add objectivity example

example/14-p-value-objectivity.ipynb

@@ -0,0 +1,113 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d94dbb54-d25a-41dc-a62d-f6b69ac62bec",
+   "metadata": {},
+   "source": [
+    "Reproduce an example from James Berger and Donald Berry's paper\n",
+    "[Statistical Analysis and the Illusion of Objectivity](https://si.biostat.washington.edu/sites/default/files/modules/BergerBerry.pdf).\n",
+    "\n",
+    "The example shows how a p-value depends on not just the data but also the intentions of the investigator."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "29ca528a-12ef-4a2a-a42b-4b9123ff5c4a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Imagine an investigator is studying a coin for bias. The investigator flips \n",
+    "# the coin 17 time and observes 13 heads.\n",
+    "n = 17\n",
+    "num_heads = 13"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "81ce0295-f667-4092-86a1-667f770995f1",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p-value 1: 0.049041748046875\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Here's the p-value the investigator would compute, if their intention\n",
+    "# with their experiment was to flip the coin 17 times and then do their analysis\n",
+    "import math\n",
+    "def p_value1(n, k):\n",
+    "    res = 0\n",
+    "    k = min(k, n - k)\n",
+    "    for j in range(k+1):\n",
+    "        t = math.comb(n, j) * 0.5 ** n\n",
+    "        if j != n - j:\n",
+    "            t *= 2\n",
+    "        res += t\n",
+    "    return res\n",
+    "print('p-value 1:', p_value1(n, num_heads))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "e264e4af-905a-4f45-a5b4-809268c3b787",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "p-value 2: 0.021270751953125\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Here's the p-value the investigator would compute if their intention\n",
+    "# was to keep flipping the coin until the got at least 4 heads and 4 tails\n",
+    "def p_value2(n, k):\n",
+    "    t = 0\n",
+    "    k = min(k, n - k)\n",
+    "    for j in range(2 * k, n):\n",
+    "        t += 2 * math.comb(j - 1, k - 1) * 0.5 ** j\n",
+    "    return 1 - t\n",
+    "print('p-value 2:', p_value2(n, num_heads))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "304ebda6-ee8e-4e48-9af9-9855972193f2",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
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+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
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