4.19 I.
! > ! Left: max !, ! = ! Right: log
! ! !! !
!!! !(!!!)
= log
! ! !! !
! ! !! !
! !!
= log
! ! !! !
! ! !! !
+ log ! ! = ! ⟹ left = right II. ! = ! Left: max !, ! = ! !" ! Right: log
! ! !! !
!!! !(!!!)
= log
! ! !! !
! ! !! ! ! !!
= log
! ! !! !
! ! !! !
+ log ! ! = ! = ! ⟹ left = right III. ! < ! Left: max !, ! = ! Right: log
! ! !! !
!!! !(!!!)
= log
⟹ left = right ! ! + !!
∴ max !, ! = log
1 + ! !(!!!)
! ! !! !
! ! !! !
! !!
= log
! ! !! !
! ! !! !
+ log ! ! = ! 4.21 From (4.28) !
!
!!
! !!!
!
!
! ! !!!
!
!!!
!
!! ! ! , ! =
! !!
!!
!! ! ! , ! = ! !(!
! ,!)
!
⟹ ! ! , ! = − log 2 2! !
for a given ! , !!! , !! !"# !!! = !!! = 1, we can rewrite !
!
+
!
(!!! )! !!!! !(!! )! !!!!
!! !
−
!
!!! !! !!! !!
!!
!
!
! ! ,! = ! +
!
! ! ,! =
!
!!! !! !!! !!
, if ignore constant ! then branch metric is !!
!
!!! !! !!! !!
!!
4.22 ! !! = log
!!
!!
∝!!! ! ! !! ! ! , ! !! !
∝!!! ! ! !! ! ! , ! !! !
−∝!!! ! ! !! ! ! , ! !! !
= − log
!!
!!
! ∝!!!
= − log
!!
! ! !!! ! ! ,! !!! !
+ log
∗
= min
[∝!!! ! ! + !! ! ! , ! + !! ! ] −
!
!
−∝!!! ! ! !! ! ! , ! !! ! + log
! ∝!!!
!!
∗
min
[∝!!!
!
!
! ! !!! ! ! ,! !!! !
! ! + !! ! ! , ! + !! ! ]