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Natural extension with irrelevance relations

Consider the effect of adding irrelevance relations, in particular the statement that the nondescendants of a variable are irrelevant to the variable given the parents of the variable. Four constraints represent this statement regarding credal sets: $0.4 \leq p(f\vert b) \leq 0.5$ and $0.4 \leq p(b\vert f) \leq 0.5$. To simplify the calculation of lower and upper bounds, Theorem 2 can be used. The upper bound is obtained by solving the program:

\begin{displaymath}
\max \frac{ 0.48 w_1 + 0.06 w_2 + 0.005 w_3 + 0.035 w_4}
{ 0.6 w_1 + 0.6 w_2 + 0.05 w_3 + 0.05 w_4 }
\end{displaymath}


\begin{displaymath}
\mbox{s.t.} \;\;
\left[ \begin{array}{cccc}
-3 & 0 & 2 & 0 ...
...}{c}
0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right],
\end{displaymath}

and $\sum w_i = 1$ (w1 = p(f,b), w2 = p(f,bc), w3 = p(fc,b), w4 = p(fc,bc)). This program produces the upper bound 0.4509. By minimization, the lower bound 0.3818 is obtained. Note that these bounds are different from the bounds obtained by type-1 extension.



Fabio Gagliardi Cozman
1998-07-03