# Correct interpretation of summary_plot shap graph

While through the various resources online to understand the shap plots, I ended up slightly confused. Find below my interpretation of the overall plot given in examples -

1. Shap value 0 for a feature corresponds to the average prediction using all the other possible combinations of features except for the feature of interest. For e.g, shap value 0 for LSTAT corresponds to the average prediction of models having different combinations of features (excluding LSTAT)
2. Shap value of 4 with respect to 0 means that there is a relative increase in prediction being 1. How do we explain shap values 4, 2, etc. with respect to 0 in layman's terms?
3. Slight confusion in interpreting the overall graph vs the single point graph ( (

) - In a single data point example, the base value/expected value is the average prediction in the entire dataset -> but in the overall plot (referenced at the top) we are saying it is the average prediction without that feature. So things don't add up? Thank you for your help! :)

I think your interpretation is not entirely correct. Loosely rephrasing Lundberg et al. [arXiv:1802.03888], the SHAP value of feature $$i$$ is $$E[f(x) \mid S \cup \{i\}] - E[f(x) \mid S]$$ averaged over all possible subsets of features $$S$$, $$i \notin S$$. Here $$f(x)$$ is the prediction of the model for inputs $$x$$. Figure 2 from the preprint is a good illustration:
We are interested in the (signed) length of arrow for a particular feature, averaged over all permutations of the features. For a linear model $$y(x) = w^Tx$$ the SHAP value of feature $$i$$ would be simply $$w_i x_i$$.
3. The summary is just a swarm plot of SHAP values for all examples. The example whose power plot you include below corresponds to the points with $$\text{SHAP}_\text{LSTAT} = 4.98$$, $$\text{SHAP}_\text{RM} = 6.575$$, and so on in the summary plot.