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 sum_(i)a_i sum_i b_i -sum_i log(a_i+b_i)

detineerlf

detineerlf

Answered question

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

Beginner2022-07-19Added 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

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