Define “Canonical Outcome Space” in plain English












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I'm trying to grasp what the below author is intending by "canonical outcome space", but I'm not quite sure what she is referring to, especially with her phrasing. If someone can offer a plain English explanation or explicate what she's already said, that would be super helpful!



Probabilistic Graphical Models



Where I'm having difficulty:
Given $$chi= { X_1,X_2,...,X_n}$$
and an assignment $$X_1=x_1, X_2=x_2,...,X_n=x_n$$



is it such that $$
Values(X_i) = {x_1,x_2,...,x_n}, iin{1,2,...,n}$$

Meaning $X_1=x_1$, or we can have $X_1=x_2$ and so on until $X_1=x_n$ as possible values for each $X_i$



or is it that
$$ Values(X_i) = { x_i }, iin{1,2,...,n}$$ which is to say only one assignment is possible for each $X_i$?



What trips me up is




...for a choice of values $x_1,...,x_n$ for all variables. Moreover,any two such events must be either identical or disjoint, since they both assign values to all the variables in $chi$




and what KO's me is this




In addition, any event defined using variables in $chi$ must be a union of a set of such events.




If someone can break this down for me/fill in the gaps, much thanks!










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    0












    $begingroup$


    I'm trying to grasp what the below author is intending by "canonical outcome space", but I'm not quite sure what she is referring to, especially with her phrasing. If someone can offer a plain English explanation or explicate what she's already said, that would be super helpful!



    Probabilistic Graphical Models



    Where I'm having difficulty:
    Given $$chi= { X_1,X_2,...,X_n}$$
    and an assignment $$X_1=x_1, X_2=x_2,...,X_n=x_n$$



    is it such that $$
    Values(X_i) = {x_1,x_2,...,x_n}, iin{1,2,...,n}$$

    Meaning $X_1=x_1$, or we can have $X_1=x_2$ and so on until $X_1=x_n$ as possible values for each $X_i$



    or is it that
    $$ Values(X_i) = { x_i }, iin{1,2,...,n}$$ which is to say only one assignment is possible for each $X_i$?



    What trips me up is




    ...for a choice of values $x_1,...,x_n$ for all variables. Moreover,any two such events must be either identical or disjoint, since they both assign values to all the variables in $chi$




    and what KO's me is this




    In addition, any event defined using variables in $chi$ must be a union of a set of such events.




    If someone can break this down for me/fill in the gaps, much thanks!










    share|cite|improve this question









    $endgroup$















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      0








      0


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      $begingroup$


      I'm trying to grasp what the below author is intending by "canonical outcome space", but I'm not quite sure what she is referring to, especially with her phrasing. If someone can offer a plain English explanation or explicate what she's already said, that would be super helpful!



      Probabilistic Graphical Models



      Where I'm having difficulty:
      Given $$chi= { X_1,X_2,...,X_n}$$
      and an assignment $$X_1=x_1, X_2=x_2,...,X_n=x_n$$



      is it such that $$
      Values(X_i) = {x_1,x_2,...,x_n}, iin{1,2,...,n}$$

      Meaning $X_1=x_1$, or we can have $X_1=x_2$ and so on until $X_1=x_n$ as possible values for each $X_i$



      or is it that
      $$ Values(X_i) = { x_i }, iin{1,2,...,n}$$ which is to say only one assignment is possible for each $X_i$?



      What trips me up is




      ...for a choice of values $x_1,...,x_n$ for all variables. Moreover,any two such events must be either identical or disjoint, since they both assign values to all the variables in $chi$




      and what KO's me is this




      In addition, any event defined using variables in $chi$ must be a union of a set of such events.




      If someone can break this down for me/fill in the gaps, much thanks!










      share|cite|improve this question









      $endgroup$




      I'm trying to grasp what the below author is intending by "canonical outcome space", but I'm not quite sure what she is referring to, especially with her phrasing. If someone can offer a plain English explanation or explicate what she's already said, that would be super helpful!



      Probabilistic Graphical Models



      Where I'm having difficulty:
      Given $$chi= { X_1,X_2,...,X_n}$$
      and an assignment $$X_1=x_1, X_2=x_2,...,X_n=x_n$$



      is it such that $$
      Values(X_i) = {x_1,x_2,...,x_n}, iin{1,2,...,n}$$

      Meaning $X_1=x_1$, or we can have $X_1=x_2$ and so on until $X_1=x_n$ as possible values for each $X_i$



      or is it that
      $$ Values(X_i) = { x_i }, iin{1,2,...,n}$$ which is to say only one assignment is possible for each $X_i$?



      What trips me up is




      ...for a choice of values $x_1,...,x_n$ for all variables. Moreover,any two such events must be either identical or disjoint, since they both assign values to all the variables in $chi$




      and what KO's me is this




      In addition, any event defined using variables in $chi$ must be a union of a set of such events.




      If someone can break this down for me/fill in the gaps, much thanks!







      probability probability-theory probability-distributions






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      share|cite|improve this question











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      share|cite|improve this question










      asked Jan 17 at 5:21









      Matt1991Matt1991

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