11.2.1 Information Mappings

Figure 11.3: Many alternative information mappings may be proposed. Each leads to a derived information space.

Consider a function that maps the history I-space into a space that is simpler to manage. Formally, let ${\kappa}: {\cal I}_{hist}\rightarrow {\cal I}_{der}$ denote a function from a history I-space, ${\cal I}_{hist}$, to a derived I-space, ${\cal I}_{der}$. The function, ${\kappa }$, is called an information mapping, or I-map. The derived I-space may be any set; hence, there is great flexibility in defining an I-map.^11.2 Figure 11.3 illustrates the idea. The starting place is ${\cal I}_{hist}$, and mappings are made to various derived I-spaces. Some generic mappings, ${\kappa}_1$, ${\kappa}_2$, and ${\kappa}_3$, are shown, along with some very important kinds, ${\cal I}_{est}$, ${\cal I}_{ndet}$ and ${\cal I}_{prob}$. The last two are the subjects of Sections 11.2.2 and 11.2.3, respectively. The other important I-map is ${\kappa}_{est}$, which uses the history to estimate the state; hence, the derived I-space is $X$ (see Example 11.11). In general, an I-map can even map any derived I-space to another, yielding ${\kappa}: {\cal I}_{der}\rightarrow {\cal I}'_{der}$, for any I-spaces ${\cal I}_{der}$ and ${\cal I}'_{der}$. Note that any composition of I-maps yields an I-map. The derived I-spaces ${\cal I}_2$ and ${\cal I}_3$ from Figure 11.3 are obtained via compositions.