How to find $min mathbbE Big[ || (X_1,X_2) - u(X_3) ||^2Big]$ when $X sim Dirichlet (alpha_1, ldots, alpha_4)$

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I'm trying to find the minimum of the following formula, where $(X_1, cdots, X_4) sim Dirich(alpha_1, cdots, alpha_4)$.



$min_u(cdot) mathbbE Big[ || (X_1,X_2) - u(X_3) ||^2Big]$



Since I know



$$
mathbbE(X_1 | X_3 ) = frac alpha_1 alpha_1 + alpha_2 + alpha_4 (1-X_3) \
mathbbE(X_2 | X_3 ) = frac alpha_2 alpha_1 + alpha_2 + alpha_4 (1-X_3)
$$



$u(cdot)$ might be $u(X_3) = Big[mathbbE(X_1 | X_3 ), mathbbE(X_1 | X_3 )Big]$.



But I can't proceed. Anybody to help me?




probability mean-square-error




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    up vote
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    I'm trying to find the minimum of the following formula, where $(X_1, cdots, X_4) sim Dirich(alpha_1, cdots, alpha_4)$.



    $min_u(cdot) mathbbE Big[ || (X_1,X_2) - u(X_3) ||^2Big]$



    Since I know



    $$
    mathbbE(X_1 | X_3 ) = frac alpha_1 alpha_1 + alpha_2 + alpha_4 (1-X_3) \
    mathbbE(X_2 | X_3 ) = frac alpha_2 alpha_1 + alpha_2 + alpha_4 (1-X_3)
    $$



    $u(cdot)$ might be $u(X_3) = Big[mathbbE(X_1 | X_3 ), mathbbE(X_1 | X_3 )Big]$.



    But I can't proceed. Anybody to help me?




    probability mean-square-error




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I'm trying to find the minimum of the following formula, where $(X_1, cdots, X_4) sim Dirich(alpha_1, cdots, alpha_4)$.



      $min_u(cdot) mathbbE Big[ || (X_1,X_2) - u(X_3) ||^2Big]$



      Since I know



      $$
      mathbbE(X_1 | X_3 ) = frac alpha_1 alpha_1 + alpha_2 + alpha_4 (1-X_3) \
      mathbbE(X_2 | X_3 ) = frac alpha_2 alpha_1 + alpha_2 + alpha_4 (1-X_3)
      $$



      $u(cdot)$ might be $u(X_3) = Big[mathbbE(X_1 | X_3 ), mathbbE(X_1 | X_3 )Big]$.



      But I can't proceed. Anybody to help me?




      probability mean-square-error




      share|cite|improve this question











      I'm trying to find the minimum of the following formula, where $(X_1, cdots, X_4) sim Dirich(alpha_1, cdots, alpha_4)$.



      $min_u(cdot) mathbbE Big[ || (X_1,X_2) - u(X_3) ||^2Big]$



      Since I know



      $$
      mathbbE(X_1 | X_3 ) = frac alpha_1 alpha_1 + alpha_2 + alpha_4 (1-X_3) \
      mathbbE(X_2 | X_3 ) = frac alpha_2 alpha_1 + alpha_2 + alpha_4 (1-X_3)
      $$



      $u(cdot)$ might be $u(X_3) = Big[mathbbE(X_1 | X_3 ), mathbbE(X_1 | X_3 )Big]$.



      But I can't proceed. Anybody to help me?





      probability mean-square-error





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 19 at 13:09









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