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paper/paper.md

@@ -67,25 +67,26 @@ of oil and gas are not familiar with decision analysis. This VOI Streamlit
 
 Decision Analysis requires an analysis of the expected outcome (e.g. weighted average) 
 of the decision without further information. This uses the probabilities of positive $Pr(\Theta = positive)$ and negative $Pr(\Theta = negative)$
-hidden geothermal as the weights multiplied by the value outcomes: $v_a(\Theta = \theta_i)$, the values input into the two by two table that typically represent dollar amounts.  <br />.
+hidden geothermal as the weights multiplied by the value outcomes: $v_a(\Theta = \theta_i)$, the values input into the two by two table that typically represent dollar amounts.  <br />
  $V_{prior} = \max\limits_a \sum_i^2 Pr(\Theta = \theta_i) v_a(\Theta = \theta_i)$
 <!-- The prior probability $Pr(\Theta = \theta_i)$ where there are two $\theta_i)$:  $i ={negative, positve}$ -->
-Also calculated is the Value with Perfect Information:  <br />.
+Also calculated is the Value with Perfect Information:  <br />
  $V_{perfect} = \sum_i^2 Pr(\Theta = \theta_i) \max\limits_a v_a(\Theta = \theta_i)$
   <!-- \Sigma_{i=1}^2 Pr(\Theta = \theta_i) \max\limits_a v_a(\theta_i) \ \  \forall a  -->
-comparing to $V_{prior}$ gives an upper bound on what *any* information could bring or the value *of* perfect information ($VOI_{perfect}$). \\
+comparing to $V_{prior}$ gives an upper bound on what *any* information could bring or the value *of* perfect information ($VOI_{perfect}$): \
 $VOI_{perfect} = V_{perfect}- V_{prior}$
 
-After the .csv files are uploaded, the code base performs a grid search on bin sizes ($x_j$) or kde bandwidths, as documented in @trainor-guitton_voi_2023. To determine the "best" bandwidth, the data are split into training and testing sets, and the accuracy of Naive Bayes classifier is evaluated . The grid search performs the Naïve Bayes classification for 20 different bandwidths then compares the predicted class with the true class.  The bandwidth that results in the highest accuracy in Naïve Bayes is deemed the ideal bandwidth.
+After the .csv files are uploaded, the code base performs a grid search on bin sizes ($x_j$) or kde bandwidths, as documented in @trainor-guitton_voi_2023. To determine the "best" bandwidth, the data are split into training and testing sets, and the accuracy of Naïve Bayes classifier is evaluated . The grid search performs the Naïve Bayes classification for 20 different bandwidths then compares the predicted class with the true class.  The bandwidth that results in the highest accuracy in Naïve Bayes is deemed the ideal bandwidth.
 
-Next, the VOI App calculates the posterior probability: \\
+Next, the VOI App calculates the posterior probability: \
 <!-- Double dollars make self-standing equations: -->
 $$Pr( \Theta = \theta_i | X =x_j ) = \color{cyan} \frac{Pr(\Theta = \theta_i ) 
-\color{purple} Pr( X=x_j | \Theta = \theta_i )}{\color{orange} Pr (X=x_j)}$$
+\color{purple} Pr( X=x_j | \Theta = \theta_i )}{\color{orange} Pr (X=x_j)}$$ \
 which scales the "ideal" likelihood from the grid search with the user-entered prior probability of success ($\Theta = \theta_i$).
-The posterior replaces the prior to become the weight in 
+The posterior replaces the prior to become the weight in the value *with* imperfect information:
 $V_{imperfect} = \sum_{j=1}^2 Pr(X = x_j) \max_a \sum_{i=1}^2  Pr(\Theta = \theta_i | X=x_j)  v_a(\Theta = \theta_i)$
 
+This value tells user the ceiling of worth for this data attribute, given the economics and prior probability entered, and the reliability of the data to discriminate between a positive and negative geothermal case.
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