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4 equations, 4 variables

  1. Feb 9, 2014 #1
    How do you solve the system of equations

    [tex]e^{x_{1}+y_{1}}+e^{x_{1}+y_{2}}=a_{1}[/tex]
    [tex]e^{x_{2}+y_{1}}+e^{x_{2}+y_{2}}=a_{2}[/tex]
    [tex]e^{x_{1}+y_{1}}+e^{x_{2}+y_{1}}=b_{1}[/tex]
    [tex]e^{x_{1}+y_{2}}+e^{x_{2}+y_{2}}=b_{2}[/tex]

    x1, x2, y1, y2 are the variables for which I want to solve the equations, a1, a2, b1, b2 are known.

    Context: I need to solve this in order to get the unknown maximum entropy joint probabilities

    [tex]p_{ij}=e^{-1-x_{i}-y_{j}}[/tex]

    [tex]\mbox{for the known marginal probabilities (}a_{i}\mbox{ and }b_{j}\mbox{).}[/tex]

    i know there is way to do this in information theory, but I need to solve it algebraically.
     
  2. jcsd
  3. Feb 9, 2014 #2

    chiro

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  4. Feb 9, 2014 #3
    I don't need a numeric solution or an approximation. I want to solve the equation for x1, x2, y1, y2, if possible. If not, I'll have to live with it.
     
  5. Feb 9, 2014 #4
    There's a problem here. First subsitute [itex] u_1 = e^{x1} [/itex] [itex] v_1 = e^{y1} [/itex] etc.

    Than the four equations become:

    [tex] u_1 v_1 + u_1 v_2 = a_1 [/tex]
    [tex] u_2 v_1 + u_2 v_2 = a_2 [/tex]
    [tex] u_1 v_1 + u_2 v_1 = b_1 [/tex]
    [tex] u_1 v_2 + u_2 v_2 = b_2 [/tex]

    No if you add the first two you get:

    [tex] u_1 v_1 + u_1 v_2 + u_2 v_1 + u_2 v_2 = a_1 + a_2 [/tex]

    and if you add the last two you get

    [tex] u_1 v_1 + u_1 v_2 + u_2 v_1 + u_2 v_2 = b_1 + b_2 [/tex]

    These can't both be true unless [itex] a_1 + a_2 = b_1 + b_2 [/itex]

    And if that is the case, you have only 3 equations left for 4 unknowns, so there won't be an unique solution.
     
  6. Feb 9, 2014 #5
    Thank you! Quick reply here: yes, a1+a2=b1+b2 because they are marginal probabilities and sum to 1. Also,

    [tex]u_{1}v_{1}+u_{1}v_{2}+u_{2}v_{1}+u_{2}v_{2}=1[/tex]

    because these are the joint probabilities. Sorry! I should have mentioned that. I will be back in half an hour to report if this gives me enough information to solve the system.
     
  7. Feb 9, 2014 #6
    If you write the equations as:

    [tex] u_1 (v_1 + v_2) = a_1 [/tex]
    [tex] v_1 (u_1 + u_2) = b_1 [/tex]
    [tex] (v_1 + v_2)(u_1 + u_2) = 1 [/tex]

    it's easy to see if [itex] (u_1,u_2,v_1,v_2) [/itex] is a solution, so is [itex] (c u_1, c u_2, \frac {v_1}{c}, \frac {v_2}{c} ) [/itex]

    to get a solution you can set u1 + u2 = 1 so v1

    this gets you

    [tex] u_1 = c a_1 [/tex]
    [tex] u_2 = c a_2 [/tex]
    [tex] v_1 = \frac {b_1}{c} [/tex]
    [tex] v_2 = \frac {b_2}{c} [/tex]

    as the complete solution.
     
  8. Feb 9, 2014 #7
    Got it. Thank you, willem2. There was information hiding here that I didn't take into account (basically, that it is sufficient to know p_ij for i=1,...,m-1 and j=1,...,n-1 in order to know the m x n dimensional joint probability matrix). Be that as it may, problem solved!
     
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