SHAP for a single data point, instead of average prediction of entire dataset

Question

I am trying to explain a regression model based on LightGBM using SHAP. I'm using the

shap.TreeExplainer(<lightgbm model>).shap_values(X) 

method to get the SHAP values, where X is the entire training dataset. These SHAP values give me comparison of an individual prediction, compared to the average prediction of the entire dataset.

In the online book by Christopher Molnar, section 5.9.4, he mentions that:

"Instead of comparing a prediction to the average prediction of the entire dataset, you could compare it to a subset or even to a single data point."

I have a couple of questions regarding this:

1. Am I correct to interpret that if, instead of passing the entire training dataset, I pass a subset of say 20 observations, then the SHAP values returned will be relative to the average of these 20 observations? This will be the equivalent of "subset" that Christopher Molnar mentioned in his book
2. Assuming that the answer to question 1 is yes, what if, instead of generating SHAP values relative to the average of 20 observations, I want to generate SHAP values relative to one specific observation. Christopher Molnar seems to imply that is possible. If it is possible, how do I do that?

Thank you in advance for the guidance!

Answer 1

1. Yes, but definition of "average" is important. If you supply a "background" dataset, your explanations will be calculated against this background, not against the whole dataset. As far as "relative to the average" of the background, one needs to understand shap values are average marginal contributions over all possible coalitions. So as far as SHAP values are concerned, you fix coalition(s), and the rest is yes, averaged. This allows fitting model once, and then passing different coalitions (with the rest averaged) through the model that was trained only once. This is where SHAP time savings come from.
   If you're interested in more you may visit original paper or this blog.

2. Yes. You supply a single data row as background, for a binary classification e.g., supply another class' data row for explanation, and see which feature, and by how much, changed class output.

Answer 2

1. Yes. By the mathematical formulation in the original paper, SHAP values are "the contribution of a feature to the difference between the actual prediction and the average prediction". The average prediction, sometimes called the "base value" or "expected model output", is relative to the background dataset you provided.

2. Yes. You can use a background dataset of 1 sample. The common choices of the background dataset is the training data, one single sample as the reference sample, or even a dataset of all zeros. From the author: “I recommend using either a single background data point, a small random subset of the true background, or for the best performance a set of k-medians (weighted by how many training points they each represent) designed to represent the background succinctly. “

Below are more details to support my answers to the two questions and how 2 can be done. So, why does the "expected model output" depend on the background dataset? To answer this questions, let's walk through how SHAP is done:

Step 1: We create a shap explainer providing two things: a trained prediction model and a background dataset. From the background dataset, SHAP creates an artificial dataset of coalitions. Each coalition is a binary vector representing the permutation of feature combinations, 1 represents a feature being present, and 0 absent. So there are 2^M possible coalitions for M features.

explainer = shap.KernelExplainer(f, background_X)

Step 2: We provide the sample(s) for which we want to compute SHAP values for. SHAP fills in values for this artificial dataset such that present features take original values of that sample, and absent features are filled with a value from the background dataset. Then the prediction is generated for this coalition. If the background dataset has n rows, the absent features are filled n times and the average of the n predictions is used as the prediction of this coalition. If the background dataset has a single sample, then the absent feature is filled with the values of that sample.

shap_values = explainer.shap_values(test_X)

Therefore, the SHAP values are relative to the average prediction of the background dataset.