Presentation for Pittsburgh Python Meetup, May 2015

PghPythonBayes.ipynb

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   "source": [
    "# Bayesian Methods in Python\n",
    "- Pittsburgh Python May 2015\n",
    "- Louis Luangkesorn <[email protected]>\n"
   ]
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   "source": [
    "%matplotlib inline"
   ]
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   "source": [
    "# Introduction\n",
    "\n",
    "-  Frequentist and Bayesian views of probability\n",
    "    -  Differences\n",
    "    -  Implications\n"
   ]
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    "# Frequentism vs. Bayesianism: a Philosophical Debate\n",
    "\n",
    "- Fundamentally, the disagreement between frequentists and Bayesians concerns the definition of probability.\n",
    "- For frequentists, probability only has meaning in terms of a limiting case of repeated measurements. \n",
    "  - In the limit of a large number of measurements, the frequency of any given value indicates the probability of measuring that value. \n",
    "  - For frequentists probabilities are fundamentally related to frequencies of events. \n",
    "  - In a strict frequentist view, it is meaningless to talk about the probability of the true value of a parameter.\n"
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    "# Bayesian philosophy\n",
    "\n",
    "-  For Bayesians, the concept of probability is extended to cover degrees of certainty about statements. \n",
    "-  The probability is a statement of my knowledge of what the measurement result will be. \n",
    "-  Probabilities are fundamentally related to our own knowledge about an event. \n",
    "-  In a Bayesian view, we can meaningfully talk about the probability that the true value of a parameters lies within a given range"
   ]
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    "#  Implications\n",
    "\n",
    "-  What if you need to estimate a probability of something never before seen?\n",
    "-  Frequentist - To estimate a probability, need to observe many events in the setting in question.\n",
    "-  Bayesian - We can use our knowledge about similar events to make statements about this event.\n",
    "-  Frequentist - We can answer if a hypothesis about a parameter is true or false.\n",
    "-  Bayesian - We return a probability distribution about a parameter."
   ]
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    "# Bayesian modeling\n",
    "\n",
    "-  What is Bayesian Statistics?\n",
    "-  Probability in Bayesian statistics.\n",
    "-  Results of Bayesian analysis.\n",
    "-  Why is this important."
   ]
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    "# Bayesian statistics\n",
    "\n",
    "[Bayes' Theorem](http://en.wikipedia.org/wiki/Bayes\\'_theorem), a fundamental law of probability:\n",
    "\n",
    "$$ P(F_{\\rm true}~|~D) = \\frac{P(D~|~F_{\\rm true})~P(F_{\\rm true})}{P(D)} $$\n",
    "\n",
    "Though Bayes' theorem is where Bayesians get their name, it is not this law itself that is controversial, but the Bayesian *interpretation of probability* implied by the term $P(F_{\\rm true}~|~D)$.\n",
    "\n",
    "Terms in Bayes' theorem\n",
    "\n",
    "- $P(F_{\\rm true})$: The **model prior**, which encodes what we knew about the model without the data $D$.\n",
    "- $P(F_{\\rm true}~|~D)$: The **posterior**, or the probability of the model parameters given the data.\n",
    "- $P(D~|~F_{\\rm true})$: The **likelihood**, which is proportional to the $\\mathcal{L}(D~|~F_{\\rm true})$.\n",
    "- $P(D)$: The **probability of the data**, usually a normalization term."
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    "- $P(F_{\\rm true})$: The **model prior**.\n",
    "- $P(F_{\\rm true}~|~D)$: The **posterior**.\n",
    "- $P(D~|~F_{\\rm true})$: The **likelihood**.\n",
    "- $P(D)$: The **probability of the data**.\n",
    "\n",
    "- Evaluating probabilities is usually done by taking integrals (areas).\n",
    "- But for most of these terms, there is no general closed form solution.\n",
    "- In practice these integrals are solved using Monte Carlo Simulation"
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    "# Specifying Bayesian models\n",
    "\n",
    "-  Priors\n",
    "-  Hyper priors\n",
    "-  Acyclic Directed graphs"
   ]
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    "# Prior\n",
    "\n",
    "-  Represents our previous knowledge about the system under considersation.\n",
    "-  It could be a probability distribution that we believe the actual parameter is within.\n",
    "-  It could be an understanding that this type of system follows a particular probability distribution.\n",
    "-  "
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    "#  Python implementations\n",
    "\n",
    "- emcee (http://dan.iel.fm/emcee/current/)\n",
    "    -  Affine Invariant Markov chain Monte Carlo (MCMC)\n",
    "- PyMC (https://github.com/pymc-devs/pymc)\n",
    "    - Metropolis-Hastings\n",
    "- PyStan (http://mc-stan.org/)\n",
    "    - Hamiltonian Monte Carlo"
   ]
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   "source": [
    "# emcee\n",
    "\n",
    "- http://dan.iel.fm/emcee/current/\n",
    "- Pure Python implementation of Affine Invariant Markov chain Monte Carlo (MCMC) Ensemble sampler \n",
    "- Also include Metropolis-Hastings sampler\n",
    "- Python 2 or Python 3\n",
    "- Advantages\n",
    "    -  Pure Python\n",
    "    -  Faster\n",
    "- Disadvantages\n",
    "    -  Less flexibility in models that it can handle\n",
    "    -  Limited modeling features"
   ]
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   "source": [
    "# PyMC\n",
    "\n",
    "-  Requires Fortran and C compilers for building\n",
    "-  Binaries available for Windows and OSX\n",
    "-  Metropolis-Hasting sampler\n",
    "-  Used in Cam Davidson-Pilon, Probabilistic Programming for Hackers by https://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/\n",
    "-  Kiran R Karkera, (2014), Building Probabilistic Graphical Models with Python, Packt Publishing\n",
    "-  Advantages:\n",
    "    -  Lots of functionality built into Python\n",
    "    -  Models in Python, but there is some PyMC specific boilerplate.\n",
    "-  Disadvantages\n",
    "    -  Requires a Fortran compiler to install. "
   ]
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   "source": [
    "# PyStan\n",
    "\n",
    "-  Interface to Stan (http://mc-stan.org)\n",
    "    -  Domain Specific Language for Bayesian Monte Carlo\n",
    "    -  Successor to Bayesian inference Using Gibbs Sampling (BUGS)\n",
    "    -  Hamiltonian Monte Carlo (No U-Turn Sampler)\n",
    "    -  Written in C++, can also be used from R, Julia, Matlab, and Command line\n",
    "-  Requires a C++ compiler to be available at runtime.\n",
    "    -  Windows users can use Visual C++ (see PyStan website for additional instructions)\n",
    "    -  Available in Python 2 and Python 3\n",
    "-  Advantages\n",
    "    -  Well documented and tested\n",
    "    -  Stan is very flexible, \n",
    "    -  APIs in multiple languages provides pathway from prototyping to deployment.\n",
    "-  Disadvantages\n",
    "    -  Requires learning a DSL (Stan, which is R-like)\n",
    "    -  Requires a C++ compiler at runtime"
   ]
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   "source": [
    "# Example\n",
    "\n",
    "-  Problem: characterizing the production disruption process of a manufacturing system.\n",
    "-  Data: monthly lost time records from comparable facilities"
   ]
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     "text": [
      "19.3363872143\n"
     ]
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   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "monthlytime = np.array([  2.49179399e+01,   1.64994543e+01,   9.24604431e+00,\n",
    "         9.63146345e+00,   3.15720802e+01,   2.60361425e+01,\n",
    "         2.99447617e+01,   1.50387235e+01,   2.23899953e+01,\n",
    "         2.22204538e+01,   1.52221442e+01,   2.21336043e+01,\n",
    "         2.77345387e+01,   6.75676904e+00,   5.00959730e+00,\n",
    "         1.43230173e+00,   2.47980619e+01,   1.32841408e+01,\n",
    "         2.16228971e+01,   4.31069909e+01,   2.48117842e+00,\n",
    "         1.55881125e+01,   1.30156262e+01,   4.14153586e+00,\n",
    "         4.91830857e+01,   1.71423527e+01,   6.72986886e+00,\n",
    "         1.25622619e+01,   3.04378224e+01,   3.94550102e+01,\n",
    "         1.64819634e+01,   0.00000001e+00,   4.07651601e+00,\n",
    "         6.82764325e+00,   2.69020758e+00,   1.33089319e+01,\n",
    "         1.61896126e+01,   0.00000001e+00,   3.19534026e+00,\n",
    "         1.18964033e+01,   3.56682091e+01,   5.63098156e+00,\n",
    "         3.98295925e+01,   3.02406018e+01,   2.80450681e+00,\n",
    "         8.69825190e+00,   8.96207765e-01,   1.74601502e+01,\n",
    "         2.99631808e+01,   2.66285005e+01,   1.51351362e+01,\n",
    "         2.06077708e+01,   8.12959939e+00,   1.80830912e+01,\n",
    "         2.73655836e+01,   2.52443775e+01,   2.42019651e+01,\n",
    "         1.85799207e+01,   2.78341166e+01,   9.58892117e+00,\n",
    "         1.41181959e+01,   9.99556068e+00,   3.61166832e+01,\n",
    "         3.28856192e+01,   1.82385027e+01,   2.60818192e+01,\n",
    "         9.55287183e+00,   1.71748057e+01,   3.69953552e+01,\n",
    "         1.77356368e+01,   1.63532490e+00,   1.74326279e+01,\n",
    "         4.33029360e+00,   4.57859415e+01,   3.37503132e+01,\n",
    "         7.64806162e+00,   1.21112918e+01,   1.04237173e+01,\n",
    "         1.61632554e+01,   2.67462911e+01,   3.86148230e+01,\n",
    "         1.35489943e+01,   2.67133708e+01,   1.07708285e+01,\n",
    "         3.50328192e+01,   0.00000001e+00,   1.23633893e+01,\n",
    "         1.81384274e+01,   2.00759877e+01,   4.08866304e+00,\n",
    "         5.97695583e+00,   6.06898915e+01,   2.49031102e+01,\n",
    "         2.52384006e+01,   3.32386071e+00,   2.24843776e+01,\n",
    "         1.68112507e+01,   3.97525241e+01,   1.75996255e+01,\n",
    "         3.79904489e+01,   3.06854403e+01,   4.85847744e+01,\n",
    "         1.85630616e+01,   1.71993799e+01,   1.83448434e+01,\n",
    "         1.56628516e+01,   0.00000001e+00,   2.41727188e+01,\n",
    "         4.42192515e+01,   2.38988782e+01,   1.42807208e+01,\n",
    "         7.08182337e+00,   4.92507959e+01,   3.50497922e+01,\n",
    "         9.43717693e+00,   5.75335913e+01,   3.28264400e+01,\n",
    "         1.81904415e+01,   7.50897544e-02,   1.38308498e+01,\n",
    "         3.68165837e+01,   3.76625657e+01,   2.45909362e+01,\n",
    "         1.35663121e+01,   1.06222416e+01,   6.61981076e+00,\n",
    "         1.49212998e+01,   2.48572601e+01,   5.37511636e+00,\n",
    "         4.00065481e+01,   3.34261910e+01,   1.65527992e+00,\n",
    "         4.27803816e+01,   1.90730180e+01,   1.26760294e+01,\n",
    "         1.05443893e+01,   1.11937050e+01,   5.10569815e+00,\n",
    "         2.15048656e+01,   4.25717685e+01,   4.13509594e+00,\n",
    "         1.72295175e+00,   9.05902838e+00,   7.60242688e+00,\n",
    "         1.65091770e+01,   2.54722991e+01,   1.72082765e+01,\n",
    "         1.51420024e+01,   3.27011995e+01,   8.07647247e+00,\n",
    "         2.85499825e+01,   1.05079289e+01,   1.00641429e+01,\n",
    "         1.50327603e+01,   1.02016207e+01,   2.83550792e+01,\n",
    "         3.70882037e+01,   5.77708927e+00,   9.80647023e+00,\n",
    "         2.95922618e+01,   9.71884089e+00,   2.32819941e+01,\n",
    "         9.04751512e+00,   7.93725690e+00,   1.86800551e+01,\n",
    "         0.00000001e+00,   1.65171448e+01,   9.86632517e+00,\n",
    "         3.70076954e+01,   5.62946130e+01,   8.79972911e+00,\n",
    "         2.97529529e+00,   9.36525852e+00,   8.09489263e+01,\n",
    "         1.10125346e+01,   3.63186018e+01,   7.10514953e+00,\n",
    "         1.94170427e+01,   2.32266727e+01,   8.99519114e+00,\n",
    "         8.35596998e+00,   1.23461020e+01,   2.17115648e+01,\n",
    "         1.35079021e+01,   1.73331305e+01,   2.59980413e+01,\n",
    "         1.32845317e+01,   2.53120899e+01,   1.88686858e+01,\n",
    "         2.80040023e+01,   1.00171749e+01,   2.86367553e+01])\n",
    "averagemonthly = np.mean(monthlytime)\n",
    "print(averagemonthly)\n",
    "minincidents = 1\n",
    "maxincidents=5"
   ]
  },
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7feb46a1c8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(monthlytime)\n",
    "plt.title(\"Distribution of lost hours per month\")\n",
    "plt.xlabel(\"Hours lost per month\")\n",
    "plt.ylabel(\"Months (out of 192)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#  Why Bayesian\n",
    "\n",
    "-  The facility does not exist, so no *correct* data is available.\n",
    "-  We do not have data on disruption frequency or duration, only the total time per month.\n",
    "-  Bayesian methods say *this is the model of the process that is most likely to have created the data seen*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#  Bayesian model\n",
    "\n",
    "-  Draw directed acyclic graph using `pydot`\n",
    "    - Python interface to Graphviz library\n",
    "-  Assume that each month disruptions follow a Poisson process.\n",
    "    - Exponential interevent times\n",
    "    - Memoryless\n",
    "-  Assume that a disruption takes an exponential amount to time.\n",
    "-  From discussions with subject matter experts, they say that during an event, it is not possible to determine how much longer the event will take place and it is not possible to predict when the next event will occur.\n",
    "    - Consistant with the mathematical description of exponential distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Directed Acylcic graphs\n",
    "\n",
    "-  Standard representation of a Bayesian model is a DAG.\n",
    "-  Hyperpriors are used to determine priors.\n",
    "-  Priors and likelihood are combined to determine posterior probability distributions.\n",
    "-  `pydot` is a Python interface to Graphviz and can be used to create a DAG."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#  Hyperpriors\n",
    "\n",
    "-  Need an average incident duration and average number of incidents per month\n",
    "-  Because we have the average number of lost hours per month, these are related according to:\n",
    "$$lost hours = average duration * average number$$\n",
    "-  We can create a *hyperprior* on the average number of incidents\n",
    "    - Uniform distribution with lower and upper bounds of 2 and 5.\n",
    "    - Chosen to be well beyond the range of reasonableness.\n",
    "-  MCMC determines what combination of parameters of average duration and average number of incidents per month can create the monthly interuption pattern observed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Couldn't import dot_parser, loading of dot files will not be possible.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pydot\n",
    "disruptiongraph = pydot.Dot(graph_type='digraph')\n",
    "losthours = pydot.Node(\"Disruption time \\n per month\")\n",
    "minincidents = pydot.Node(\"Minimum incidents \\n per month\", shape=\"box\")\n",
    "maxincidents = pydot.Node(\"Maximum incidents \\n per month\", shape=\"box\")\n",
    "permonth = pydot.Node(\"Incidents \\n per month\")\n",
    "incidentlength = pydot.Node(\"Incident \\n duration\")\n",
    "disruptiongraph.add_node(losthours)\n",
    "disruptiongraph.add_node(minincidents)\n",
    "disruptiongraph.add_node(maxincidents)\n",
    "disruptiongraph.add_node(permonth)\n",
    "disruptiongraph.add_node(incidentlength)\n",
    "disruptiongraph.add_edge(pydot.Edge(losthours, permonth))\n",
    "disruptiongraph.add_edge(pydot.Edge(minincidents, permonth))\n",
    "disruptiongraph.add_edge(pydot.Edge(maxincidents, permonth))\n",
    "disruptiongraph.add_edge(pydot.Edge(losthours, incidentlength))\n",
    "disruptiongraph.add_edge(pydot.Edge(permonth, incidentlength))\n",
    "disruptiongraph.write_png('disruptiondoodle.png')\n",
    "                         \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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       "BrF8EREREWmQHMAmqUMQERERtRD5/w94a3v076gA1AAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='disruptiondoodle.png') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# PyMC implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pymc\n",
    "import numpy as np\n",
    "\n",
    "avgpermonth = pymc.Uniform('Lambda', 0.5, 5)\n",
    "\n",
    "@pymc.stochastic(observed=False)\n",
    "def numincidents(mu=avgpermonth, value=1):\n",
    "    numincidents = np.random.poisson(mu)\n",
    "    return numincidents\n",
    "\n",
    "@pymc.deterministic(plot=False)\n",
    "def durationbeta(averagetime=averagemonthly, avgnum=avgpermonth):\n",
    "    if avgnum >0:\n",
    "        duration= averagetime/avgnum\n",
    "    else:\n",
    "        duration = 0\n",
    "    return duration\n",
    "\n",
    "losttime = pymc.Gamma(\"monthly lost time\", \n",
    "                      alpha=numincidents, beta=1/durationbeta,\n",
    "                      observed=True, value=monthlytime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 100000 of 100000 complete in 25.9 sec"
     ]
    }
   ],
   "source": [
    "modelpymc = pymc.Model([avgpermonth, numincidents, durationbeta, losttime])  \n",
    "# package the full model in a dictionary\n",
    "#modelpymc = dict(avgpermonth=avgpermonth, numincidents=numincidents, avgduration=durationbeta, \n",
    "#             losttime=losttime)   \n",
    "# run the basic MCMC: we'll do 100000 iterations to match emcee above\n",
    "pymcmodel = pymc.MCMC(modelpymc)\n",
    "pymcmodel.sample(iter=100000, burn=50000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "durationbeta:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t[[ 25.277]]      [[ 3.087]]       [[ 0.088]]     [ 19.52   31.107]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t[[ 19.999]]      [[ 22.985]]     [[ 25.093]]    [[ 27.228]]   [[ 31.871]]\n",
      "\t\n",
      "\n",
      "numincidents:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t[[ 0.763]]       [[ 0.068]]       [[ 0.002]]       [ 0.64   0.901]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t[[ 0.635]]       [[ 0.714]]      [[ 0.761]]     [[ 0.806]]    [[ 0.901]]\n",
      "\t\n",
      "\n",
      "Lambda:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t[[ 0.776]]       [[ 0.094]]       [[ 0.003]]       [ 0.61   0.967]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t[[ 0.607]]       [[ 0.71]]       [[ 0.771]]     [[ 0.841]]    [[ 0.967]]\n",
      "\t\n"
     ]
    }
   ],
   "source": [
    "pymcmodel.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# A look at another example in two implementations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Car stopping example\n",
    "\n",
    "-  What is the relationship of the stopping distance of a car to speed?\n",
    "-  We'll use some fake data\n",
    "-  Adapted from Jake VanderPlas https://jakevdp.github.io/blog/2014/03/11/frequentism-and-bayesianism-a-practical-intro/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7feb46a1c410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "theta_true = (-17.6, 3.9)\n",
    "xdata = 100 * np.random.random(20)\n",
    "ydata = theta_true[0] + theta_true[1] * xdata\n",
    "# add scatter to points\n",
    "xdata = np.random.normal(xdata, 10)\n",
    "ydata = np.random.normal(ydata, 10)\n",
    "plt.plot(xdata, ydata, 'ok')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# The model\n",
    "\n",
    "-  Treat this as a linear regression model\n",
    "-  slope $\\beta$ and a y-intercept $\\alpha$:\n",
    "\n",
    "$$\n",
    "\\hat{y}(x_i~|~\\alpha,\\beta) = \\alpha + \\beta x_i\n",
    "$$\n",
    "\n",
    "-  Normally distributed errors with standard deviation $\\sigma$\n",
    "$$\n",
    "P(x_i, y_i~|~\\alpha, \\beta, \\sigma) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp\\left[\\frac{-[y_i - \\hat{y}(x_i~|~\\alpha, \\beta)]^2}{2\\sigma^2}\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Create some convenience routines for plotting\n",
    "\n",
    "def compute_sigma_level(trace1, trace2, nbins=20):\n",
    "    \"\"\"From a set of traces, bin by number of standard deviations\"\"\"\n",
    "    L, xbins, ybins = np.histogram2d(trace1, trace2, nbins)\n",
    "    L[L == 0] = 1E-16\n",
    "    logL = np.log(L)\n",
    "\n",
    "    shape = L.shape\n",
    "    L = L.ravel()\n",
    "\n",
    "    # obtain the indices to sort and unsort the flattened array\n",
    "    i_sort = np.argsort(L)[::-1]\n",
    "    i_unsort = np.argsort(i_sort)\n",
    "\n",
    "    L_cumsum = L[i_sort].cumsum()\n",
    "    L_cumsum /= L_cumsum[-1]\n",
    "    \n",
    "    xbins = 0.5 * (xbins[1:] + xbins[:-1])\n",
    "    ybins = 0.5 * (ybins[1:] + ybins[:-1])\n",
    "\n",
    "    return xbins, ybins, L_cumsum[i_unsort].reshape(shape)\n",
    "\n",
    "\n",
    "def plot_MCMC_trace(ax, xdata, ydata, trace, scatter=False, **kwargs):\n",
    "    \"\"\"Plot traces and contours\"\"\"\n",
    "    xbins, ybins, sigma = compute_sigma_level(trace[0], trace[1])\n",
    "    ax.contour(xbins, ybins, sigma.T, levels=[0.683, 0.955], **kwargs)\n",
    "    if scatter:\n",
    "        ax.plot(trace[0], trace[1], ',k', alpha=0.1)\n",
    "    ax.set_xlabel(r'$\\alpha$')\n",
    "    ax.set_ylabel(r'$\\beta$')\n",
    "    \n",
    "    \n",
    "def plot_MCMC_model(ax, xdata, ydata, trace):\n",
    "    \"\"\"Plot the linear model and 2sigma contours\"\"\"\n",
    "    ax.plot(xdata, ydata, 'ok')\n",
    "\n",
    "    alpha, beta = trace[:2]\n",
    "    xfit = np.linspace(-20, 120, 10)\n",
    "    yfit = alpha[:, None] + beta[:, None] * xfit\n",
    "    mu = yfit.mean(0)\n",
    "    sig = 2 * yfit.std(0)\n",
    "\n",
    "    ax.plot(xfit, mu, '-k')\n",
    "    ax.fill_between(xfit, mu - sig, mu + sig, color='lightgray')\n",
    "\n",
    "    ax.set_xlabel('x')\n",
    "    ax.set_ylabel('y')\n",
    "\n",
    "\n",
    "def plot_MCMC_results(xdata, ydata, trace, colors='k'):\n",
    "    \"\"\"Plot both the trace and the model together\"\"\"\n",
    "    fig, ax = plt.subplots(1, 2, figsize=(10, 4))\n",
    "    plot_MCMC_trace(ax[0], xdata, ydata, trace, True, colors=colors)\n",
    "    plot_MCMC_model(ax[1], xdata, ydata, trace)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# emcee"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.1.0\n"
     ]
    }
   ],
   "source": [
    "import emcee\n",
    "print(emcee.__version__)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Define our posterior using Python functions\n",
    "# for clarity, I've separated-out the prior and likelihood\n",
    "# but this is not necessary. Note that emcee requires log-posterior\n",
    "\n",
    "def log_prior(theta):\n",
    "    alpha, beta, sigma = theta\n",
    "    if sigma < 0:\n",
    "        return -np.inf  # log(0)\n",
    "    else:\n",
    "        return -1.5 * np.log(1 + beta ** 2) - np.log(sigma)\n",
    "\n",
    "def log_likelihood(theta, x, y):\n",
    "    alpha, beta, sigma = theta\n",
    "    y_model = alpha + beta * x\n",
    "    return -0.5 * np.sum(np.log(2 * np.pi * sigma ** 2) + (y - y_model) ** 2 / sigma ** 2)\n",
    "\n",
    "def log_posterior(theta, x, y):\n",
    "    return log_prior(theta) + log_likelihood(theta, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Here we'll set up the computation. emcee combines multiple \"walkers\",\n",
    "# each of which is its own MCMC chain. The number of trace results will\n",
    "# be nwalkers * nsteps\n",
    "\n",
    "ndim = 3  # number of parameters in the model\n",
    "nwalkers = 50  # number of MCMC walkers\n",
    "nburn = 1000  # \"burn-in\" period to let chains stabilize\n",
    "nsteps = 2000  # number of MCMC steps to take\n",
    "\n",
    "# set theta near the maximum likelihood, with \n",
    "np.random.seed(0)\n",
    "starting_guesses = np.random.random((nwalkers, ndim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 5.24 s, sys: 0 ns, total: 5.24 s\n",
      "Wall time: 5.23 s\n",
      "done\n"
     ]
    }
   ],
   "source": [
    "# Here's the function call where all the work happens:\n",
    "# we'll time it using IPython's %time magic\n",
    "\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[xdata, ydata])\n",
    "%time sampler.run_mcmc(starting_guesses, nsteps)\n",
    "print(\"done\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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       "GIZhGEaR8v8B6NAajcrpsZ8AAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7feb242f4c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sampler.chain is of shape (nwalkers, nsteps, ndim)\n",
    "# we'll throw-out the burn-in points and reshape:\n",
    "emcee_trace = sampler.chain[:, nburn:, :].reshape(-1, ndim).T\n",
    "plot_MCMC_results(xdata, ydata, emcee_trace)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# PyMC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Define the variables needed for the routine, with their prior distributions\n",
    "alpha = pymc.Uniform('alpha', -100, 100)\n",
    "\n",
    "@pymc.stochastic(observed=False)\n",
    "def beta(value=0):\n",
    "    return -1.5 * np.log(1 + value ** 2)\n",
    "\n",
    "@pymc.stochastic(observed=False)\n",
    "def sigma(value=1):\n",
    "    return -np.log(abs(value))\n",
    "\n",
    "# Define the form of the model and likelihood\n",
    "@pymc.deterministic\n",
    "def y_model(x=xdata, alpha=alpha, beta=beta):\n",
    "    return alpha + beta * x\n",
    "\n",
    "y = pymc.Normal('y', mu=y_model, tau=1. / sigma ** 2, observed=True, value=ydata)\n",
    "\n",
    "# package the full model in a dictionary\n",
    "model1 = dict(alpha=alpha, beta=beta, sigma=sigma,\n",
    "              y_model=y_model, y=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 100000 of 100000 complete in 23.9 sec"
     ]
    }
   ],
   "source": [
    "# run the basic MCMC: we'll do 100000 iterations to match emcee above\n",
    "S = pymc.MCMC(model1)\n",
    "S.sample(iter=100000, burn=50000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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       "76vqrSKyA/YZNlhVHy7z0pxWgMeoOY7jOI7jVCju+nQcx3Ecx6lQXKg5juM4juNUKC7UHMdxHMdx\n",
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       "yPtTigAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7feb232ab1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# extract the traces and plot the results\n",
    "pymc_trace = [S.trace('alpha')[:],\n",
    "              S.trace('beta')[:],\n",
    "              S.trace('sigma')[:]]\n",
    "\n",
    "plot_MCMC_results(xdata, ydata, pymc_trace)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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       "UpAXEREpKAV5ERGRgvo/mtksELbv2RAAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7feb231b0810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 8))\n",
    "plot_MCMC_trace(ax, xdata, ydata, emcee_trace, True,\n",
    "                colors='blue', linewidths=2)\n",
    "plot_MCMC_trace(ax, xdata, ydata, pymc_trace,\n",
    "                colors='red', linewidths=2)\n",
    "#plot_MCMC_trace(ax, xdata, ydata, pystan_trace,\n",
    "#                colors='green', linewidths=2)\n",
    "ax.legend(ax.collections[::1], ['emcee', 'pymc'], fontsize=16);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# PyStan\n",
    "\n",
    "- **Not working**"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "import pystan\n",
    "print(pystan.__version__)"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "%%file pystan_example.py\n",
    "\n",
    "import numpy as np\n",
    "import pystan\n",
    "\n",
    "#---------------------------------------------\n",
    "# Generate data (same as used for emcee and pymc)\n",
    "\n",
    "np.random.seed(42)\n",
    "theta_true = (-17.6, 3.9)\n",
    "xdata = 100 * np.random.random(20)\n",
    "ydata = theta_true[0] + theta_true[1] * xdata\n",
    "\n",
    "# add scatter to points\n",
    "xdata = np.random.normal(xdata, 10)\n",
    "ydata = np.random.normal(ydata, 10)\n",
    "\n",
    "#----------------------------------------------\n",
    "# Create the Stan model\n",
    "#  this is done by defining a string of Stan code.\n",
    "\n",
    "fit_code = \"\"\"\n",
    "data {\n",
    "    int<lower=0> N; // number of points\n",
    "    real x[N]; // x values\n",
    "    real y[N]; // y values\n",
    "}\n",
    "\n",
    "parameters {\n",
    "    real alpha_perp;\n",
    "    real<lower=-pi()/2, upper=pi()/2> theta;\n",
    "    real log_sigma;\n",
    "}\n",
    "\n",
    "transformed parameters {\n",
    "    real alpha;\n",
    "    real beta;\n",
    "    real sigma;\n",
    "    real ymodel[N];\n",
    "    \n",
    "    alpha <- alpha_perp / cos(theta);\n",
    "    beta <- sin(theta);\n",
    "    sigma <- exp(log_sigma);\n",
    "    for (j in 1:N)\n",
    "    ymodel[j] <- alpha + beta * x[j];\n",
    "}\n",
    "\n",
    "model {\n",
    "    y ~ normal(ymodel, sigma);\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "# perform the fit\n",
    "fit_data = {'N': len(xdata), 'x': xdata, 'y': ydata}\n",
    "fit = pystan.stan(model_code=fit_code, data=fit_data, iter=25000, chains=4)\n",
    "\n",
    "# extract the traces\n",
    "traces = fit.extract()\n",
    "pystan_trace = [traces['alpha'], traces['beta'], traces['sigma']]\n",
    "\n",
    "# save the traces with numpy\n",
    "np.save(\"pystan_trace.npy\", pystan_trace)"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# run the code we've created on the command-line\n",
    "!python pystan_example.py"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# load the results from file; plot as above\n",
    "pystan_trace = np.load('pystan_trace.npy')\n",
    "plot_MCMC_results(xdata, ydata, pystan_trace)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Some resources\n",
    "\n",
    "## Bayesian modeling\n",
    "\n",
    "-  Allan Downey, (2013), Think Bayes, http://greenteapress.com/thinkbayes/index.html\n",
    "\n",
    "## emcee\n",
    "- http://dan.iel.fm/emcee/current/\n",
    "- Goodman & Weare’s Affine Invariant Markov chain Monte Carlo (MCMC) Ensemble sampler \n",
    "- http://msp.org/camcos/2010/5-1/p04.xhtml\n",
    "\n",
    "## PyMC\n",
    "-  https://pymc-devs.github.io/pymc/index.html\n",
    "-  Cam Davidson-Pilon https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers\n",
    "-  Kiran R Karkera, (2014), Building Probabilistic Graphical Models with Python\n",
    "\n",
    "## PyStan\n",
    "-  Stan http://mc-stan.org/\n",
    "-  Gelman et al., (2014), Bayesian Data Analysis (3rd ed)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Code content is BSD licenced."
   ]
  }
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