11.4.3.1 Nondeterministic and probabilistic I-spaces for discrete stages

The concepts of I-maps and derived I-spaces from Section 11.2 extend directly to continuous spaces. In the nondeterministic case, ${{\kappa}_{ndet}}$ once again transforms the initial condition and history into a subset of $X$. In the probabilistic case, ${{\kappa}_{prob}}$ yields a probability density function over $X$. First, consider the discrete-stage case.

The nondeterministic I-states are obtained exactly as defined in Section 11.2.2, except that the discrete sets are replaced by their continuous counterparts. For example, $F(x,u)$ as defined in (11.28) is now a continuous set, as are $X$ and $\Theta(x,u)$. Since probabilistic I-states are probability density functions, the derivation in Section 11.2.3 needs to be modified slightly. There are, however, no important conceptual differences. Follow the derivation of Section 11.2.3 and consider which parts need to be replaced.

The replacement for (11.35) is

$$p(x_k\vert y_k) = {p(y_k\vert x_k) p(x_k) \over \displaystyle\strut \int_X p(y_k\vert x_k) p(x_k) dx_k } ,$$ (11.56)

which is based in part on deriving $p(y_k\vert x_k)$ from $p(\psi_k\vert x_k)$. The base of the induction, which replaces (11.36), is obtained by letting $k=1$ in (11.56). By following the explanation given from (11.37) to (11.40), but using instead probability density functions, the following update equations are obtained:

$$\begin{split}p(x_{k+1}\vert{\eta}_k,u_k) & = \int_X p(x_{k+1}... ... p(x_{k+1}\vert x_k,u_k) p(x_k\vert{\eta}_k) dx_k , \end{split}$$ (11.57)

and

$$p(x_{k+1}\vert y_{k+1},{\eta}_k,u_k) = {p(y_{k+1}\vert x_{k+1}) p... ...le\strut \int_X p(y_{k+1}\vert x_{k+1}) p(x_{k+1}\vert{\eta}_k,u_k) dx_{k+1}} .$$ (11.58)