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   "source": [
    "# 17 — Generative calibration hooks (MMD)\n",
    "\n",
    "Doc page: [generative_calibration_hooks.rst](../../docs/source/algorithms/generative_calibration_hooks.rst).\n"
   ]
  },
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   "id": "d98275e1",
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from optimizr import _core as opt\n",
    "plt.rcParams['figure.figsize'] = (8.5, 4.5)\n",
    "plt.rcParams['figure.dpi'] = 110\n",
    "plt.rcParams['axes.grid'] = True\n",
    "plt.rcParams['grid.alpha'] = 0.3\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "759b0068",
   "metadata": {},
   "source": [
    "## Cellule 1 — MMD nulle entre échantillons identiques\n",
    "\n",
    "**Théorème (Gretton et al. 2012).** Pour le noyau gaussien\n",
    "$k(x, y) = \\exp(-(x-y)^2 / 2\\sigma^2)$, la MMD au carré empirique\n",
    "entre $X = (x_i)$ et $Y = (y_j)$ est\n",
    "$$\\widehat{\\text{MMD}}^2(X, Y) = \\frac{1}{m^2}\\sum_{ij} k(x_i, x_j)\n",
    "   + \\frac{1}{n^2}\\sum_{ij} k(y_i, y_j)\n",
    "   - \\frac{2}{mn}\\sum_{ij} k(x_i, y_j).$$\n",
    "\n",
    "**Équation pivot.** $X = Y \\Rightarrow \\widehat{\\text{MMD}}^2 = 0$.\n",
    "\n",
    "**Démonstration.** Les trois sommes coïncident terme à terme.  $\\square$\n",
    "\n",
    "**Ce que la cellule vérifie.** `mmd_gaussian(X, X, sigma)` retourne $0$\n",
    "à précision machine.\n"
   ]
  },
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MMD(X, X) = 0.000e+00  (attendu : 0)\n"
     ]
    },
    {
     "data": {
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       "<Figure size 935x495 with 1 Axes>"
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   ],
   "source": [
    "rng = np.random.default_rng(0)\n",
    "X = rng.standard_normal(200).tolist()\n",
    "m = float(opt.mmd_gaussian(X, X, sigma=1.0))\n",
    "print(f\"MMD(X, X) = {m:.3e}  (attendu : 0)\")\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.hist(X, bins=30, density=True, color='C0',\n",
    "        edgecolor='white', alpha=0.85)\n",
    "ax.set_title(r'Échantillon $X = Y$')\n",
    "ax.set_xlabel('x'); ax.set_ylabel('densité')\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01f32911",
   "metadata": {},
   "source": [
    "**Résultat attendu.** $\\sim 0$.\n",
    "\n",
    "**Lecture du graphique.** Histogramme gaussien standard.\n",
    "\n",
    "**Conclusion.** L'auto-MMD nulle est validée.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bb48f55",
   "metadata": {},
   "source": [
    "## Cellule 2 — MMD entre gaussienne et Laplace\n",
    "\n",
    "**Théorème.** Deux distributions de moyenne et variance identiques\n",
    "mais formes différentes ont une MMD strictement positive.\n",
    "\n",
    "**Équation pivot.**\n",
    "$$\\text{MMD}^2(\\mathcal{N}(0, 1), \\text{Laplace}(0, 1/\\sqrt{2})) > 0.$$\n",
    "\n",
    "**Ce que la cellule vérifie.** Sweep en $\\sigma$ : la MMD est\n",
    "maximale autour de $\\sigma$ commensurable à l'échelle des\n",
    "distributions.\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 3,
   "id": "a73d2bf4",
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MMD min : 1.182e-02, MMD max : 1.603e-01\n",
      "σ optimal : 0.38\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 1320x440 with 2 Axes>"
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    }
   ],
   "source": [
    "rng = np.random.default_rng(2)\n",
    "n = 500\n",
    "X = rng.standard_normal(n).tolist()  # N(0, 1)\n",
    "Y = rng.laplace(0.0, 1.0 / np.sqrt(2.0), n).tolist()  # même variance\n",
    "\n",
    "sigmas = np.geomspace(0.1, 10.0, 25)\n",
    "mmds = [float(opt.mmd_gaussian(X, Y, sigma=s)) for s in sigmas]\n",
    "print(f\"MMD min : {min(mmds):.3e}, MMD max : {max(mmds):.3e}\")\n",
    "print(f\"σ optimal : {sigmas[int(np.argmax(mmds))]:.2f}\")\n",
    "\n",
    "fig, axes = plt.subplots(1, 2, figsize=(12, 4))\n",
    "xs = np.linspace(-5, 5, 400)\n",
    "axes[0].hist(X, bins=40, density=True, alpha=0.5,\n",
    "             label=r'$\\mathcal{N}(0,1)$', color='C0')\n",
    "axes[0].hist(Y, bins=40, density=True, alpha=0.5,\n",
    "             label='Laplace', color='C3')\n",
    "axes[0].plot(xs, np.exp(-xs**2/2)/np.sqrt(2*np.pi),\n",
    "             'C0--', lw=1.5)\n",
    "axes[0].plot(xs, np.exp(-np.abs(xs)*np.sqrt(2))*np.sqrt(2)/2,\n",
    "             'C3--', lw=1.5)\n",
    "axes[0].set_xlabel('x'); axes[0].set_ylabel('densité')\n",
    "axes[0].set_title(\"Densités comparées\"); axes[0].legend()\n",
    "axes[1].semilogx(sigmas, mmds, 'o-', lw=2, color='C2')\n",
    "axes[1].set_xlabel(r'bandwidth $\\sigma$')\n",
    "axes[1].set_ylabel(r'$\\widehat{\\text{MMD}}^2$')\n",
    "axes[1].set_title(\"Sensibilité à la bandwidth\")\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6403c0a",
   "metadata": {},
   "source": [
    "**Résultat attendu.** MMD strictement positive, courbe en cloche.\n",
    "\n",
    "**Lecture du graphique.** Gauche : la Laplace est plus pointue ;\n",
    "droite : MMD maximale lorsque $\\sigma \\sim 1$.\n",
    "\n",
    "**Conclusion.** Le choix de bandwidth est crucial ; règle pratique :\n",
    "$\\sigma \\approx \\text{médiane des distances}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdf034d4",
   "metadata": {},
   "source": [
    "## Cellule 3 — Exemple concret : MMD entre mélange et gaussienne\n",
    "\n",
    "**Modèle.** $P = \\mathcal{N}(0, 1)$ vs\n",
    "$Q_\\mu = \\frac{1}{2}\\mathcal{N}(-\\mu, 1) + \\frac{1}{2}\\mathcal{N}(+\\mu, 1)$.\n",
    "La MMD doit *croître* avec $\\mu$.\n",
    "\n",
    "**Équation pivot.**\n",
    "$$\\text{MMD}^2(P, Q_\\mu) \\xrightarrow[\\mu \\to 0]{} 0,\n",
    "   \\qquad \\nearrow \\text{ en } \\mu.$$\n",
    "\n",
    "**Ce que la cellule vérifie.** Pour $\\mu \\in [0, 2.5]$, la courbe\n",
    "MMD est croissante.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "45b8aa9b",
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "μ = 0.00  ->  MMD² = 8.317e-02\n",
      "μ = 0.25  ->  MMD² = 3.606e-02\n",
      "μ = 0.50  ->  MMD² = 9.864e-02\n",
      "μ = 0.75  ->  MMD² = 1.881e-01\n",
      "μ = 1.00  ->  MMD² = 2.075e-01\n",
      "μ = 1.25  ->  MMD² = 3.452e-01\n",
      "μ = 1.50  ->  MMD² = 3.628e-01\n",
      "μ = 1.75  ->  MMD² = 4.763e-01\n",
      "μ = 2.00  ->  MMD² = 5.565e-01\n",
      "μ = 2.25  ->  MMD² = 6.350e-01\n",
      "μ = 2.50  ->  MMD² = 6.936e-01\n"
     ]
    },
    {
     "data": {
      "image/png": "[encoded data, 61708 characters]",
      "text/plain": [
       "<Figure size 1320x440 with 2 Axes>"
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   ],
   "source": [
    "rng = np.random.default_rng(11)\n",
    "n = 400\n",
    "P = rng.standard_normal(n).tolist()\n",
    "mus = np.linspace(0.0, 2.5, 11)\n",
    "\n",
    "mmd_vals = []\n",
    "for mu in mus:\n",
    "    half = n // 2\n",
    "    Q = np.concatenate([\n",
    "        rng.standard_normal(half) - mu,\n",
    "        rng.standard_normal(n - half) + mu,\n",
    "    ]).tolist()\n",
    "    mmd_vals.append(float(opt.mmd_gaussian(P, Q, sigma=1.0)))\n",
    "\n",
    "for mu, m in zip(mus, mmd_vals):\n",
    "    print(f\"μ = {mu:.2f}  ->  MMD² = {m:.3e}\")\n",
    "\n",
    "fig, axes = plt.subplots(1, 2, figsize=(12, 4))\n",
    "axes[0].plot(mus, mmd_vals, 'o-', lw=2, color='C2')\n",
    "axes[0].set_xlabel(r'séparation $\\mu$')\n",
    "axes[0].set_ylabel(r'$\\widehat{\\text{MMD}}^2$')\n",
    "axes[0].set_title(\"MMD croissante avec la séparation\")\n",
    "\n",
    "mu_show = mus[-1]\n",
    "half = n // 2\n",
    "Q_show = np.concatenate([\n",
    "    rng.standard_normal(half) - mu_show,\n",
    "    rng.standard_normal(n - half) + mu_show,\n",
    "])\n",
    "axes[1].hist(P, bins=30, density=True, alpha=0.5,\n",
    "             label=r'P = $\\mathcal{N}(0,1)$', color='C0')\n",
    "axes[1].hist(Q_show, bins=30, density=True, alpha=0.5,\n",
    "             label=fr'Q (μ = {mu_show:.1f})', color='C3')\n",
    "axes[1].set_xlabel('x'); axes[1].set_ylabel('densité')\n",
    "axes[1].set_title(\"P vs Q (mélange séparé)\")\n",
    "axes[1].legend()\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3d9b723",
   "metadata": {},
   "source": [
    "**Résultat attendu.** MMD croît monotone en $\\mu$.\n",
    "\n",
    "**Lecture du graphique.** Gauche : courbe croissante.  Droite : la\n",
    "gaussienne unique vs le mélange bimodal sont visuellement\n",
    "distincts à $\\mu = 2.5$.\n",
    "\n",
    "**Conclusion.** MMD avec noyau gaussien est un détecteur efficace de\n",
    "différence distributionnelle, applicable à la calibration de modèles\n",
    "génératifs (GAN, normalizing flows) en boucle d'apprentissage.\n"
   ]
  }
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