Prove the Chapman–Kolmogorov relation: r P r ( w t | x 1 ...t − 1 ) = P r ( w t | w t − 1 ) P r ( w t − 1 | x 1 ...t − 1 ) d w t − 1 r = Norm w t [ µ p + Ψw t − 1 , Σ p ] Norm w t − 1 [ µ t − 1 , Σ t...






    1. Prove the Chapman–Kolmogorov relation:






r





P




r



(



w



t



|




x


1

...t




1

)




=





P




r



(



w



t



|




w



t




1

)



P




r



(



w



t




1


|




x


1

...t




1

)





d




w



t




1



r




=




Norm


w



t







[




µ





p







+





Ψw



t




1


,








Σ



p


]


Norm


w



t








1




[




µ





t








1



,








Σ



t




1

]





d




w



t




1




=




Norm


w



t







[




µ





p







+





Ψ





µ





t








1



,








Σ



p




+





ΨΣ



t




1


Ψ











]



.





T







May 12, 2022
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