vault backup: 2025-10-28 17:01:24

uncertainty.md

@@ -25,14 +25,32 @@ In statistical inference and [[strategy]],
 > is the amount a decision maker would be willing to pay
 > for information prior to making a decision.
 
-It is the value of the reduction in uncertainty
-that the information provides.
+Suppose information $I$ is available to a decision maker
+Consider these two scenarios:
 
-In a monetary context, it is the reduction of
-expected opportunity loss.
+1. the decision maker does not purchase the information
+  and makes \$9,000. $P(D)=9000$
+
+2. the decision maker purchases the information
+  and makes \$10,000 $P(D)|I=10000$
+
+The monetary value of $I$ is the difference between the payout
+without ($P(D)$) and with ($P(D)|I$) the information $I$.
 
 $$
-\text{EVI} = \text{EOL} - \text{EOL}|I
+\begin{align*}
+V(I) &= P(D)|I - P(D) \\
+&= (10000) - (9000) \\
+&= 1000
+\end{align*}
+$$
+
+> [!info] Expectation Notation
+> When forecasting, the payout of decisions is unknown,
+> thus
+
+$$
+\mathbb{E}\left[V(I)\right] = \mathbb{E}\left[P(D)\right] - \mathbb{E}\left[P(D)|I\right]
 $$
 
 ### Expected Value of Perfect Information