tmp

$f(y_1, y_2|x) = f(y_1|x) f(y_2|x)$ ~ $f(y_1|x) = \frac{1}{2 \pi \sigma ^2} exp(- \frac{(x-y_1)^2}{2 \sigma^2})$ ~ $f(y_2|x) = \frac{1}{2 \pi \sigma ^2} exp(- \frac{((50-x)-y_2)^2}{2 \sigma^2})$ ~ $\frac{\partial}{\partial x}(log f(y_1, y_2|x)) = -\frac{2}{\sigma^2}(x - \frac{50-y_2+y_1}{2})$ ~

~ ~ ~ ~ $\begin{bmatrix}\theta_t \\ b_t \end{matrix} = \begin{bmatrix}\theta_{t-1}+(\omega_t - b_t) \Delta t \\ b_{t_1} \end{matrix} = \begin{bmatrix} 1 & -\Delta t \\ 0 & 1 \end{matrix} \begin{bmatrix}\theta_{t-1} \\ b_{t-1} \end{matrix} + \begin{bmatrix} \Delta t \\ 0 \end{matrix} \omega_t$ ~

$a = g sin(\theta)$

$H_t = \begin{bmatrix} g & 0 \end{matrix}$ ~