How to interpret LIME results?

Question

LIME is a recent method that claims to help explaining individual predictions from classifiers agnostically. See e.g. arxiv or its implementation on github for details.

I am trying to understand what exactly it outputs. For that, I am using a trivial example: logistic regression.

Consider the following set of events:

data  = []
for t in range(100000):
    a = 1 - 2*numpy.random.random()  # U(-1, 1)
    b = 1 - 2*numpy.random.random()  # U(-1, 1)
    noise = numpy.random.logistic()
    c = int(a + b + noise > 0)  # the target
    data.append([a, b, c])
data = numpy.array(data)

x = data[:, :-1]
y = data[:, -1]

This is a latent logistic process with parameters $a_0 = 0$, $a_1 = a_2 = 1$, of which logistic regression assymptotically fits.

Let us fit the data using logistic regression:

classifier = sklearn.linear_model.LogisticRegression(C=1e10)  # C=inf => no regularization    
classifier.fit(x, y)
print(classifier.coef_)  # [[ 0.99092809  1.00551462]]

Now, lets apply LIME to it:

explainer = lime.lime_tabular.LimeTabularExplainer(x, feature_names=['a', 'b'])

instance = numpy.array([1, 1])
explanation = explainer.explain_instance(instance, classifier.predict_proba, num_samples=100000)
print(explanation.as_list())

The result I get is something like this:

[
 ('a > 0.50', 0.2216), 
 ('b > 0.50', 0.2170)
]

the question is: what is this supposed to mean?

Answer 1

In the code implementaion repo you linked there's an example code which claims that explains, see cell 8

Note that LIME has discretized the features in the explanation. This is because we let discretize_continuous=True in the constructor (this is the default).

Lime Tabular Data Tutorial

If you modify the constructor to

explainer = lime.lime_tabular.LimeTabularExplainer(x, feature_names=['a', 'b'],discretize_continuous=False)

then the output is something like

[('b', 0.11223168027269199), ('a', 0.11110292313683988)]

I understood that both instances - discretized or not - are supposed to approximate a linear model in the vicinity of the feature example [1,1].