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  {
   "cell_type": "markdown",
   "id": "97a673be",
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   "source": [
    "# IRT modeling\n",
    "\n",
    "MathArena benchmark measures the performance of different LLMs on a variety of mathematical problems.\n",
    "\n",
    "We use the [results](https://matharena.ai/?view=detailed&comp=usamo--usamo_2026) from USAMO 2026, which is a math competition for high school students in the United States.\n",
    "\n",
    "There are **6 models**, **6 problems**, each problem is scored on a **scale of 0 to 7** and each model has **4 attempts** at each problem.\n",
    "\n",
    "Let's try to fit an IRT model to this data."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03d8a0bd",
   "metadata": {},
   "source": [
    "IRT model assumes that each LLM $m$ has a latent ability $\\theta_m$ and each question has a difficulty $b_q$ and discrimination $a_q$. The probability of LLM $i$ solving problem $j$ is given by the logistic function:\n",
    "$$P(X_{mq} = 1) = \\frac{1}{1 + e^{-a_q(\\theta_m - b_q)}}$$\n",
    "\n",
    "Where $X_{mq}$ is the binary outcome of whether LLM $m$ solved problem $q$ (1 if solved, 0 if not).\n",
    "\n",
    "Well... our outcome is not binary, but as a first approximation we can treat the score as binomial rv with $n=7$ trials and $p$ being the probability of solving each point of the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a22a5535",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import pymc as pm\n",
    "import arviz as az\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7c631000",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>model</th>\n",
       "      <th>problem</th>\n",
       "      <th>run</th>\n",
       "      <th>score</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>GPT-5.4</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>GPT-5.4</td>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>7</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>GPT-5.4</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>GPT-5.4</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>GPT-5.4</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     model  problem  run  score\n",
       "0  GPT-5.4        1    1      7\n",
       "1  GPT-5.4        1    2      7\n",
       "2  GPT-5.4        1    3      7\n",
       "3  GPT-5.4        1    4      7\n",
       "4  GPT-5.4        2    1      7"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"models_scores.csv\")\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "5cf04682",
   "metadata": {},
   "outputs": [],
   "source": [
    "model_idx, model_labels = pd.factorize(df[\"model\"])\n",
    "problem_idx, problem_labels = pd.factorize(df[\"problem\"])"
   ]
  }
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