{
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   "cell_type": "code",
   "execution_count": 1,
   "id": "fe446bd5-ea33-43a8-af48-b973617563a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8c532763-c8ea-451b-b9a5-10e71e4bf542",
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_lambd_optimal(X):\n",
    "    def neg_log_likelihood(lambd):\n",
    "        return -np.sum(-lambd * X + np.log(lambd))\n",
    "\n",
    "    # use scipy to find the optimum lambd for likelihood\n",
    "    return scipy.optimize.minimize_scalar(\n",
    "        neg_log_likelihood, bounds=(1e-9,1000)\n",
    "    ).x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1ffb73eb-4fcd-4124-9018-56a0bf097cfc",
   "metadata": {},
   "outputs": [],
   "source": [
    "def dataset_analysis(X):\n",
    "    lambd_optimal = calc_lambd_optimal(X)\n",
    "\n",
    "    # bootstrapping\n",
    "    lambd_bootstrapped = [\n",
    "        calc_lambd_optimal(np.random.choice(X, size=len(X)))\n",
    "        for i in range(1000)\n",
    "    ]\n",
    "\n",
    "    confidence_level = 0.95\n",
    "    lambd_lower, lambd_upper = np.quantile(\n",
    "        lambd_bootstrapped, [(1-confidence_level)/2, (1+confidence_level)/2]\n",
    "    )\n",
    "\n",
    "    # reporting\n",
    "    print(f\"Inferred optimal value of lambda: {lambd_optimal:.3f}\")\n",
    "    print(f\"{confidence_level*100:.0f}% confidence interval: from {lambd_lower:.3f} to {lambd_upper:.3f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d8d86f66-25f2-4c6e-a453-7c0999b37699",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True value of lambda: 10.0\n",
      "Inferred optimal value of lambda: 9.725\n",
      "95% confidence interval: from 9.090 to 10.409\n"
     ]
    }
   ],
   "source": [
    "lambd_true = 10\n",
    "print(f\"True value of lambda: {lambd_true:.1f}\")\n",
    "dataset_analysis(np.random.exponential(scale=1/lambd_true, size=1000))"
   ]
  }
 ],
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