Filtering theory
Assume a dynamical system that is governed by a stochastic difference equation:
\[dx_t = f(x_t, t)+G(x_t,t)d \beta_t\]
for all times, \(t \geq 0\). Observations occur at discrete times:
\[y_k=h(x_k,t_k)+\nu_k\]
where \(k=1,2,...;\) and \(t_{k+1} > t_k \geq t_0\).
The observational error is white in time and is Gaussian (this latter assumption is not essential).
\[\nu_k \rightarrow N(0,R_k)\]
The complete history of observations is:
\[Y_\tau=\{y_l;t_l \leq \tau\}\]
Our goal is to find the probability distribution for the state at time \(t\).
\[p(x,t|Y_t)\]
The state between observation times is obtained from the difference equation. We need to update the state given new observations:
\[p(x,t_k | Y_{t_k}) = p(x,t_k |y_k, Y_{t_{k-1}})\]
We do so by applying Bayes’ rule:
\[p(x,t_k | Y_{t_k}) = \frac{p(y_k |x_k, Y_{t_{k-1}}) p(x,t_k | Y_{t_{k-1}})}{p(y_k, Y_{t_{k-1}})}\]
Since the error is white in time:
\[p(y_k | x_k, Y_{t_{k-1}})=p(y_k|x_k)\]
We integrate the numerator to obtain a normalizing denominator:
\[p(y_k | x_k, Y_{t_{k-1}})= \int p(y_k|x) p(x,t_k |Y_{t_{k-1}})dx\]
This allows us to update the probability after a new observation:
\[p(x,t_k | Y_{t_k}) = \frac{p(y_k|x) p(x,t_k |Y_{t_{k-1}})}{\int p(y_k|\xi) p(\xi,t_k |Y_{t_{k-1}})d\xi}\]