Equation solving question

• To: mathgroup at smc.vnet.net
• Subject: [mg111680] Equation solving question
• From: Yaroslav Bulatov <yaroslavvb at gmail.com>
• Date: Tue, 10 Aug 2010 03:55:05 -0400 (EDT)

```I'd like to solve the system of equations below and Mathematica gets
stuck, are the tricks I can do to help it solve this kind of system?
It should produce solution q1=p1, q2=p2, q3=p3, q4=p4

lagr = p1 (Log[q1/(q1 + q3)] + Log[q1/(q1 + q2)]) +
p2 (Log[q2/(q1 + q2)] + Log[q2/(q2 + q4)]) +
p3 (Log[q3/(q1 + q3)] + Log[q3/(q3 + q4)]) +
p4 (Log[q4/(q2 + q4)] + Log[q4/(q3 + q4)]) - \[Lambda] (q1 + q2 +
q3 + q4 - 1);
vars = {q1, q2, q3, q4, \[Lambda]};

Assuming[{p1 > 0 && p2 > 0 && p3 > 0 && p4 > 0 &&
p1 + p2 + p3 + p4 == 1 && q1 > 0 && q2 > 0 && q3 > 0 && q4 > 0},
Solve[D[lagr, #] == 0 & /@ vars, vars]]

Motivation: I'm trying to show consistency of some pseudo-likelihood
estimators, this example is the simplest case where PL applies

```

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