Grammar on "as is the case for Gaussian observation noise"

Hi,
When I read a Kalman filter similar model (see below please), I am not clear what it points in the last within quotation mark (as is the case for Gaussian observation noise). "as is... noise" means that Gaussian density is abadoned in observation case. Thus, the probablity densities of the system is not Gaussian.

"as is... noise" can be substituted as "because of the abandon of Gaussian probabilities in observation noise"?

Please clarify my question if you could.

Thanks in advance.


///////////////////
"For the general state space filtering formulation with non-Gaussian measurement noise, Masreliez proposed an approximation to this optimal filter that grealy reduces complexity. In particular, Masreliez proposed that some, but not all, of the Gaussian assumptions used in the derivation of the Kalman-Bucy filter be retained in defining a nonlinear recursively updated filter. He abandoned the requirement that the observation noise be Gaussian. However, he retained a Gaussian distribution for the conditional mean, although it is not a consequence of the probability densities of the system (as is the case for Gaussian observation noise)"

On Fri, 15 Feb 2013 07:28:43 -0800, fl wrote:

> Hi,
> When I read a Kalman filter similar model (see below please), I am not
> clear what it points in the last within quotation mark (as is the case
> for Gaussian observation noise). "as is... noise" means that Gaussian
> density is abadoned in observation case. Thus, the probablity densities
> of the system is not Gaussian.
> 
> "as is... noise" can be substituted as "because of the abandon of
> Gaussian probabilities in observation noise"?
> 
> Please clarify my question if you could.
> 
> Thanks in advance.
> 
> 
> ///////////////////
> "For the general state space filtering formulation with non-Gaussian
> measurement noise, Masreliez proposed an approximation to this optimal
> filter that grealy reduces complexity. In particular, Masreliez proposed
> that some, but not all, of the Gaussian assumptions used in the
> derivation of the Kalman-Bucy filter be retained in defining a nonlinear
> recursively updated filter. He abandoned the requirement that the
> observation noise be Gaussian. However, he retained a Gaussian
> distribution for the conditional mean, although it is not a consequence
> of the probability densities of the system (as is the case for Gaussian
> observation noise)"

Without reading more of the paper I cannot say for sure.

In the case of estimating the states of a linear system with Gaussian 
probabilities for the initial values of the states and Gaussian noise on 
the inputs and at the measurement, the state estimate error always has a 
Gaussian distribution.

I'm pretty sure that is what the clause of the last sentence beginning 
with "although" is referring to.

HTH.  I'm not sure about your rephrasing, so I'm just giving you mine.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com