nb08b-model-comparison.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "05080d5f-640d-4a95-bba4-f0dadda33ce5",
   "metadata": {},
   "source": [
    "# `nb08b`: Model comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "e72dc22d-a680-4c48-aebe-20cd35508c77",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ec084d85-4568-4c66-8688-8a08f8823560",
   "metadata": {},
   "source": [
    "# Model comparison using Bayes factors\n",
    "\n",
    "## Marginal likelihood"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e028f340-2850-438a-84f3-f281522a9dd5",
   "metadata": {},
   "source": [
    "$$p(\\theta | x, \\mathcal{M}_1) = \\frac{p(x | \\theta, \\mathcal{M}_1) p(\\theta | \\mathcal{M}_1)}{p(x | \\mathcal{M}_1)}$$\n",
    "\n",
    "$$p(x | \\mathcal{M}_1) = \\int p(x | \\theta, \\mathcal{M}_1) p(\\theta | \\mathcal{M}_1) d\\theta$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb3357a0-b777-4db9-b54b-131bfede3e18",
   "metadata": {},
   "source": [
    "We assume a study where we assess the number of successes observed in a fixed number of trials. For example, we have 80 successes out of 100 trials. \n",
    "\n",
    "A simple model of this data can be built by assuming that the data is distributed according to a binomial distribution. In a binomial distribution, $n$ independent experiments are performed and the result of each experiment is succesful with probability $\\theta$. Then, the binomial distribution is the probability distribution of the number of successes $k$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d50fef37-c596-4886-b3bb-9ae7f6b61120",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Model 1\n",
    "from scipy.stats import binom\n",
    "\n",
    "def likelihood1(theta, k, n):\n",
    "    return binom.pmf(k, n, theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "8c2ddb90-9c1c-46fd-96e9-7c939b77a032",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = plt.axes()\n",
    "\n",
    "theta = 0.6\n",
    "n = 100\n",
    "\n",
    "ks = range(0, n+1)\n",
    "ax.plot(ks, likelihood1(theta, ks, n))\n",
    "ax.set(xlabel=r\"$k$\", ylabel=r\"$p(k | n={}, \\theta={})$\".format(n, theta))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "402e038d-a425-4a92-87f5-70855f8dff0f",
   "metadata": {},
   "source": [
    "Suppose we have prior information about the probability parameter $\\theta$, encoded as a Beta distribution of parameters $a$ and $b$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "44d04c84-fcaf-4edf-ae3d-f1ba104093be",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import beta\n",
    "\n",
    "a = 4\n",
    "b = 2\n",
    "\n",
    "def prior1(theta):\n",
    "    return beta.pdf(theta, a=a, b=b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "4667df1c-db68-4d60-af33-a4d9c8ec952a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = plt.axes()\n",
    "\n",
    "thetas = np.linspace(0, 1, num=100)\n",
    "ax.plot(thetas, prior1(thetas))\n",
    "ax.set(xlabel=r\"$\\theta$\", ylabel=r\"$p(\\theta)$\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40e1e8bf-89b2-4878-a4c6-37c5f24ee054",
   "metadata": {},
   "source": [
    "Then the marginal likelihood is given by the integral of the product of the likelihood and the prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "2395cd7b-2c40-466e-a918-e6c217ccc5bb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.019982271038244702"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import quad\n",
    "ml1, _ = quad(lambda theta: likelihood1(theta, k=80, n=n) * prior1(theta), 0, 1)\n",
    "ml1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "a170d37b-10cb-422c-8f98-d0ddec099879",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.009900990099009892"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Model 2: wider prior\n",
    "def prior2(theta):\n",
    "    return beta.pdf(theta, a=1, b=1)\n",
    "\n",
    "ml2, _ = quad(lambda theta: likelihood1(theta, k=80, n=n) * prior2(theta), 0, 1)\n",
    "ml2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "a5d32917-e629-4e81-b644-53b84b95448a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Model 3: more complex likelihood\n",
    "from scipy.stats import betabinom, lognorm\n",
    "\n",
    "def likelihood3(alpha, beta, k, n):\n",
    "    return betabinom.pmf(k, n, alpha, beta)\n",
    "\n",
    "def prior3(alpha, beta):\n",
    "    return lognorm.pdf(alpha, 100) * lognorm.pdf(beta, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "9582cd84-df61-40a3-871c-558249d40087",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = plt.axes()\n",
    "\n",
    "vs = np.linspace(0, 100)\n",
    "ax.plot(vs, lognorm.pdf(vs, 100))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "600c6588-4d31-4795-8f4a-1e3403864287",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/glouppe/anaconda3/envs/dats0001/lib/python3.9/site-packages/scipy/integrate/_quadpack_py.py:1233: IntegrationWarning: The integral is probably divergent, or slowly convergent.\n",
      "  quad_r = quad(f, low, high, args=args, full_output=self.full_output,\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.750979990594095e-06"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import dblquad\n",
    "ml3, _ = dblquad(lambda alpha, beta: likelihood3(alpha, beta, k=80, n=n) * prior3(alpha, beta), 0, np.inf, 0, np.inf)\n",
    "ml3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "53b049dc-78c5-478c-a8f4-ab548f68a331",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model 1: 0.019982271038244702\n",
      "Model 2: 0.009900990099009892\n",
      "Model 3: 5.750979990594095e-06\n"
     ]
    }
   ],
   "source": [
    "print(\"Model 1:\", ml1)\n",
    "print(\"Model 2:\", ml2)\n",
    "print(\"Model 3:\", ml3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d005ffc-8c56-4ce2-96a8-7cb615fe114f",
   "metadata": {},
   "source": [
    "## Bayes factor"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d5e8f4b-23dc-4b34-b1a4-5aea50ae74c2",
   "metadata": {},
   "source": [
    "$$\\text{BF}_{1,2} = \\frac{p(x | \\mathcal{M}_1)}{p(x | \\mathcal{M}_2)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "68ab30c5-3e39-4a1b-9819-9e2f09472c59",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "BF(1,2): 2.0182093748627166 0.7022106701979013\n",
      "BF(1,3): 3474.585387347256 8.153230437886936\n",
      "BF(2,3): 1721.617900810517 7.451019767689035\n"
     ]
    }
   ],
   "source": [
    "print(\"BF(1,2):\", ml1 / ml2, np.log(ml1 / ml2))\n",
    "print(\"BF(1,3):\", ml1 / ml3, np.log(ml1 / ml3))\n",
    "print(\"BF(2,3):\", ml2 / ml3, np.log(ml2 / ml3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8571a89f-a9d8-4d85-a512-1983f9ad7a16",
   "metadata": {},
   "source": [
    "If we want to know, given the data $x$, how much more probable model $\\mathcal{M}_1$ is than model $\\mathcal{M}_2$, then we need the prior odds:\n",
    "\n",
    "$$\\frac{p(\\mathcal{M}_1 | x)}{p(\\mathcal{M}_2 | x)} = \\frac{p(\\mathcal{M}_1)}{p(\\mathcal{M}_2)} \\frac{p(x | \\mathcal{M}_1)}{p(x | \\mathcal{M}_2)} = \\frac{p(\\mathcal{M}_1)}{p(\\mathcal{M}_2)} \\text{BF}_{1,2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51d20df7-e069-4209-bf54-8fc080e08e91",
   "metadata": {},
   "source": [
    "![](figures/nb11/bf.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f97fed4c-f739-46e0-ae18-77a5c8104857",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    \n",
    "**Exercise**. Using the light speed experiment data from `nb08a`, compare:\n",
    "- $\\mathcal{M}_1: p(x | \\mu, \\sigma) = \\mathcal{N}(x | \\mu, \\sigma), p(\\mu, \\sigma) = \\mathcal{U}$\n",
    "- $\\mathcal{M}_2: p(x | \\mu, \\sigma, a) = \\text{skewnorm}(x | \\mu, \\sigma, a), p(\\mu, \\sigma, a) = \\mathcal{U}$.\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4126d5ad-a9d0-4c2d-a05a-7dc3cb72180f",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "ff3f4fa7-c60d-4b9b-ac62-a162f396f606",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    \n",
    "**Exercise**. How to compute the Bayes factor for more complicated models?\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e890697c-8a53-404c-95c2-7faeb2d9a3e5",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "bfed2a7d-bc08-4fc5-aeab-4e9c48ae1b4a",
   "metadata": {},
   "source": [
    "# Cross-validation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2027fedb-3bcc-4f89-8232-fcbad320e398",
   "metadata": {},
   "source": [
    "A popular way to evaluate and compare models is on their ability to make predictions for future or unseen observations $x_\\text{pred}$, using what we learned from the observed data $\\{x\\} = \\{x_1, ..., x_n\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4237e60a-f71c-4889-ba3d-8d078c579977",
   "metadata": {},
   "source": [
    "## Expected log predictive density"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e26e8a8c-4099-491f-b96b-53d6f10e5931",
   "metadata": {},
   "source": [
    "Let $$\\log p(x_\\text{pred} | \\{x\\}, \\mathcal{M}_1) = \\log \\int p(x_\\text{pred} | \\theta, \\mathcal{M}_1) p(\\theta | \\{x\\}, \\mathcal{M}_1) d\\theta $$\n",
    "\n",
    "$$\\text{elpd} = \\mathbb{E}_{p_\\text{true}(x_\\text{pred})} [ \\log p(x_\\text{pred} | \\{x\\}, \\mathcal{M}_1) ]$$\n",
    "\n",
    "where $p_\\text{true}$ is the true data generating distribution. \n",
    "\n",
    "The model with the highest $\\text{elpd}$ is the the model that best explains data from the true data generating distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eeb90917-766b-4244-8ca2-eb102facd4e8",
   "metadata": {},
   "source": [
    "But we we don't know $p_\\text{true}$! Solution: use the $n$ observations from $\\{x\\}$:\n",
    "\n",
    "$$\\widehat{\\text{elpd}} = \\frac{1}{n} \\sum_{i=1}^n \\log p(x_i | \\{x\\}, \\mathcal{M}_1)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "b348864b-0913-400d-97a1-06cf71d467ea",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Example\n",
    "theta_true = np.array([1.0, 3.0])\n",
    "p_true = beta(a=theta_true[0], b=theta_true[1])\n",
    "\n",
    "np.random.seed(42)\n",
    "n = 50\n",
    "xs = p_true.rvs(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "eae78204-705f-40f9-acbe-127d88bf1ddf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = plt.axes()\n",
    "\n",
    "vs = np.linspace(0, 1)\n",
    "ax.plot(vs, p_true.pdf(vs))\n",
    "ax.hist(xs, density=True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "18744c51-c17c-4414-b5f3-bb2b0626e963",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Model 1\n",
    "def log_likelihood(theta, x):\n",
    "    return beta(theta[0], theta[1]).logpdf(x).sum()\n",
    "\n",
    "def log_prior(theta):\n",
    "    return lognorm.pdf(theta[0], 10) * lognorm.pdf(theta[1], 10)\n",
    "\n",
    "def log_posterior(theta, x):\n",
    "    if theta[0] <= 0 or theta[1] <= 0:\n",
    "        return -np.inf\n",
    "    else:                          \n",
    "        return log_likelihood(theta, x) + log_prior(theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "16ae458e-198a-464b-af6a-6d431d9d4cde",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 1000/1000 [00:03<00:00, 307.48it/s]\n"
     ]
    }
   ],
   "source": [
    "# Fit the posterior\n",
    "import emcee\n",
    "pos = np.array([1.0, 1.0]) + 0.1 * np.random.randn(5, 2)\n",
    "nwalkers, ndim = pos.shape\n",
    "\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=(xs,))\n",
    "sampler.run_mcmc(pos, 1000, progress=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "2a6883bc-4f0d-4e46-bf8b-a6d9aa67ed9c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4500\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 550x550 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import corner\n",
    "thetas = sampler.get_chain(flat=True, discard=100)\n",
    "print(len(thetas))\n",
    "fig = corner.corner(thetas, labels=[\"a\", \"b\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "861476c5-aa46-4e23-a561-555da2d7c458",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5680835505118703"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# elpd\n",
    "def log_posterior_predictive(x_i, thetas):\n",
    "    return np.mean([log_likelihood(theta, x_i) for theta in thetas])\n",
    "\n",
    "def elpd(xs, thetas):\n",
    "    return np.mean([log_posterior_predictive(x_i, thetas) for x_i in xs])\n",
    "\n",
    "subset = np.random.permutation(len(thetas))[:500]\n",
    "elpd(xs, thetas[subset])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0d10b96-1397-4bc5-8030-336880937d95",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    \n",
    "**Exercise**. What's wrong with this procedure?\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "806bf9fc-f9f3-4f08-bd5b-2dc6a49c32e3",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "cafde425-e4cc-411b-a8f6-0833e1e11eea",
   "metadata": {},
   "source": [
    "## K-fold validation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "9b12efe9-bb13-411f-8b33-6eb61633fff8",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 1000/1000 [00:03<00:00, 277.18it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 309.98it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 308.21it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 299.44it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 300.74it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 307.45it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 299.70it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 306.38it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 304.93it/s]\n",
      "100%|██████████| 1000/1000 [00:03<00:00, 305.82it/s]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.model_selection import KFold\n",
    "kf = KFold(n_splits=10, shuffle=True)\n",
    "elpd_test = []\n",
    "\n",
    "for train_idx, test_idx in kf.split(xs):\n",
    "    # posterior fit\n",
    "    pos = np.array([1.0, 1.0]) + 0.1 * np.random.randn(5, 2)\n",
    "    nwalkers, ndim = pos.shape\n",
    "    sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=(xs[train_idx],))\n",
    "    sampler.run_mcmc(pos, 1000, progress=True);\n",
    "    thetas = sampler.get_chain(flat=True, discard=100)\n",
    "    \n",
    "    # epld\n",
    "    subset = np.random.permutation(len(thetas))[:500]\n",
    "    elpd_test.append(elpd(xs[test_idx], thetas[subset]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "0a32f432-fcac-43ef-af81-7214ccfca624",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.05999472269935764,\n",
       " 0.4545898351603725,\n",
       " 0.6365955085439219,\n",
       " 0.7492708843483227,\n",
       " 0.604772863122996,\n",
       " 0.5974314578347603,\n",
       " 0.6511026894641978,\n",
       " 0.5477261478813987,\n",
       " 0.5739421571578041,\n",
       " -0.2091810595966206]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "elpd_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "5626a279-8ca8-4327-8e02-8d7203a1dd00",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.46662452066165105, 0.28623540579811957)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(elpd_test), np.std(elpd_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2800ca6e-88b5-4806-b51e-6733c679b39b",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    \n",
    "**Exercise**. Evaluate the K-Fold estimate as a function of $K$. \n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "93777fd5-6c3b-4c24-b778-4207d6642543",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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