Before revision: By "based on those observations" it is meant that Ŷt is measurable with respect to the σ-algebra Gt generated by the observations Zs, 0 ≤ s ≤ t. Denote by K = K(Z, t) be collection of all Rn-valued random variables Y that are square-integrable and Gt-measurable: By "best estimate", it is meant that Ŷt minimizes the mean-square distance between Yt and all candidates in K: Basic result: orthogonal projection The space K(Z, t) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷt of the minimization problem (M) is given by where PK(Z,t) denotes the orthogonal projection of L2(Ω, Σ, P; Rn) onto the linear subspace K(Z, t) = L2(Ω, Gt, P; Rn).
After revision: By "based on those observations" it is meant that Ŷt is measurable with respect to the σ-algebra Gt generated by the observations Zs, 0 ≤ s ≤ t. Denote by K = K(Z, t) the collection of all Rn-valued random variables Y that are square-integrable and Gt-measurable: By "best estimate", it is meant that Ŷt minimizes the mean-square distance between Yt and all candidates in K: Basic result: orthogonal projection The space K(Z, t) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷt of the minimization problem (M) is given by where PK(Z,t) denotes the orthogonal projection of L2(Ω, Σ, P; Rn) onto the linear subspace K(Z, t) = L2(Ω, Gt, P; Rn).