{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Section 1: Estimating model parameters\n", "In this section we will discuss how Bayesians think about data, and how we can estimate model parameters using a technique called MCMC." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "WARNING (theano.configdefaults): g++ not detected ! Theano will be unable to execute optimized C-implementations (for both CPU and GPU) and will default to Python implementations. Performance will be severely degraded. To remove this warning, set Theano flags cxx to an empty string.\n" ] } ], "source": [ "from IPython.display import Image\n", "\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pandas as pd\n", "import pymc3 as pm\n", "import scipy\n", "import scipy.stats as stats\n", "import scipy.optimize as opt\n", "import statsmodels.api as sm\n", "\n", "%matplotlib inline\n", "plt.style.use('bmh')\n", "colors = ['#348ABD', '#A60628', '#7A68A6', '#467821', '#D55E00', \n", " '#CC79A7', '#56B4E9', '#009E73', '#F0E442', '#0072B2']\n", "\n", "messages = pd.read_csv('data/hangout_chat_data.csv')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### How do Bayesians think about data?\n", "When I started to learn how to apply Bayesian methods, I found it very useful to understand how Bayesians think about data. Imagine the following scenario:\n", "> A curious boy watches the number of cars that pass by his house every day. He diligently notes down the total count of cars that pass per day. Over the past week, his notebook contains the following counts: 12, 33, 20, 29, 20, 30, 18\n", "\n", "From a Bayesian's perspective, this data is generated by a random process. However, now that the data is observed, it is fixed and does not change. This random process has some model parameters that are fixed. However, the Bayesian uses probability distributions to represent his/her uncertainty in these parameters.\n", "\n", "Because the boy is measuring counts (non-negative integers), it is common practice to use a Poisson distribution to model the data (eg. the random process). A Poisson distribution takes a single parameter $\\mu$ which describes both the mean and variance of the data. You can see 3 Poisson distributions below with different values of $\\mu$.\n", "\n", "$$p(x \\ | \\ \\mu) = \\frac{e^{-\\mu}\\mu^{x}} {x!} \\mbox{ for } \n", "x = 0, 1, 2, \\cdots$$\n", "\n", "$$\\lambda = E(x) = Var(\\mu)$$" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ajj/+eCZNmsRpp51WJ/Gmy2tyJjPramYDyrX1MbPlZrbKzP5oZo08Y/oKKAHK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\nN/FXk9tvITu34D47eRJn0TqjY9REiDF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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(11,3))\n", "ax = fig.add_subplot(111)\n", "x_lim = 60\n", "mu = [5, 20, 40]\n", "for i in np.arange(x_lim):\n", " plt.bar(i, stats.poisson.pmf(mu[0], i), color=colors[3])\n", " plt.bar(i, stats.poisson.pmf(mu[1], i), color=colors[4])\n", " plt.bar(i, stats.poisson.pmf(mu[2], i), color=colors[5])\n", " \n", "_ = ax.set_xlim(0, x_lim)\n", "_ = ax.set_ylim(0, 0.2)\n", "_ = ax.set_ylabel('Probability mass')\n", "_ = ax.set_title('Poisson distribution')\n", "_ = plt.legend(['$\\mu$ = %s' % mu[0], '$\\mu$ = %s' % mu[1], '$\\mu$ = %s' % mu[2]])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the previous section we imported my hangout chat dataset. I'm particularly interested in the time it takes me to respond to messages (`response_time`). Given that `response_time` is count data, we can model it as a Poisson distribution and estimate its parameter $\\mu$. We will explore both a frequentist and Bayesian method of estimating this parameter." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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xlDcSQ3kjaYp+Tqi7j3P30919kLvPzilf5u693X18U4MzMwN+ATzl7i/lzJoC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\nzPoqF6FmZn2IpOMktZa8miT9Q9LPJY0pOr56JWkfSedJ2rT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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(11,3))\n", "_ = plt.title('Frequency of messages by response time')\n", "_ = plt.xlabel('Response time (seconds)')\n", "_ = plt.ylabel('Number of messages')\n", "_ = plt.hist(messages['time_delay_seconds'].values, \n", " range=[0, 60], bins=60, histtype='stepfilled')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Frequentists method of estimating $\\mu$\n", "Before we jump into Bayesian techniques, lets first look at a frequentist method of estimating the parameters of a Poisson distribution. We will use an optimization technique that aims to maximize the likelihood of a function.\n", "\n", "The below function `poisson_logprob()` returns the overall likelihood of the observed data given a Poisson model and parameter value. We use the method `opt.minimize_scalar` to find the value of $\\mu$ that is most credible (maximizes the log likelihood) given the data observed. Under the hood, this optimization technique is intelligently iterating through possible values of `mu` until it finds a value with the highest likelihood." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The estimated value of mu is: 18.2307692324\n", "CPU times: user 90 µs, sys: 45 µs, total: 135 µs\n", "Wall time: 101 µs\n" ] } ], "source": [ "y_obs = messages['time_delay_seconds'].values\n", "\n", "def poisson_logprob(mu, sign=-1):\n", " return np.sum(sign*stats.poisson.logpmf(y_obs, mu=mu))\n", "\n", "freq_results = opt.minimize_scalar(poisson_logprob)\n", "%time print(\"The estimated value of mu is: %s\" % freq_results['x'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, the estimate of the value of $\\mu$ is 18.0413533867. The optimization technique doesn't provide any measure of uncertainty - it just returns a point value. And it does so very efficiently...\n", "\n", "The below plot illustrates the function that we are optimizing. At each value of $\\mu$, the plot shows the log probability at $\\mu$ given the data and the model. The optimizer works in a hill climbing fashion - starting at a random point on the curve and incrementally climbing until it cannot get to a higher point." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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mb2TZzgP7fiXHRXP6QOeDzrauMMlx0Rzbw5ndVlldy5qCcpbn72NtQQXV3mxp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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(1, 60)\n", "y_min = np.min([poisson_logprob(i, sign=1) for i in x])\n", "y_max = np.max([poisson_logprob(i, sign=1) for i in x])\n", "fig = plt.figure(figsize=(6,4))\n", "_ = plt.plot(x, [poisson_logprob(i, sign=1) for i in x])\n", "_ = plt.fill_between(x, [poisson_logprob(i, sign=1) for i in x], \n", " y_min, color=colors[0], alpha=0.3)\n", "_ = plt.title('Optimization of $\\mu$')\n", "_ = plt.xlabel('$\\mu$')\n", "_ = plt.ylabel('Log probability of $\\mu$ given data')\n", "_ = plt.vlines(freq_results['x'], y_max, y_min, colors='red', linestyles='dashed')\n", "_ = plt.scatter(freq_results['x'], y_max, s=110, c='red', zorder=3)\n", "_ = plt.ylim(ymin=y_min, ymax=0)\n", "_ = plt.xlim(xmin=1, xmax=60)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above optimization has estimated the parameter ($\\mu$) of a Poisson model to be 18. We know for any Poisson distribution, the parameter $\\mu$ represents both its mean and variance. The below plot illustrates this distribution." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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99SuyLklXjx49+PjjjzvlvdRxbcEOO+yQdgiSUbvsskvaIUgGKW8khvJGYihv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\nIiI6KoggsgRQBEMusrqgRIkJXGQTELiQXyAKaC6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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(11,3))\n", "ax = fig.add_subplot(111)\n", "x_lim = 60\n", "mu = np.int(freq_results['x'])\n", "for i in np.arange(x_lim):\n", " plt.bar(i, stats.poisson.pmf(mu, i), color=colors[3])\n", " \n", "_ = ax.set_xlim(0, x_lim)\n", "_ = ax.set_ylim(0, 0.1)\n", "_ = ax.set_xlabel('Response time in seconds')\n", "_ = ax.set_ylabel('Probability mass')\n", "_ = ax.set_title('Estimated Poisson distribution for Hangout chat response time')\n", "_ = plt.legend(['$\\lambda$ = %s' % mu])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above Poisson model and estimated value of $\\mu$ suggest that there is minimal chance of an observation less than 10 or greater than 30. The vast majority of the probability mass is between 10 and 30. However, we know this is not reflected in the data that we observed - which has observed values between 0 and 60." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Bayesian method of estimating $\\mu$\n", "\n", "If you've encountered Bayes' theorem before, the below formula will look familiar. This framework never resonated with me until I read John K. Kruschke's book \"Doing Bayesian Data Analysis\" and saw the below formula through the lens of his beautifully simple Bayesian graphical models.\n", " \n", "$$\\overbrace{p(\\mu \\ |\\ Data)}^{\\text{posterior}} = \\dfrac{\\overbrace{p(Data \\ | \\ \\mu)}^{\\text{likelihood}} \\cdot \\overbrace{p(\\mu)}^{\\text{prior}}}{\\underbrace{p(Data)}_{\\text{marginal likelihood}}}$$" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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e6GC5EuP57jjUe//vteOHk3O1P3qNg7Z+zibu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"text/plain": [ "" ] }, "execution_count": 7, "metadata": { "image/png": { "width": 320 } }, "output_type": "execute_result" } ], "source": [ "Image('graphics/Poisson-dag.png', width=320)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above schema can be interpreted as follows (from the bottom up):\n", "- We observe counts of data (y) for each conversation i (Observed Data)\n", "- This data was generated by a random process which we believe can be represented as a Poisson distribution (Likelihood)\n", "- This Poisson distribution has a single parameter $\\mu$ which we know is between 0 and 60 (Prior)\n", " - We will model $\\mu$ as a uniform distribution because we do not have an opinion as to where within this range to expect it\n", "\n", "### The magical mechanics of MCMC\n", "The process of Markov Chain Monte Carlo (MCMC) is nicely illustrated in the below animation. The MCMC sampler draws parameter values from the prior distribution and computes the likelihood that the observed data came from a distribution with these parameter values. \n", "\n", "$$\\overbrace{p(\\mu \\ |\\ Data)}^{posterior} \\varpropto \\overbrace{p(Data \\ | \\ \\mu)}^{likelihood} \\cdot \\overbrace{p(\\mu)}^{prior}$$\n", "\n", "This calculation acts as a guiding light for the MCMC sampler. As it draws values from the paramater priors, it computes the likelihood of these paramters given the data - and will try to guide the sampler towards areas of higher probability.\n", "\n", "In a conceptually similar manner to the frequentist optimization technique discussed above, the MCMC sampler wanders towards areas of highest likelihood. However, the Bayesian method is not concerned with findings the absolute maximum values - but rather to traverse and collect samples around the area of highest probability. All of the samples collected are considered to be a credible parameter." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Image(url='graphics/mcmc-animate.gif')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Applied interval-transform to mu and added transformed mu_interval_ to model.\n", " 2%|▏ | 3769/200000 [01:13<1:05:21, 50.04it/s]" ] } ], "source": [ "with pm.Model() as model:\n", " mu = pm.Uniform('mu', lower=0, upper=60)\n", " likelihood = pm.Poisson('likelihood', mu=mu, observed=messages['time_delay_seconds'].values)\n", " \n", " start = pm.find_MAP()\n", " step = pm.Metropolis()\n", " trace = pm.sample(200000, step, start=start, progressbar=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above code has just gathered 200,000 credible samples of $\\mu$ by traversing over the areas of high likelihood of the posterior distribution of $\\mu$. The below plot (left) shows the distribution of values collected for $\\mu$. The mean of this distribution is almost identical to the frequentist estimate (red line). However, we also get a measure of uncertainty and can see that there are credible values of $\\mu$ between 17 and 19. This measure of uncertainty is incredibly valuable as we will see later." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "_ = pm.traceplot(trace, varnames=['mu'], lines={'mu': freq_results['x']})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Discarding early samples (burnin)\n", "You may have wondered what the purpose of `pm.find_MAP()` is in the above MCMC code. MAP stands for maximum a posteriori estimation. It helps the MCMC sampler find a good place from which to start sampling. Ideally this will start the model off in an area of high likelihood - but sometimes that doesn't happen. As a result, the samples collected early in the trace (burnin samples) are often discarded." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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BRwwT3Szb6TkpPsMEi+XiM9Z9l6LDry4yxHLbgN5tn9MCq5+39pXWteNvGyvw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\n3PX8O0B170GSevWBtRjIfeuSHuZtf5/vPpNcL13OmI5BrP+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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(11,3))\n", "plt.subplot(121)\n", "_ = plt.title('Burnin trace')\n", "_ = plt.ylim(ymin=16.5, ymax=19.5)\n", "_ = plt.plot(trace.get_values('mu')[:1000])\n", "fig = plt.subplot(122)\n", "_ = plt.title('Full trace')\n", "_ = plt.ylim(ymin=16.5, ymax=19.5)\n", "_ = plt.plot(trace.get_values('mu'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Model convergence\n", "#### Trace\n", "Just because the above model estimated a value for $\\mu$, doesn't mean the model estimated a good value given the data. There are some recommended checks that you can make. Firstly, look at the trace output. You should see the trace jumping around and generally looking like a hairy caterpillar. If you see the trace snake up and down or appear to be stuck in any one location - it is a sign that you have convergence issues and the estimations from the MCMC sampler cannot be trusted.\n", "\n", "#### Autocorrelation plot\n", "The second test you can perform is the autocorrelation test (see below plot). It is a measure of correlation between successive samples in the MCMC sampling chain. When samples have low correlation with each other, they are adding more \"information\" to the estimate of your parameter value than samples that are highly correlated.\n", "\n", "Visually, you are looking for an autocorrelation plot that tapers off to zero relatively quickly and then oscilates above and below zero correlation. If your autocorrelation plot does not taper off - it is generally a sign of poor mixing and you should revisit your model selection (eg. likelihood) and sampling methods (eg. Metropolis)." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "_ = pm.autocorrplot(trace[:2000], varnames=['mu'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### [>> Go to the Next Section](http://nbviewer.ipython.org/github/markdregan/Bayesian-Modelling-in-Python/blob/master/Section%202.%20Model%20checking.ipynb)\n", "\n", "### References\n", "- [MCMC animation by Maxwell Joeseph](http://blog.revolutionanalytics.com/2013/09/an-animated-peek-into-the-workings-of-Bayesian-statistics.html)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Apply pretty styles\n", "from IPython.core.display import HTML\n", "\n", "def css_styling():\n", " styles = open(\"styles/custom.css\", \"r\").read()\n", " return HTML(styles)\n", "css_styling()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python [conda root]", "language": "python", "name": "conda-root-py" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.12" } }, "nbformat": 4, "nbformat_minor": 0 }