Baysian derivation of kalman filter

kalmanDeriv.m

Theorems:
1. Linear transform of a normal random variable (matrix cookbook, 8.1.4):
  Assume:
    X ~ N(x,Px),
    f(v) = A*v
  Then:
    f(X) ~ N(A*x,A*Px*A')

2. Conditional Distribution (matrix cookbook, 8.1.3):
  [X;Y]~N([x0;y_expected],[Pxx,Pxy;Pxy';Pyy]);
  p(x1|y_observed) ~ N(x1,Pxx_new))
  K = Pxy*(Pyy)^-1
  x1 = x0-K*(y_expected-y_observed)
  Pxx_new = Pxx-K*(Pxy')

We have discrete time linear system:
  [X1;Y0]=[A,B;C,D]*[X0;R]
  
Random variables:
  Uncertainty:      X0~N(x0,P0)
  Noise:             R~N(0,I)

Uncorrelated, so we can say:
  [X0;R]~N([x0;0],[P0,0;0,I])]

Apply theorem 1:
  f(v)=[A,B;C,D]*v

  [X1;Y0]~f([X0;R])
  [X1;Y0]~N([A,B;C,D]*[x0,0],[A,B;C,D]*[P0,0;0,I]*[A,B;C,D]');
  [X1;Y0]~N([A*x0;C*x0],[A*P0*A'+B*B',A*P0*C'+B*D';
                          C*P0*A'+D*B',C*P0*C'+D*D']);

Apply theorem 2:
  p(x1|y0) = N(x1,P1)
  K  = (A*P0*C'+B*D')*(C*P0*C'+D*D')^-1
  x1 = A*x0 - K*(C*x0 - y0)
  P1 = A*P0*A'+B*B - K*(C*P0*A'+D*B')

For seperate update and propagate:
Update:
  A=I
  B=0
  K  = (P0*C')*(C*P0*C'+D*D')^-1
  P(x1|y0) ~ N(x0 - K*(C*x0 - y0), P0 - K*(C*P0))
  
Propagate:
  P(x1|x0) ~ N(A*x0, A*P0*A'+B*B')

References:
  Matrix cookbook - https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf