Update

Author: ZebinYang

docs/_build/html/_sources/guides/explain/shap.rst.txt

@@ -34,7 +34,7 @@ The exact solution is obtained using the Shapley value formula, which requires e
 
 .. math::
    \begin{align}     
-     \phi_{i}= \sum_{S \subseteq \{1, \ldots, p\} \{ i \}} \frac{|S|!(p-|S|-1)!}{p!}(val(S \cup \{i\}) - val(S)).  \tag{2}
+     \phi_{i}= \sum_{S \subseteq \{1, \ldots, p\} \backslash \{ i \}} \frac{|S|!(p-|S|-1)!}{p!}(val(S \cup \{i\}) - val(S)).  \tag{2}
    \end{align}
 
 where :math:`val` is the value function that returns the prediction of each coalition. The marginal contribution of feature :math:`i` to the coalition :math:`S` is calculated as the difference between the value of the coalition with the addition of feature :math:`i` and the value of the original coalition, i.e., :math:`val(S \cup \{i\}) - val(S)`. The term :math:`\frac{|S|!(p-|S|-1)!}{p!}` is a normalization factor. When the number of features is small, this exact estimation approach is acceptable. However, as the number of features increases, the exact solution may become problematic.

docs/_build/html/guides/explain/shap.html

@@ -230,7 +230,7 @@ <h3><span class="section-number">4.2.2.1.1. </span>Exact Solution<a class="heade
 <p>The exact solution is obtained using the Shapley value formula, which requires evaluating all possible coalitions of features with and without the <span class="math notranslate nohighlight">\(i\)</span>-th feature.</p>
 <div class="math notranslate nohighlight">
 \[\begin{align}
-  \phi_{i}= \sum_{S \subseteq \{1, \ldots, p\} \{ i \}} \frac{|S|!(p-|S|-1)!}{p!}(val(S \cup \{i\}) - val(S)).  \tag{2}
+  \phi_{i}= \sum_{S \subseteq \{1, \ldots, p\} \backslash \{ i \}} \frac{|S|!(p-|S|-1)!}{p!}(val(S \cup \{i\}) - val(S)).  \tag{2}
 \end{align}\]</div>
 <p>where <span class="math notranslate nohighlight">\(val\)</span> is the value function that returns the prediction of each coalition. The marginal contribution of feature <span class="math notranslate nohighlight">\(i\)</span> to the coalition <span class="math notranslate nohighlight">\(S\)</span> is calculated as the difference between the value of the coalition with the addition of feature <span class="math notranslate nohighlight">\(i\)</span> and the value of the original coalition, i.e., <span class="math notranslate nohighlight">\(val(S \cup \{i\}) - val(S)\)</span>. The term <span class="math notranslate nohighlight">\(\frac{|S|!(p-|S|-1)!}{p!}\)</span> is a normalization factor. When the number of features is small, this exact estimation approach is acceptable. However, as the number of features increases, the exact solution may become problematic.</p>
 <p>It’s worth noting that the value function <span class="math notranslate nohighlight">\(val\)</span> takes the feature coalition <span class="math notranslate nohighlight">\(S\)</span> as input. However, in machine learning models, the prediction is not solely based on the feature coalition but on the entire feature vector. Therefore, we need to specify how removing a feature from the feature vector affects the prediction. Two common approaches are available, both of which depend on a pre-defined background distribution instead of merely replacing the “missing” features with a fixed value.</p>