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Math Help - Bayesian Statistics

  1. #1
    Member
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    Bayesian Statistics

    Hello,

    I have a small question which I find hard to handle:

    X~U(0,a)
    a is unknown and his prior distribution is:

    p c
    g(a)=
    1-p 2c

    find the posterior distribution of a. for which range of x, the posterior distribution of a=c is bigger ? show your calculations.

    I found the posterior:

    p/c a=c
    f(a|x)=
    (1-p)/2c a=2c

    how can I found the second part ?
    cheers !
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  2. #2
    Grand Panjandrum
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    someplace
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    Quote Originally Posted by WeeG View Post
    Hello,

    I have a small question which I find hard to handle:

    X~U(0,a)
    a is unknown and his prior distribution is:

    p c
    g(a)=
    1-p 2c

    find the posterior distribution of a. for which range of x, the posterior distribution of a=c is bigger ? show your calculations.

    I found the posterior:

    p/c a=c
    f(a|x)=
    (1-p)/2c a=2c

    how can I found the second part ?
    cheers !
    Because spaces are not preserved in posts I find your expressions for g(a) and f(a|x) dificult to understand.

    Could you either post them in [code]..[/code] block or as an image file of some kind.

    example of use of [code]..[/code] block:

    Code:
            a+b
    f(x) =  ----
             c
    (you may have to addjust the number of spaces to get correct allighment due to vatriable width charaters)

    RonL
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  3. #3
    Member
    Joined
    Mar 2006
    Posts
    125
    another try:
    ( if it doesn't work I'll try the image idea )

    Code:
    I have a small question which I find hard to handle:
    
    X~U(0,a)
    a is unknown and his prior distribution is:
    
              p    c 
    g(a)= 
             1-p  2c
    
    find the posterior distribution of a. for which range of x, the posterior distribution of a=c is bigger ? show your calculations.
    
    I found the posterior:
    
                p/c        a=c
    f(a|x)=
               (1-p)/2c  a=2c
    
    the functions are defined in two different regions.
    
    how can I found the second part ?
    cheers !
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  4. #4
    Senior Member
    Joined
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    Quote Originally Posted by WeeG View Post
    another try:
    ( if it doesn't work I'll try the image idea )

    Code:
    I have a small question which I find hard to handle:
    
    X~U(0,a)
    a is unknown and his prior distribution is:
    
              p    c 
    g(a)= 
             1-p  2c
    
    find the posterior distribution of a. for which range of x, the posterior distribution of a=c is bigger ? show your calculations.
    
    I found the posterior:
    
                p/c        a=c
    f(a|x)=
               (1-p)/2c  a=2c
    
    the functions are defined in two different regions.
    
    how can I found the second part ?
    cheers !
    Is this what you mean?

    g(a) = \begin{cases}<br />
p &  \text{if } a = c \\<br />
1-p & \text{if } a = 2c.<br />
\end{cases}

    f(a|x) = \begin{cases}<br />
p/c &  \text{if } a = c \\<br />
(1-p)/2c & \text{if } a = 2c.<br />
\end{cases}

    If so, there is a problem: the definition of f(a|x) does not include x! If it is a posterior distribution, it has to.

    Actually, the joint density is

    f(a,x) = \begin{cases}<br />
p/c &  \text{if } a = c \text{ and } x \in [0,c] \\<br />
(1-p)/2c & \text{if } a = 2c \text{ and } x \in [0,2c].<br />
\end{cases}

    The marginal density of x is

    f_x (x) = \begin{cases}<br />
p/c + (1-p)/2c &  \text{if } x \in [0,c] \\<br />
(1-p)/2c & \text{if } x \in [c,2c].<br />
\end{cases}

    And the posterior density is

    f(a|x) = \frac{f(a,x)}{f_x (x)} = \begin{cases}<br />
\frac{p}{p + (1-p)/2} &  \text{if } a = c \text{ and } x \in [0,c] \\<br />
\frac{(1-p)/2}{p + (1-p)/2} &  \text{if } a = 2c \text{ and } x \in [0,c] \\<br />
1 & \text{if } a = 2c \text{ and } x \in [c,2c].<br />
\end{cases}
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  5. #5
    Member
    Joined
    Mar 2006
    Posts
    125
    you are right, the quesion is taken from an old exam I was trying to solve.
    there is something wrong with it...I don't know
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