190 add a tutorial notebook for data valuation (#243)

docs/source/index.rst

@@ -30,6 +30,7 @@ Contents
    :maxdepth: 1
    :caption: TUTORIALS
 
+   notebooks/sv_calculation
    notebooks/shapiq_scikit_learn
    notebooks/treeshapiq_lightgbm
    notebooks/visualizing_shapley_interactions
@@ -38,6 +39,7 @@ Contents
    notebooks/conditional_imputer
    notebooks/parallel_computation
    notebooks/benchmark_approximators
+   notebooks/data_valuation
    notebooks/core
 
 .. toctree::

docs/source/notebooks/sv_calculation.ipynb

@@ -0,0 +1,84 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "source": [
+    "# Shapley Value Calculation\n",
+    "A popular approach to tackle the problem of XAI is to use concepts from game theory in particular cooperative game theory.\n",
+    "The most popular method is to use the **Shapley Values** named after Lloyd Shapley, who introduced it in 1951 with his work *\"II: The Value of an n-Person Game\"*.\n",
+    "\n",
+    "## Cooperative Game Theory\n",
+    "Cooperative game theory deals with the study of games in which players/participants can form groups to achieve a collective payoff. More formally a cooperative game is defined as a tuple $(N,\\nu)$ where:\n",
+    "- $N$ is a finite set of players\n",
+    "- $\\nu$ is a characteristic function that maps every coalition of players to a real number, i.e. $\\nu:2^N \\rightarrow \\mathbb{R}$\n",
+    "\n",
+    "Of particular interest is to find a concept that distributes the payoff of $\\nu(N)$ among the players, as it is assumed that the *grand coalition* $N$ is formed.\n",
+    "The distribution of the payoff among the players is called a *solution concept*.\n",
+    "\n",
+    "## Shapley Values: A Unique Solution Concept\n",
+    "Given a cooperative game $(N,\\nu)$, the Shapley value is a payoff vector dividing the total payoff $\\nu(N)$ among the players. The Shapley value of player $i$ is denoted by $\\phi_i(\\nu)$ and is defined as:\n",
+    "$$\n",
+    "\\phi_i(\\nu) := \\sum_{S \\subseteq N \\setminus \\{i\\}} \\frac{|S|!(|N|-|S|-1)!}{|N|!} [\\nu(S \\cup \\{i\\}) - \\nu(S)]\n",
+    "$$\n",
+    "and can be interpreted as the  average marginal contribution of player $i$ across all possible permutations of the players.\n",
+    "Its popularity arises from uniquely satisfies the following properties:\n",
+    "- **Efficiency**: The sum of the Shapley values equals the total payoff, i.e. $\\sum_{i \\in N} \\phi_i(\\nu) = \\nu(N)$\n",
+    "- **Symmetry**: If two players $i$ and $j$ are such that for all coalitions $S \\subseteq N \\setminus \\{i,j\\}$, $\\nu(S \\cup \\{i\\}) = \\nu(S \\cup \\{j\\})$, then $\\phi_i(\\nu) = \\phi_j(\\nu)$\n",
+    "- **Additivity**: For a game $(N,\\nu + \\mu)$ based on two games $(N,\\nu)$ and $(N,\\mu)$, the Shapley value of the sum of the games is the sum of the Shapley values, i.e. $\\phi_i(\\nu + \\mu) = \\phi_i(\\nu) + \\phi_i(\\mu)$\n",
+    "- **Dummy Player**: If for a player $i$ is holds for all coalitions $S \\subseteq N \\setminus \\{i\\}$, $\\nu(S \\cup \\{i\\}) - \\nu(S) = \\nu(\\{i\\})$ then $\\phi_i(\\nu) = \\nu(\\{i\\})$\n",
+    "\n",
+    "## Shapley Values: Cooking Game\n",
+    "To illustrate the concept of Shapley values, we consider a simple example of a cooking game.\n",
+    " The game consists of three players(cooks), Alice, Bob, and Charlie, who are cooking a meal together.\n",
+    " The characteristic function $\\nu$ maps each coalition of players to the quality of the meal."
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