what's the relationship with log(sum) and sum(log)?hi I'm a little confused about the log(sum) function...

detineerlf

detineerlf

Answered

2022-07-18

what's the relationship with log(sum) and sum(log)?
hi I'm a little confused about the log(sum) function and sum(log) function. In special, what's the relationship between these two terms?
log i a i i b i
i log ( a i + b i )
given a negative log-likelihood of an observation set:
L = i , j log ( π a M i , j + π b N i , j )
where C is the constant parameter. π a + π b =1 are proportion of the two component, given the instance O i j
Lemma1
log k = 1 K f k ( x ) = min Φ ( x ) Δ + k = 1 K { Φ k ( x ) [ log Φ k ( x ) l o g ( f k ( x ) ] } s . t . Φ k ( x ) = 1 , Φ k ( x ) ( 0 , 1 )
proof
R H S = k = 1 K Φ k ( x ) log Φ k ( x ) f k ( x ) >= k = 1 K Φ k ( x ) log k = 1 K Φ k ( x ) k = 1 K f k ( x ) ( l o g s u m   i n e q u a l i t y ) = log k = 1 K f k ( x ) ( Φ k ( x ) = 1 )
Let:
C = i , j Φ a i , j ( log Φ a i , j log ( π a M i , j ) ) + Φ b i , j ( log Φ B i , j log ( π b N i , j ) )
given the constraint, that for each ( i , j ), Φ a i , j + Φ b i , j = 1
Then how to prove:
Minimize C equals minimize L?
we have
min C = log ( π a M i , j ) log ( π b N i , j )
then the next step is how to prove the relationship between min C and L?

Answer & Explanation

eri1ti0m

eri1ti0m

Expert

2022-07-19Added 11 answers

sorry, I forgot one constraint, that is, for each i,j, we have Φ a i , j + Φ b i , j = 1. So this should be straightforward, i.e.,
for each coordinate ( i , j ), Φ a i , j + Φ b i , j = 1, then,
C i , j = Φ a i , j ( log Φ a i , j log ( π a M i , j ) ) + Φ b i , j ( log Φ b i , j log ( π b N i , j ) )
e.g., π a = π g , π b = π u , M i , j = N o r m a l i , j ( O i , j | θ ) , N i , j = 1 256
Apply L e m m a   1
min C i , j = log ( π a M i , j + π b N i , j )
integrating out RHS of C,
min C = i , j min C i , j = i , j log ( π a M i , j + π b N i , j ) = L

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?