Kalman smoother derivation subsitution

[MikeLowri123]

Hi I am working through the derivation of Kalman smoothing and am stuck on probably a simple substitution problem for the conditional variance of the hidden state x_{t}|x{t+1}, conditioned on all previous observations

Firstly the variance is given by:

[itex]Var[x_{t}|x_{t+1},y_{o},..,y_{t}]=P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}AP_{t|t}[/itex]

with A being the system updat equation and P represent the respective covariance in the relevant states

The substitution is of [itex]L{t}=P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

And the variance is thus supposed to equal.

Thanks in advance

[itex]P_{t|t}-L_{t}P_{t+1|t}L^T_{t}[/itex]

But working through I obtain:

[itex]P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}P_{t+1|t}P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

which condenses to:

[itex]P_{t|t}-P_{t|t}A^{T}P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

first three terms agree and can be canceled so I am left with:

[itex]P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

which should equal

[itex]P^{-1}_{t+1|t}AP_{t|t}[/itex]

Due to symmetry however

[itex]P_{t|t}A^{T}=AP_{t|t}[/itex]

So:

[itex]AP_{t|t}P^{-1}_{t+1|t}=P^{-1}_{t+1|t}AP_{t|t}[/itex]

The question is, does this final term hold, I am not sure if this relationship is commutative or not.

 

[Asim Riaz]

No, this final term does not hold. This relationship is not commutative. The correct expression for the variance of x_{t}|x_{t+1}, conditioned on all previous observations is: Var[x_{t}|x_{t+1},y_{o},..,y_{t}]=P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}AP_{t|t}.