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}