:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \]
\form#73:$f_i(x)$
\form#74:$p$
\form#75:$p\times$
\form#76:$n$
\form#77:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \]
\form#78:$\gamma=\sum_i \beta_i$
\form#79:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \]
\form#80:$\beta$
\form#81:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \]
\form#82:$ \mu=A*\mbox{rvc}+\mu_0 $
\form#83:$\mu$
\form#84:$k$
\form#85:$\alpha=k$
\form#86:$\beta=k/\mu$
\form#87:$\mu/\sqrt(k)$
\form#88:$ \mu $
\form#89:$ k $
\form#90:$ \alpha=\mu/k^2+2 $
\form#91:$ \beta=\mu(\alpha-1)$
\form#92:$ \mu/\sqrt(k)$
\form#93:$l$
\form#94:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\]
\form#95:$ \log(x)\sim \mathcal{N}(\mu,\sigma^2) $
\form#96:\[ x \sim \frac{1}{x\sigma\sqrt{2\pi}}\exp{-\frac{1}{2\sigma^2}(\log(x)-\mu)} \]
\form#97:$\mathcal{I}$
\form#98:$\theta$
\form#99:$\alpha$
\form#100:$ \Lambda $
\form#101:$ R $
\form#102:$ R_e $
\form#103:$ \Psi $
\form#104:$ \nu $
\form#105:$ \nu-p-1 $
\form#106:$w$
\form#107:$x^{(i)}, i=1..n$
\form#108:\[ f(x_i|y_i), i=1..n \]
\form#109:$ \cup [x_i,y_i] $
\form#110:\[ f(z_i|y_i,x_i) f(x_i|y_i) f(y_i) i=1..n \]
\form#111:$ z_i $
\form#112:$ y_i={}, z_i={}, \forall i $
\form#113:$ f(z_i|x_i,y_i) $
\form#114:$ f(D) $
\form#115:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]
\form#116:$ f(a|b,c) $
\form#117:$ f(b) $
\form#118:$ f(c) $
\form#119:\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#120:$ x_t $
\form#121:$ A, B, C, D$
\form#122:$v_t, w_t$
\form#123:$Q, R$
\form#124:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray}
\form#125:$ g(), h() $
\form#126:\[ y_t = \theta' \psi_t + \rho e_t \]
\form#127:$[\theta,\rho]$
\form#128:$\psi_t$
\form#129:$\mathcal{N}(0,1)$
\form#130:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \]
\form#131:\[ \nu_t = \sum_{i=0}^{n} 1 \]
\form#132:$ \theta_t , r_t $
\form#133:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \]
\form#134:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \]
\form#135:$ \phi $
\form#136:$ \phi \in [0,1]$
\form#137:\[ \mathrm{win_length} = \frac{1}{1-\phi}\]
\form#138:$ \phi=0.9 $
\form#139:$ V_0 , \nu_0 $
\form#140:$ V_t , \nu_t $
\form#141:$ \phi<1 $
\form#142:$ f(a)$
\form#143:$ a $
\form#144:$ f(a) $
\form#145:$ f(x_t |d_1 \ldots d_t)$
\form#146:$ d $
\form#147:\[ y_t \sim \mathcal{N}( a y_{t-3} + b u_{t-1}, r) \]
\form#148:$ a,b $
\form#149:$ r $
\form#150:$ y_{t-3}$
\form#151:$ u_{t-1}$
\form#152:$ u $
\form#153:$ f(y_{t}|y_{t-3},u_{t-1})$
\form#154:\[ u_t \sim \mathcal{N}(0, r_u) \]
\form#155:\[ f(y_{t},u_{t}|y_{t-3},u_{t-1}) = f(y_{t}|y_{t-3},u_{t-1})f(u_{t}) \]
\form#156:$ f(a|b)$
\form#157:$ f(u_t)$
\form#158:$ f(u_t| \{\})$
\form#159:$ _t $
\form#160:\[ f(a) = \mathcal{U}(-1,1) \]
\form#161:\[ f(y_t|y_{t-3},u_{t-1}) = \mathcal{N}( a y_{t-3} + b u_{t-1}, r) \]
\form#162:\[ f(u_t) = \mathcal{N}(0, r_u) \]
\form#163:$ r_u $
\form#164:\[ f(rv) = N(\mu, R) \]
\form#165:\[ f(rv) = GiW(V,\nu) \]
\form#166:\[ f(rv|rvc) = \Gamma(\alpha, \beta) \]
\form#167:\[ f(rv) = U(low,high) \]
\form#168:\[ f(rv|rvc) = N(A*rvc+const, R) \]
\form#169:\[ f(rv|rvc) = N( g(rvc), R) \]
\form#170:\[ f(rv|rvc) = \Gamma(k, k/rvc) \]
\form#171:\[ f(rv|rvc) = i\Gamma(k, k/(rvc^l \circ ref^{(1-l)}) \]
\form#172:\[ f(rv|rvc) = log\mathcal{N}( \log(rvc)-0.5\log(k^2+1), k I) \]
\form#173:$ [d_1, d_2, \ldots d_t] $
\form#174:\[ f(x_t|d_1\ldots d_t) \propto f(d_t|x_t,d_1\ldots d_{t-1}) f(x_t| d_1\ldots d_{t-1}) \]
\form#175:$ d_t $
\form#176:$ f(d_t|d_1\ldots d_{t-1})$
\form#177:\[ f(d_{t+1}| d_1 \ldots d_{t} \]
\form#178:$ x_{t-2} $
\form#179:$ [a_{t-1}', b_{t+1}] $
\form#180:\[ f(d_{t+1}| d_1 \ldots d_{t}), \]
\form#181:\[ f(x_t|d_1\ldots d_t)=f(x_{1,t}|x_{2,t},d_1\ldots d_t)f(x_{2,t}|d_1\ldots d_t) \]
\form#182:$ x_{1,t}$
\form#183:$x_{2,t}$
\form#184:$ [\theta_t, r_t, \phi_t]$
\form#185:\[ f(\theta_t, r_t, \phi_t) = f(\theta_t, r_t| \phi_t) f(\phi_t) \]
\form#186:$ \phi_t $
\form#187:\[ f(\theta| d_1 \ldots d_t , \phi_t) \]
\form#188:\[ \beta = rvc / k + \beta_c \]
\form#189:$ \beta_c $
\form#190:\[ f(rv|rvc) = Di(rvc*k) \]
\form#191:$ x_{2,t}$
\form#192:$[\phi, 1-\phi]$
\form#193:\[ f(\phi_t|\phi_{t-1}) = Di (\phi_{t-1}/k + \beta_c) \]