python/ch13/13.5-bayesian-inference-unknown-variance.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "dcd252ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch.distributions import Normal, Gamma\n",
    "import math\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f8c615f",
   "metadata": {},
   "source": [
    "## Bayesian Inferencing of Precision of Gaussian likelihood, known Mean\n",
    "\n",
    "We will study the case where the mean is known but the variance is unknown and expressed as a random variable. The computations become quite a bit simpler if we use precision (inverse of variance).\n",
    "\n",
    "We will only express the variance as a random variable while the mean will be a constant.\n",
    "\n",
    "Precision, $\\lambda$ which is related to variance $\\sigma$ as $\\lambda = \\frac{ 1 } { \\sigma^{2} }$.\n",
    "\n",
    "The likelihood term becomes\n",
    "$$\n",
    "p\\left( X \\middle\\vert \\lambda\\right) = \\frac{ \\lambda^{ \\frac{ n }{ 2 } } } { { \\sqrt {2\\pi }  } } e^{ -\\frac{ \\lambda }{ 2 } \\sum_{i=1}^n \\left( {x^{ \\left( i \\right) } - \\mu } \\right)^2 }$$\n",
    "\n",
    "The prior for the precision is a Gamma distribution\n",
    "$$\n",
    "p\\left( \\lambda \\right)  = \\frac{ \\beta_{0}^{ \\alpha_{0} } }{ \\Gamma\\left( \\alpha_{0} \\right) } \\lambda ^{ \\left( \\alpha_{0} - 1 \\right)} e^{ - \\beta_{0} \\lambda  }\n",
    "$$\n",
    "\n",
    "The corresponding posterior is also a Gamma distribution, such that \n",
    "\n",
    "$$p\\left( \\lambda \\middle\\vert X \\right) = \\frac{ \\beta_{n}^{ \\alpha_{n} } }{ \\Gamma\\left( \\alpha_{n} \\right) } \\lambda ^{ \\left( \\alpha_{n} - 1 \\right)} e^{ - \\beta_{n} \\lambda} $$\n",
    "\n",
    "where \n",
    "$$ \\alpha_{n}  = \\frac{ n }{ 2 } + \\alpha_{0} \\\\\n",
    "\\beta_{n} = \\frac{ 1 }{ 2 } \\sum_{i=1}^{n} \\left( {x^{ \\left( i \\right) } - \\mu } \\right)^2 + \\beta_{0} = \\frac{ n }{ 2 } s + \\beta_{0} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b93b1898",
   "metadata": {},
   "outputs": [],
   "source": [
    "def inference_unknown_variance(X, prior_dist):\n",
    "    sigma_mle = torch.std(X)\n",
    "    n = X.shape[0]\n",
    "    \n",
    "    # Parameters of the prior\n",
    "    alpha_0 = prior_dist.concentration\n",
    "    beta_0 = prior_dist.rate\n",
    "    \n",
    "    # Parameters of posterior\n",
    "    alpha_n = n / 2 + alpha_0\n",
    "    beta_n = n / 2 * sigma_mle ** 2 + beta_0\n",
    "    posterior_dist = Gamma(alpha_n, beta_n)\n",
    "    return posterior_dist"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f48042c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_distribution(dists, legend, precisions, precision_legend, title, xlim=(0, 4)):\n",
    "    fig, ax = plt.subplots(dpi=400)\n",
    "    ax.set_title(title)\n",
    "    ax.set_ylabel(\"PDF(X)\")\n",
    "    ax.set_xlabel(\"X\")\n",
    "    y_lim = 0\n",
    "    for dist in dists:\n",
    "        x = torch.linspace(0.01, 10, 1000)\n",
    "        pdf = dist.log_prob(x).exp()\n",
    "        ax.plot(x, pdf)\n",
    "        y_lim = max(torch.max(pdf), y_lim)\n",
    "    for precision in precisions:\n",
    "        ax.plot([precision] * len(x) , x, '--')\n",
    "    ax.set_xlim(*xlim)\n",
    "    ax.set_ylim(0, y_lim+0.01)\n",
    "    legend.extend(precision_legend)\n",
    "    ax.legend(legend, loc='upper right')\n",
    "    return ax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2bc34b09",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Let us assume that the true distribution is a normal distribution. The true distribution corresponds \n",
    "# to a single class.\n",
    "mu_known = 20\n",
    "true_dist = Normal(mu_known, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9290059c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True precision: 0.04\n",
      "MAP precision: 0.14\n",
      "MLE precision: 3.70\n"
     ]
    }
   ],
   "source": [
    "# Case 1\n",
    "# Let us assume our prior is a Gamma distribution with a good estimate of the variance\n",
    "prior_dist = Gamma(1, 10)\n",
    "\n",
    "# Let us set a seed for reproducability\n",
    "torch.manual_seed(42)\n",
    "\n",
    "# Number of samples is low. \n",
    "n = 3\n",
    "X = true_dist.sample((n,))\n",
    "posterior_dist_low_n = inference_unknown_variance(X, prior_dist)\n",
    "\n",
    "precision_mle = (1 / X.std()**2)\n",
    "\n",
    "true_precision = (1/ true_dist.scale**2)\n",
    "\n",
    "max_posterior_precision = (posterior_dist_low_n.concentration-1) / posterior_dist_low_n.rate\n",
    "\n",
    "# When n is low, the posterior is dominated by the prior. Thus, a good prior can help offset the lack of data.\n",
    "# We can see this in the following case. \n",
    "\n",
    "# With a small sample (n=3), the MLE estimate of precision is 3.70, which is way off from the true value of 0.04\n",
    "# Using a good prior here helps offset it. The MAP estimate of precision, 0.14 is much better.  \n",
    "\n",
    "print(f\"True precision: {true_precision:0.2f}\")\n",
    "print(f\"MAP precision: {max_posterior_precision:0.2f}\")\n",
    "print(f\"MLE precision: {precision_mle:0.2f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c910669c",
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 2560x1920 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dists = [prior_dist, posterior_dist_low_n]\n",
    "legend = [\"Prior\", \"Posterior n=3\"]\n",
    "\n",
    "precisions = [true_precision, precision_mle, max_posterior_precision]\n",
    "legend_p = [\"True Precision\", \"MLE Precision\", \"MAP Precision\"]\n",
    "\n",
    "plot_distribution(dists, legend, precisions, legend_p, \"Low data n=3\",)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "04d050cd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True precision: 0.04\n",
      "MAP precision: 0.04\n",
      "MLE precision: 0.04\n"
     ]
    }
   ],
   "source": [
    "# Case 2\n",
    "# Let us assume our prior is a Gamma distribution with a good estimate of the variance\n",
    "prior_dist = Gamma(1, 10)\n",
    "\n",
    "# Let us set a seed for reproducability\n",
    "torch.manual_seed(42)\n",
    "\n",
    "# Number of samples is high. \n",
    "n = 100\n",
    "X = true_dist.sample((n,))\n",
    "posterior_dist_high_n = inference_unknown_variance(X, prior_dist)\n",
    "\n",
    "precision_mle = (1 / X.std()**2)\n",
    "\n",
    "true_precision = (1/ true_dist.scale**2)\n",
    "\n",
    "max_posterior_precision = (posterior_dist_high_n.concentration-1) / posterior_dist_high_n.rate\n",
    "\n",
    "\n",
    "# When n is high, the MLE tends to converge to the true distribution. The MAP also tends to converge to the MLE, \n",
    "# and in turn converges to the true distribution\n",
    "print(f\"True precision: {true_precision:0.2f}\")\n",
    "print(f\"MAP precision: {max_posterior_precision:0.2f}\")\n",
    "print(f\"MLE precision: {precision_mle:0.2f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1d2ba1d",
   "metadata": {},
   "source": [
    "### How to use the  estimated variance parameter?\n",
    "\n",
    "As before, we will find the parameter value that maximizes the posterior density. Here we will maximize the posterior probability density function for the precision $p\\left( \\lambda \\middle\\vert X \\right) = \\lambda^{ \\left( \\alpha_{n} - 1 \\right) } e^{ -\\beta_{n} \\lambda }$ by taking the derivative and equating to zero.\n",
    "\n",
    "This gives us $$\\lambda = \\frac{ \\alpha - 1 }{ \\beta }$$\n",
    "\n",
    "Thus, our estimated distribution for the class of points corresponding to the training data is $\\mathcal{N}\\left( x ; \\;\\; \\mu, \\sigma_{n} \\right)$ where $\\frac{1}{ \\sigma_{n}^{2} } = \\left( \\frac{ \\alpha_{n} - 1 }{ \\beta_{n}  } \\right)^{ -\\frac{1}{2} }$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a931a554",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MAP distribution mu: 20.00 std:4.95\n"
     ]
    }
   ],
   "source": [
    "map_precision = (posterior_dist_high_n.concentration-1) / posterior_dist_high_n.rate\n",
    "map_dist = Normal(mu_known, 1/ math.sqrt(map_precision))\n",
    "print(f\"MAP distribution mu: {map_dist.mean:0.2f} std:{map_dist.scale:0.2f}\")"
   ]
  }
 ],
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   "language": "python",
   "name": "python3"
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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