Apply the sum-product algorithm (Section XXXXXXXXXXto the Kalman filter model and show that the result is equivalent to applying the Kalman filter recursions. Prove the Kalman smoother recursions: µ t...

1. 
   

   
   Apply the sum-product algorithm (Section 11.4.3) to the Kalman filter model and show that the result is equivalent to applying the Kalman filter recursions.
   

   
   


2. 
   

   
   Prove the Kalman smoother recursions:
   

   
   

^µ




t


|


T




<sup>=






µ</sup>




t




<sup>+





C




t



(




µ</sup>




t

+1

|


T







<sup>−









µ</sup>



+

|


t


⁾

|





<sup>t








t</sup>







t

+1

|


T



+

|


t




t

where

Σ



<sub>t




T</sub>


=

Σ

+

C



(

Σ


−




Σ

)

C



^T







,

C



_t






=

Σ



<sub>t




|




t</sub>



Ψ

T

−1

+

|


t

Σ




.

Hint

: It may help to examine the proof of the forward-backward algorithm for HMMs (Section 11.4.2).