Why having the loss matrix to be 1 - identity function yields a loss of $1-p(C_l|x)$ ? (Bishop exercise 1.24)











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In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:




For a loss matrix $L_{kj}=1-I_{kj}$ we have
$sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$




Now I understand where the $1$ is coming from:



$sum_kL_{kl}P(C_k|x)=underbrace{sum_kp(C_k|x)} - sum_kp(C_k|x)$



The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?










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    In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:




    For a loss matrix $L_{kj}=1-I_{kj}$ we have
    $sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$




    Now I understand where the $1$ is coming from:



    $sum_kL_{kl}P(C_k|x)=underbrace{sum_kp(C_k|x)} - sum_kp(C_k|x)$



    The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:




      For a loss matrix $L_{kj}=1-I_{kj}$ we have
      $sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$




      Now I understand where the $1$ is coming from:



      $sum_kL_{kl}P(C_k|x)=underbrace{sum_kp(C_k|x)} - sum_kp(C_k|x)$



      The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?










      share|cite|improve this question













      In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:




      For a loss matrix $L_{kj}=1-I_{kj}$ we have
      $sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$




      Now I understand where the $1$ is coming from:



      $sum_kL_{kl}P(C_k|x)=underbrace{sum_kp(C_k|x)} - sum_kp(C_k|x)$



      The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?







      probability matrices machine-learning






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      asked Dec 1 at 13:10









      Alaa M.

      15717




      15717






















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          The second term is actually $sum_kcolor{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.






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            1 Answer
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            1 Answer
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            active

            oldest

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            active

            oldest

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            up vote
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            down vote



            accepted










            The second term is actually $sum_kcolor{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              The second term is actually $sum_kcolor{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                The second term is actually $sum_kcolor{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.






                share|cite|improve this answer












                The second term is actually $sum_kcolor{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.







                share|cite|improve this answer












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                answered Dec 1 at 13:31









                Alijah Ahmed

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                9,86421217






























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