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## 因子图引入

$$P(X, Z) = P(x_0)\prod_{i=1}P(x_i|x_{i-1}, u)\prod_{k=1}P(z_k|x_{i_k},l_{j_k})$$


\begin{aligned} x_i &= f_i(x_{i-1}, u_i) + w_i\\ z_k &= h_k(x_{i_k}, l_{j_k}) + v_k \end{aligned}


\begin{aligned} P(x_i | x_{i-1}, u_i) &\propto \exp{-\frac{1}{2}||f_i(x_{i-1}, u_i) - x_i||_{\Delta i}^2}\\ P(z_i | x_{i_k}, l_{j_k}) &\propto \exp{-\frac{1}{2}||h_k(x_{i_k}, l_{j_k}) - z_k||_{\Delta k}^2} \end{aligned}


\begin{aligned} P(\Theta) &\propto \prod_i\phi_i(\theta_i)\prod_{\{i, j\}, i <ｊ}\phi_{ij}(\theta_i, \theta_j)\\ \phi_0(x_o) &\propto P(x_0)\\ \phi_j(l_j) &\propto P(l_j)\\ \phi_{(i-1),i}(x_{i-1}, x_i) &\propto P(x_i|x_{i-1}, u_i)\\ \phi_{i_kj_k}(x_{i_k}, l_{j_k}) &\propto P(z_k|x_{i_k}, l_{j_k}) \end{aligned}


\begin{aligned} \Theta^* &= \arg\max_{\Theta}f(\Theta)\\ f(\Theta) &= \prod_if_i(\theta)\\ f_i(\theta) &\propto\exp{\left(-\frac{1}{2}||h_i(\Theta_i) - z_i||_{\Sigma i}^2\right)} \end{aligned}


\begin{aligned} \Theta^* &= \arg\max_{\Theta}f(\Theta)\\ &= \arg\max_{\Theta}\left(\prod_if_i(\theta)\right)\\ &= \arg\max_{\Theta}\left(\prod_i\exp{\left(-\frac{1}{2}||h_i(\Theta_i) - z_i||_{\Sigma i}^2\right)}\right)\\ &= \arg\max_{\Theta}\left(\sum_i\left(-\frac{1}{2}||h_i(\Theta_i) - z_i||_{\Sigma i}^2\right)\right)\\ &= \arg\min_{\Theta}\left(\frac{1}{2}\sum_i||h_i(\Theta_i) - z_i||_{\Sigma i}^2\right)\\ \end{aligned}


## 参考资料

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