This is the prediction step of Kalman filter. I'm not sure how (18.25) derives into (18.26).. Does anyone have ideas about this?
2026-03-28 11:42:44.1774698164
How is this composite Gaussian Distribution derivated?
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(Much) more simply, (18.25) is assuming that, conditionally on $z'$, the random vector $z$ is normal with mean $Az'+Bu$ and covariance matrix $Q$, and that the random vector $z'$ is normal with mean $\mu$ and covariance matrix $R$.
Thus there exists $x$ and $x'$ independent centered normal such that $$z=Az'+Bu+Lx$$ with $LL^t=Q$ and $$z'=\mu+Tx'$$ with $TT^t=R$. One sees that $$z=(ATx'+Lx)+(A\mu+Bu)$$ is normal with mean $A\mu+Bu$ and covariance matrix $$E((ATx'+Lx)(ATx'+Lx)^t)=ATT^tA^t+LL^t=ARA^t+Q$$ as desired. Elementary linear algebra all along...