I imagine the systems's behavior in time as an unfolding in the dark recesses of 
.
Each time step brings a new element to the basis, a new axis. The State Sequence goes
twisting and writhing along, while we, in some hyperdimensional Allegory of the Cave,
peer at it's flickering images in the space of our observations. While the Probability
Space spanned by the system builds over time into some baroque geometry of causality,
we poke and peer at its shadow, trying to glean some hint of the Truth which underlies
surface appearances, rife as they are with noise and error.
We begin this process by performing a Gramm-Schmidt orthogonalization of both our State and
Observation sequences. This is a recursive decomposition of the Random Variables. By
Theorem 1 we may decompose 
, the 
observation, into its projections
on 
and its complement 
for any n.
for notational convenience, and more or less in keeping with Kalman's
original notation, 
To generate an orthogonal basis for 
we merely note that if we assume
some knowledge of initial conditions we may set
are pairwise independent, and thus 
.
We will thus use the 
's for our ``reference basis''. Our next step is to perform
a similar decomposition on the state variable sequence.
Since 
we call 
the estimate of 
based on the (orthogonalized) observations 
, and 
the error.
Now, by orthogonality of the 
we have
so we may decompose the projections:
That is, we may split the process of calculating this turn's estimate into
a sum of an estimate based on last turn's estimate and one based entirely on the
current observation. This has obvious computational as well as aesthetic merit, and the
fact that the two spaces are orthogonal complements allows us to treat them seperately; the two
estimates are independent random variables 
. If we consider the ``current'' observation, we may
term the projection of on 
the a priori estimate, that on 
the update,
and on 
the posteriori estimate, because it is our best estimate
based on all the information availible at that time.