Add AOD explanation

Author: ruivieira

examples/Group fairness metrics.ipynb

@@ -519,12 +519,24 @@
    "id": "7e9ca225",
    "metadata": {},
    "source": [
-    "# Average Odds Difference"
+    "## Average Odds Difference\n",
+    "\n",
+    "_Average Odds Difference_ measures the difference between the True Positive Rates ($TPR$) for the privileged and unprivileged groups, and the False Positive Rates ($FPR$) for the same groups. Formally, the definition is:\n",
+    "\n",
+    "$$\n",
+    "AOD=\\dfrac{(FPR_{u}-FPR_{p})+(TPR_{u}-TPR_{p})}{2}\n",
+    "$$\n",
+    "\n",
+    "Typically:\n",
+    "\n",
+    "- A fair model will have $AOD=0$\n",
+    "- A positive value indicates the model benefits the unprivileged group\n",
+    "- A negative value indicates the model benefits the privileged group"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 22,
    "id": "50814c63",
    "metadata": {},
    "outputs": [],
@@ -533,14 +545,14 @@
     "\n",
     "score = average_odds_difference(test=bias,\n",
     "                                truth=nobias,\n",
-    "                                privilege_columns=[2],\n",
-    "                                privilege_values=[1],\n",
-    "                                positive_class=[1])"
+    "                                privilege_columns=[\"gender\"],\n",
+    "                                privilege_values=[1], # privileged gender value, gender = 1\n",
+    "                                positive_class=[1]) # positive class, income = 1"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 23,
    "id": "cedbd492",
    "metadata": {},
    "outputs": [
@@ -557,36 +569,13 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 20,
-   "id": "00232e97",
+   "cell_type": "markdown",
+   "id": "3568343c-523d-4a37-b534-e7ae2bf01406",
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "0.12220095246218571\n"
-     ]
-    }
-   ],
    "source": [
-    "score = average_odds_difference(test=bias,\n",
-    "                                truth=nobias,\n",
-    "                                privilege_columns=[2],\n",
-    "                                privilege_values=[0],\n",
-    "                                positive_class=[1])\n",
-    "print(score)"
+    "As we can see, the $AOD$ indicates that the privileged group (`gender = 1`) is at an advantage in this model."
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "0e7d4026",
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
   {
    "cell_type": "markdown",
    "id": "74cf44c7",