Kalman filter: state estimate

Question

Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + w_{k} $$

If I understand correctly, having observations $z_{0}, \dots, z_{k}$, in filtering problem the goal is to get the estimate $\hat{x}_{k|k} $ of $E[x_{k}|z_{k}, \dots, z_{0}]$.

The question: why do we call $\hat{x}_{k|k}$ the state estimate if state is itself random?

In wiki, it is even more confusing:

In what follows, the notation $\hat{\mathbf{x}}_{n|m}$ represents the estimate of $x$ at time $n$ given observations up to and including at time $m \leq n$.

Answer 1

Each measurement is associated with a certain time index and this then also fixes the associated estimate of the state with respect to an index. For example $\hat{x}_{k|k} = E[x_{k}|z_{k}, \dots, z_{0}]$ is a different problem compared to $\hat{x}_{k+1|k} = E[x_{k+1}|z_{k}, \dots, z_{0}]$. Namely, in that case the measurement $z_{k+1}$ is not used, so during that time step one can only predict. For example if $w_k\sim\mathcal{N}(0,W)$ then the expected value of the prediction would yield $\hat{x}_{k+1|k} = F\,\hat{x}_{k|k}$.

Also note that the state itself is a stochastic process, which does not have to be completely "random" since it can be modeled by $x_{k} = F\,x_{k-1} + w_{k}$. Only if $F=0$ would the state inherit all the stochastic properties from $w_k$.

Answer 2

The notation $\hat{ x}_{n|m}$ means $E(x_n|z_m,\cdots,z_0)$. When $m=n$, estimating $x_t$ is the filtering problem, when $m>n$, it is the prediction problem, and it is the smoothing problem for the cases where $n<m$.