Inference.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<torch._C.Generator at 0x7f19681b6b70>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import torch\n",
    "import pyro\n",
    "import pyro.infer as infer\n",
    "import pyro.optim\n",
    "import pyro.distributions as dist\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline \n",
    "\n",
    "torch.manual_seed(101)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def scale(guess):\n",
    "    # The prior over the weights encodes the uncertainity about our guess\n",
    "    w  = pyro.sample('weight', dist.Normal(guess, 1.0))\n",
    "    # The measurement fluctuates around the true weight\n",
    "    m = pyro.sample('measurement', dist.Normal(w, 0.75))\n",
    "    return m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Marginalization in pyro\n",
    "posterior = infer.Importance(scale, num_samples=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(8.0281)\n"
     ]
    }
   ],
   "source": [
    "# Run inference for a single input value\n",
    "\n",
    "\n",
    "guess = 8.5\n",
    "\n",
    "marginal = infer.EmpiricalMarginal(posterior.run(guess))\n",
    "print(marginal())\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'#')"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1916c15588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist([marginal().item() for _ in range(100)], range=(5.0, 12.0))\n",
    "plt.title(\"P(measurement | guess)\")\n",
    "plt.xlabel(\"weight\")\n",
    "plt.ylabel(\"#\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Conditioning Models on Data\n",
    "\n",
    "\n",
    "conditioned_scale = pyro.condition(\n",
    "    scale, data={\"measurement\": 9.5})\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def deferred_conditioned_scale(measurement, *args, **kwargs):\n",
    "    return pyro.condition(scale, data={\"measurement\": measurement})(*args, **kwargs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def scale2(guess):\n",
    "    weight = pyro.sample(\"weight\", dist.Normal(guess, 1.))\n",
    "    tolerance = torch.abs(pyro.sample(\"tolerance\", dist.Normal(0., 1.)))\n",
    "    return pyro.sample(\"measurement\", dist.Normal(weight, tolerance))\n",
    "\n",
    "\n",
    "# conditioning composes:\n",
    "# the following are all equivalent and do not interfere with each other\n",
    "conditioned_scale2_1 = pyro.condition(\n",
    "    pyro.condition(scale2, data={\"weight\": 9.2}),\n",
    "    data={\"measurement\": 9.5})\n",
    "\n",
    "conditioned_scale2_2 = pyro.condition(\n",
    "    pyro.condition(scale2, data={\"measurement\": 9.5}),\n",
    "    data={\"weight\": 9.2})\n",
    "\n",
    "conditioned_scale2_3 = pyro.condition(\n",
    "    scale2, data={\"weight\": 9.2, \"measurement\": 9.5})\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(9.2143)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'#')"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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6AHgtcK2kJWXYe4GjJc0GDCwHjm2xhoiImECbZw1dCmiUUd9sq82IiFh/ubI4IqJyCYKI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\ngoiIyiUIIiIqlyCIiKjcfwHUm5kzsMFAxwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1916bfb518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "guess = 8.5\n",
    "measurement = 9.5\n",
    "\n",
    "conditioned_scale = pyro.condition(scale, data={\"measurement\": measurement})\n",
    "\n",
    "marginal = pyro.infer.EmpiricalMarginal(\n",
    "    pyro.infer.Importance(conditioned_scale, num_samples=100).run(guess), sites=\"weight\")\n",
    "\n",
    "# The marginal distribution concentrates around the data\n",
    "print(marginal())\n",
    "plt.hist([marginal().item() for _ in range(100)], range=(5.0, 12.0))\n",
    "plt.title(\"P(weight | measurement, guess)\")\n",
    "plt.xlabel(\"weight\")\n",
    "plt.ylabel(\"#\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The above demonstrated method is slow so we would instead use a guide distribution that try to sample from it\n",
    "\"\"\"\n",
    "Guide function should generally be chosen so that it closely approximates the distribution over all unobserved sample statements in the model. \n",
    "The simplest guide for deferred_conditioned_scale matches the prior distribution over weight\n",
    "\"\"\"\n",
    "\n",
    "def scale_prior_guide(guess):\n",
    "    return pyro.sample(\"weight\", dist.Normal(guess, 1.))\n",
    "\n",
    "posterior = pyro.infer.Importance(conditioned_scale,\n",
    "                                  guide=scale_prior_guide,\n",
    "                                  num_samples=10)\n",
    "\n",
    "marginal = pyro.infer.EmpiricalMarginal(posterior.run(guess), sites=\"weight\")\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# We can actually do better since we can write the true posterior analytically\n",
    "\n",
    "\n",
    "def scale_posterior_guide(measurement, guess):\n",
    "    # note that torch.size(measurement, 0) is the total number of measurements\n",
    "    # that we're conditioning on\n",
    "    a = (guess + torch.sum(measurement)) / (measurement.size(0) + 1.0)\n",
    "    b = 1. / (measurement.size(0) + 1.0)\n",
    "    return pyro.sample(\"weight\", dist.Normal(a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dPd3bTqYBSTsBnwXeavvuic5ne5HtebbnDQwMTKbJiIiYhDGDQNLbJT0DeGnP4KsmunBJ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\nSxBERFQuQRARUbn/D3YjRd+Vq/LaAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1916bfbd30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "posterior = pyro.infer.Importance(deferred_conditioned_scale,\n",
    "                                  guide=scale_posterior_guide,\n",
    "                                  num_samples=20)\n",
    "\n",
    "marginal = pyro.infer.EmpiricalMarginal(posterior.run(torch.tensor([measurement]), guess), sites=\"weight\")\n",
    "plt.hist([marginal().item() for _ in range(100)], range=(5.0, 12.0))\n",
    "plt.title(\"P(weight | measurement, guess)\")\n",
    "plt.xlabel(\"weight\")\n",
    "plt.ylabel(\"#\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before guide parameter optimization \ntensor([ 9.1251])\ntensor([ 0.6701])\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " After applying stochastic variational inference \ntensor([ 9.1216])\ntensor([-0.5954])\n"
     ]
    },
    {
     "data": {
      "image/png": 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jrvrfWUklKU9pSm6mMHbsaWDFpt1c/7T3JKBBTJm3jtPunMarizbkMWWFYwHD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\ngGGMMSYQCxjGGGMC+f8ipzz21FBcvwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f19169d5dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Parametrized stochastic functions and variational inference\n",
    "\"\"\"\n",
    "Although we could write out the exact posterior distribution for scale, in general it is intractable to specify a guide that is a good approximation to\n",
    "the posterior distribution of an arbitrary conditioned stochastic function. What we can do instead is use the top-level function pyro.param to specify\n",
    "a family of guides indexed by named parameters, and search for \n",
    "the member of that family that is the best approximation. This approach to approximate posterior inference is called variational inference.\n",
    "\n",
    "\"\"\"\n",
    "\n",
    "\n",
    "\n",
    "def scale_parametrized_guide(guess):\n",
    "    a = pyro.param(\"a\", torch.tensor(torch.randn(1) + guess))\n",
    "    b = pyro.param(\"b\", torch.randn(1))\n",
    "    return pyro.sample(\"weight\", dist.Normal(a, torch.abs(b)))\n",
    "\n",
    "print(\"Before guide parameter optimization \")\n",
    "print(pyro.param(\"a\"))\n",
    "print(pyro.param(\"b\"))\n",
    "\n",
    "\n",
    "pyro.clear_param_store()\n",
    "svi = pyro.infer.SVI(model=conditioned_scale,\n",
    "                     guide=scale_parametrized_guide,\n",
    "                     optim=pyro.optim.SGD({\"lr\": 0.001}),\n",
    "                     loss=pyro.infer.Trace_ELBO())\n",
    "\n",
    "losses = []\n",
    "for t in range(1000):\n",
    "    losses.append(svi.step(guess))\n",
    "\n",
    "plt.plot(losses)\n",
    "plt.title(\"ELBO\")\n",
    "plt.xlabel(\"step\")\n",
    "plt.ylabel(\"loss\");\n",
    "\n",
    "\n",
    "print(\" After applying stochastic variational inference \")\n",
    "print(pyro.param(\"a\"))\n",
    "print(pyro.param(\"b\"))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f191d3bc940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "posterior = pyro.infer.Importance(conditioned_scale, scale_parametrized_guide, num_samples=10)\n",
    "marginal = pyro.infer.EmpiricalMarginal(posterior.run(guess), sites=\"weight\")\n",
    "\n",
    "plt.hist([marginal().item() for _ in range(100)], range=(5.0, 12.0))\n",
    "plt.title(\"P(weight | measurement, guess)\")\n",
    "plt.xlabel(\"weight\")\n",
    "plt.ylabel(\"#\")\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f19150988d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# We can also sample from the guide as an approximate posterior\n",
    "plt.hist([scale_parametrized_guide(guess).item() for _ in range(100)], range=(5.0, 12.0))\n",
    "plt.title(\"P(weight | measurement, guess)\")\n",
    "plt.xlabel(\"weight\")\n",
    "plt.ylabel(\"#\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}