Question for Dr. Lin
Published:
I am perhaps trying to be too technical. By I am curious, even if this doesn’t belong in the note that I am writing about spectral factorization. In the context of a state-space model,
\[\left\{ \begin{array}{rl} \theta_{i+1} &= F_i\theta_i + G_iu_i \\ X_i &= H_i\theta_i + v_i \end{array} \right.\]I have the state random variables $\theta_i \in \mathbb{C}^m$ and observations $X_i \in \mathbb{C}^d$. I want to condition the state variable $\theta_i$ on the observation $X_j$. I would like to think of this geometrically. And I am wondering what is going on formally, when I project the state onto the observation. I would like to write
\[\hat{\theta}_i = \mathbb{E}[\theta_i | X_j] = \langle \theta_i, X_j \rangle R^{-1}_{X,j}X_j\]where $\langle x,y \rangle = \mathbb{E} xy^*$ and $R_{X,j} = \langle X_j,X_j \rangle$ and say I am projecting $\theta_i$ onto $\text{span}(X_j)$. But, I am discouraged from doing so since in (2) $\langle \cdot,\cdot \rangle$ is not an inner-product and these variables, though are over the same probability space, are not in a common module or Hilbert space.
My question is if you know of…