Kalman filter

Overview

Kalman filter can be applied to aggregate multiple sources of data with Gaussian errors, while allowing addition of two noisy data. This is due to two closed properties of normal distribution.

Let a, b be two independent random variables. s.t.
a ~ Normal( u1, sd1 )
b ~ Normal( u2, sd2 )
then

1.  a+b ~ Normal( u1 + u2,  sqrt( sd1^2 + sd2^2) )

2. P( a == x && b == x) = P( a == x ) * P( b == x ) = normal(x, u3, sd3)
where
normal(x,u,sd) = exp( (x-u)^2/sd^2) / sqrt(2*pi) / sd
u3 = (u1*sd2^2 + u2*sd1^2)/(sd1^2+sd2^2)
sd3 = 1/ sqrt( 1/sd1^2 + 1/sd2^2 )

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