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 ) &gt;= ∑ 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 = 1Then 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?

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what&#039;s the relationship with log(sum) and sum(log)?hi I&#039;m a little confused about the log(sum) function and sum(log) function. In special, what&#039;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      )          &amp;gt;=      ∑          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        =  1Then 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?

eri1ti0m · Accepted Answer

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

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)

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