A geometrical derivation of the fixed interval smoothing algorithm

where y(i) and x(i) are p x 1 and q x 1 vectors respectively, with the rest of the vectors and matrices in (1) and (2) dimensioned conformally. We observe the y(i) (i = 1, ..., n) but not the x(i), nor the disturbances e(i) and u(i). Equations (1) and (2) can be interpreted either as observing the signal x(i) with noise, with (2) the equation of evolution for the signal; alternatively, we can view (1) as a regression equation, with Hi a matrix of exogenous variables and x(i) (i = 1, ..., n) a sequence of stochastic coefficients having the autoregressive structure (2); see, for example, Duncan & Horn (1972). Let w(i) = {e(i)', u(i)'}'. We assume that w(i) is serially uncorrelated in time, w(i) is uncorrelated with x(j) (j < i), and that the matrices (Hi, Fi, i = 1, ..., n) are known, as are the variance-covariance matrices of the w(i). For convenience of notation we assume that all variables are Gaussian and have zero mean. This enables us to work with conditional expectations, but it is clear that our results will also hold for best linear unbiased prediction in the more general case. Clearly the zero mean assumption is innocuous. For 1 s j n, let