Lower Bounds on the Probability of Benefit

Impossible Area

  • \(P(y_x)\), \(P(y_{x'})\): \((0.99, 0.99)\)
  • \(0.99 \leqslant P(y_x > y_{x'}) \leqslant 0.99\)
  • Range: \(0\)
\(P(x) = 0.5\), \(P(x') = 0.5\)
\(P(y|x) = 0.5\)
\(P(y|x') = 0.5\)
\(P(y) = 0.5\), \(P(x, y) = 0.25\), \(P(x, y') = 0.25\), \(P(x', y) = 0.25\), \(P(x', y') = 0.25\)
\(P(y_x) = 0.5\)
\(P(y_{x'}) = 0.5\)
\(0 \leqslant P(y_x > y_{x'}) \leqslant 0.5\), range: \(0.5\) impossible