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Generating Informative Indicators with Conditional Trees: Worked Examples‌‌​⁠‌​‌⁠‌​⁠​​⁠​‌​​​‌⁠‍⁠‌‍⁠‌​‌‍⁠‌‌⁠‍​‍​‌⁠⁠⁠‍‌‌⁠‍⁠⁠⁠‌​‍⁠​‍‍‍‌‌​⁠‍‌

All numeric cases below are synthetic teaching examples, not empirical research findings.

Example: informative milestone

P(C)=0.4, P(U C)=0.6, P(U not C)=0.1 imply P(U)=0.3. IG=h(0.3)-0.4h(0.6)-0.6h(0.1), about 0.1466098512 nats, or 24.0004% of maximum information.‌‌​⁠‌​‌⁠‌​⁠​​⁠​‌​​​‌⁠‍⁠‌‍⁠‌​‌‍⁠‌‌⁠‍​‍​‌⁠⁠⁠‍‌‌⁠‍⁠⁠⁠‌​‍⁠​‍‍‍‌‌​⁠‍‌

Example: dramatic but rare

A spectacular indicator with a large conditional effect may have low expected value when it almost never occurs. Include the negative branch and indicator probability before ranking.

Example: incoherent elicitation‌‌​⁠‌​‌⁠‌​⁠​​⁠​‌​​​‌⁠‍⁠‌‍⁠‌​‌‍⁠‌‌⁠‍​‍​‌⁠⁠⁠‍‌‌⁠‍⁠⁠⁠‌​‍⁠​‍‍‍‌‌​⁠‍‌

A prior of 0.7 with c=0.5,a=0.8,b=0.2 contradicts total probability, which gives 0.5. Stop and reconcile; the helper must not return a fabricated validated VOI.

Counterexample

“Indicator A has 10% information and B has 10%, therefore together they provide 20%” ignores redundancy.