Example 3: Custom value functions

These examples are from section 5.1 of the Faith-Shap paper: https://arxiv.org/abs/2203.00870

import seaborn as sns

sns.set_style("whitegrid")
sns.set_context("notebook", rc={'axes.linewidth': 2, 'grid.linewidth': 1},  font_scale=1.5)

import numpy as np
import math

import nshap

Example 1

A value function takes two arguments: A single data point x (a numpy.ndarray) and a python list S with the indices the the coordinates that belong to the coaltion.

In this example, the value does not actually depend on the point x

p = 0.1
def v_func(x, S):
    """ The value function from Example 1 in the Faith-Shap paper.
    """
    if len(S) <= 1:
        return 0
    return len(S) - p * math.comb(len(S), 2)


v_func(np.zeros((1,11)), [1,2,3,4])

3.4

Equipped with the value function, we can compute different kinds of interaction indices

faith_shap = nshap.faith_shap(np.zeros((1,11)), v_func, 1)
faith_shap[(0,1)], faith_shap[(1,2)]

We can replicate Table 1 in the Faith-Shap paper

print('Table 1 in the Faith-Shap paper:\n')
for p in [0.1, 0.2]:
    for l in [1, 2]:
        print(f'p={p} l={l}\n')

        # define the value function
        def v_func(x, S):
            if len(S) <= 1:
                return 0
            return len(S) - p * math.comb(len(S), 2)

        # compute interaction indices
        faith_shap = nshap.faith_shap(np.zeros((1,11)), v_func, l)
        shapley_taylor = nshap.shapley_taylor(np.zeros((1,11)), v_func, l)
        shapley_interaction = nshap.shapley_interaction_index(np.zeros((1,11)), v_func, l)
        banzhaf_interaction = nshap.banzhaf_interaction_index(np.zeros((1,11)), v_func, l)
        faith_banzhaf = nshap.faith_banzhaf(np.zeros((1,11)), v_func, l)

        # print result
        if l == 1:
            print('Faith-Shap: ', faith_shap[(0,)])
            print('Shapley Taylor: ', shapley_taylor[(0,)])
            print('Interaction Shapley: ', shapley_interaction[(0,)])
            print('Banzhaf Interaction: ', banzhaf_interaction[(0,)])
            print('Faith-Banzhaf: ', faith_banzhaf[(0,)])
        else:
            print('Faith-Shap: ', faith_shap[(0,)], faith_shap[(0,1)])
            print('Shapley Taylor: ', shapley_taylor[(0,)], shapley_taylor[(0,1)])
            print('Interaction Shapley: ', shapley_interaction[(0,)], shapley_interaction[(0,1)])
            print('Banzhaf Interaction: ', banzhaf_interaction[(0,)], banzhaf_interaction[(0,1)])
            print('Faith-Banzhaf: ', faith_banzhaf[(0,)], faith_banzhaf[(0,1)])
        print('')

Table 1 in the Faith-Shap paper:

p=0.1 l=1

Faith-Shap:  0.5000000000001867
Shapley Taylor:  0.500000000000185
Interaction Shapley:  0.4999999999999939
Banzhaf Interaction:  0.5087890624999979
Faith-Banzhaf:  0.5087890624999989

p=0.1 l=2

Faith-Shap:  0.9545454545463214 -0.09090909090923938
Shapley Taylor:  0 0.10000000000002415
Interaction Shapley:  0.4999999999999939 -5.759281940243e-16
Banzhaf Interaction:  0.5087890624999979 -0.11367187500000073
Faith-Banzhaf:  1.0771484374999503 -0.11367187499999737

p=0.2 l=1

Faith-Shap:  1.6486811915683575e-13
Shapley Taylor:  1.7552626019323725e-13
Interaction Shapley:  2.7755575615628914e-17
Banzhaf Interaction:  0.0087890625
Faith-Banzhaf:  0.008789062500008604

p=0.2 l=2

Faith-Shap:  0.9545454545460967 -0.19090909090925617
Shapley Taylor:  0 -3.7941871866564725e-14
Interaction Shapley:  2.7755575615628914e-17 -0.10000000000000021
Banzhaf Interaction:  0.0087890625 -0.21367187500000198
Faith-Banzhaf:  1.0771484374999574 -0.21367187499998952