{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "#What is Bayes Theorem?\n", "\n", "Bayes theorem is what allows us to go from a **sampling (or likelihood) distribution** and a **prior distribution** to a **posterior distribution**.\n", "\n", "##What is a Sampling Distribution?\n", "\n", "A sampling distribution is the probability of seeing **our data (X)** given our **parameters ($\\theta$).** This is written as $p(X|\\theta)$.\n", "\n", "For example, we might have data on 1,000 coin flips. Where 1 indicates a head. This can be represented in python as" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.46800000000000003" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "data_coin_flips = np.random.randint(2, size=1000)\n", "np.mean(data_coin_flips)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A sampling distribution allows us to specify how we think these data were generated. For our coin flips, we can think of our data as being generated from a [Bernoulli Distribution](http://en.wikipedia.org/wiki/Bernoulli_distribution). This distribution takes one **parameter** p which is the probability of getting a 1 (or a head for a coin flip). It then returns a value of 1 with probablility p and a value of 0 with probablility (1-p).\n", "\n", "You can see how this is perfect for a coin flip. With a fair coin we know our p = .5 because we are equally likely to get a 1 (head) or 0 (tail). We can create samples from this distribution like this:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.46500000000000002" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bernoulli_flips = np.random.binomial(n=1, p=.5, size=1000)\n", "np.mean(bernoulli_flips)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have defined how we believe our data were generated, we can calculate the probability of seeing our data given our parameters $p(X|\\theta)$. Since we have selected a Bernoulli distribution, we only have one parameter: p. \n", "\n", "We can use the **probability mass function (PMF)** of the Bernoulli distribution to get our desired probability for a single coin flip. The PMF takes a single observed data point and then given the parameters (p in our case) returns the probablility of seeing that data point given those parameters. For a Bernoulli distribution it is simple: if the data point is a 1 the PMF returns p, if the data point is a 0 it returns (1-p). We could write a quick function to do this:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def bern_pmf(x, p):\n", " if (x == 1):\n", " return p\n", " elif (x == 0):\n", " return 1 - p\n", " else:\n", " return \"Value Not in Support of Distribution\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now use this function to get the probability of a data point give our parameters. You probably see that with p = .5 this function always returns .5" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.5\n", "0.5\n" ] } ], "source": [ "print(bern_pmf(1, .5))\n", "print(bern_pmf(0, .5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a pretty simple PMF, but other distributions can get much more complicated. So it is good to know that Scipy has most of these built in. We can draw from the PMF as follows:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.5\n", "0.5\n" ] } ], "source": [ "import scipy.stats as st\n", "print(st.bernoulli.pmf(1, .5))\n", "print(st.bernoulli.pmf(0, .5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is nice, but what we really want to know is the probability of see all 1,000 of our data points. How do we do that? The trick here is to assume that our data are [independent and identically distributed](http://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables). This assumption allows us to say the probability of seeing all of our data is just the product of each individual probability: $p(x_{1}, ..., x_{n}|\\beta) = p(x_{1}|\\beta) * ... * p(x_{n}|\\beta)$. This is easy to do:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "9.3326361850321888e-302" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.product(st.bernoulli.pmf(data_coin_flips, .5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How does that number help us? Well by itself, it doesn't really help too much. What we need to do now is get more of a distribution for our sampling model. Currently, we have only tested our model with p = .5, but what if p = .8? or .2? What would the probablility of our data look like then? This can be done by defining a grid of values for our p. Below I will make a grid of 100 values between 0 and 1 (because p has to be between 0 and 1) and then I will calculate the probability of seeing our data given each of these values:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeQAAAFhCAYAAACh09mSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAgHmtSwePQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "%matplotlib inline\n", "sns.set(style='ticks', palette='Set2')\n", "\n", "params = np.linspace(0, 1, 100)\n", "p_x = [np.product(st.bernoulli.pmf(data_coin_flips, p)) for p in params]\n", "plt.plot(params, p_x)\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we are getting somewhere. We can see that the probablility of seeing our data peaks at p=.5 and almost certainly is between p=.4 and p=.6. Nice. So now we have a good idea of what p value generated our data assuming it was drawn from a Bernoulli distribution. We're done, right? Not quite...\n", "\n", "##Prior Distribution\n", "\n", "Bayes theorem says that we need to think about both our sampling distribution and our prior distribution. What do I mean by prior distribution? It is the $p(\\theta)$ or the probability of seeing a specific value for our parameter. In our sampling distribution we defined 100 values from 0 to 1 for our parameter p. Now we must define the prior probability of seeing each of those values. That is the probability we would have assumed before seeing any data. Most likely, we would have assumed a fair coin, which looks like the distribution above. Lets see how we can do this:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfIAAAFbCAYAAAAuggz2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzt3XuMrHd93/H33HZ299x8fAFsCD5cf4As0zZ2MXYIWMRK\nlWLJhKiSI0FqcNKYFCFSqXEjUZ+KSomgRkkjOSHGEFUhimopThW32ALTxuIQWVxKTZrwMz6+1cYK\nx7ezZ2+zO5f+MfPsjte7c9mdmeeZed4vydKZeeaZ+fnR7n7m+7s9hVarhSRJmk7FtBsgSZL2zyCX\nJGmKGeSSJE0xg1ySpClmkEuSNMUMckmSpli518EQQhG4A7gcqAE3xxhPdx3/EPBbQAv4SozxP/c7\nR5IkjU6/ivwGYC7GeDVwK3B7ciCEUAJ+B3g/8G7g4yGECzrnVHc7R5IkjVa/IL8GuA8gxvgQcEVy\nIMbYAN4WYzwHXASUgI3OOV/d7RxJkjRa/YL8KLDU9bjR6ToHIMbYDCH8IvC/gf8JrPQ7Z1AhhHII\n4UQIoWf3vyRJedYvJJeAI12PizHGZvcLYox/EUK4B/gT4CODnLNTCOEkcNtuxx544IE+TZQkaWYU\nhj2hX5CfAq4H7g4hXAU8nBwIIRwF/gq4Lsa4EUJYARq9ztlLjPEkcLL7uRDCCeDxQf9HJEnKo35B\nfg9wXQjhVOfxTSGEG4HDMcY7Qwh/CjwYQtgE/g/wp53XveyckbdakiQBUMjq3c+SivyBBx7gda97\nXdrNkSRpEobuWndDGEmSpphBLknSFDPIJUmaYga5JElTzCCXJGmKGeSSJE0xg1ySpClmkEuSNMUM\nckmSpphBLknSFDPIJUmaYga5JElTzCCXJGmKGeSSJE0xg1ySpClmkEuSNMUMckmSpphBLknSFDPI\nJUmaYga5JElTzCCXJGmKGeSSJE0xg1ySpClmkEuSNMUMckmSpphBLknSFDPIJUmaYga5JElTzCCX\nJGmKGeSSJE0xg1ySpClmkEuSNMUMckmSpphBLknSFDPIJUmaYga5JElTzCCXJGmKGeSSJE0xg1yS\npClmkEuSNMUMckmSpphBLknSFDPIJUmaYuVeB0MIReAO4HKgBtwcYzzddfxG4JNAHfgB8PEYYyuE\n8D3gbOdlj8UYPzaOxkuSlHc9gxy4AZiLMV4dQngXcHvnOUIIC8BngMtijOshhD8DPhBC+BpAjPHa\nMbZbkiTRv2v9GuA+gBjjQ8AVXcfWgXfHGNc7j8vAGvBOYDGEcH8I4YHOFwBJGVdr1Nlo1NNuhqQh\n9Qvyo8BS1+NGp7udGGMrxngGIITwCeBQjPHrwArwuRjjzwO/DnwlOUdSdv3O9+/nC3//zbSbIWlI\n/brWl4AjXY+LMcZm8qAT0J8F3gx8qPP0I8CjADHGH4UQngcuBp7Z60NCCCeB24ZtvKTRaLaaPLt6\nlp+snWOjUWeu1O9Pg6Ss6Pfbegq4Hrg7hHAV8PCO41+g3cX+wRhjq/PcTbQnx/1GCOES2lX9s70+\nJMZ4EjjZ/VwI4QTweN//A0kHtlZvd6k3Wk2eWn6BNx97VcotkjSofkF+D3BdCOFU5/FNnZnqh4Hv\nAB8FHgS+EUIA+D3gLuDLIYQHk3O6q3hJ2bPW2Nj69+ml5wxyaYr0DPJOlX3Ljqcf6fp3aY9TP3yQ\nRkmarNV6V5Cfey7FlkgalpPQJLFW39z692NLZ2i1Wj1eLSlLDHJJL6vIz23WOLO+nGJrJA3DIJe0\nFeSvP3wcgNNLZ9JsjqQhGOSStoL8suOXAPDYkuPk0rQwyCVtjZG/9dirmSuWOG2QS1PDIJfEamf5\n2eFKlRNHLuDHqy+x1jVuLim7DHJJW6G9WJ7jTUcvogU8fu75dBslaSAGuSRWO13ri+UKbzp6IeCE\nN2laGOSSWK1vUACqpQpvOJIEuePk0jQwyCWxVt9koVyhWChwuFLlNQtHefzcczRb7q4sZZ1BLonV\n+gaL5bmtx286eiHrjTo/Xj2bYqskDcIgl8RqY4OF0naQv/HoRYDd69I0MMilnGu0mtQa9VdU5NDe\nd11SthnkUs4lm8EslCtbz7164SiL5TkrcmkKGORSznWvIU8UCwXeeORCzqwvs7SxllbTJA3AIJdy\nbnWXihzgjUcvAODJ5Rcm3iZJgzPIpZxLbpiy2DXZDeBIZeFlxyVlk0Eu5dzqLl3rAPOlMgDrjfrE\n2yRpcAa5lHNrjb2CvN3VXjPIpUwzyKWc22uMvLpVkW9OvE2SBmeQSzm3d9d6O9gNcinbDHIp53Zb\nfgbbY+R2rUvZZpBLOdeva90gl7LNIJdybq/lZ3atS9PBIJdybq2xQZHCVgWemLMil6aCQS7l3Gp9\nk4XyHIVC4WXPFwsFqsWyFbmUcQa5lHNr9Q0Wd4yPJ6qlshW5lHEGuZRzq/UNFnbMWE/Ml8ru7CZl\nnEEu5Vi92WCj2ehRkVfsWpcyziCXciy5F/nOGeuJ+VKFjUadZqs1yWZJGoJBLuVYsvRsr671aqlM\nC9ho2r0uZZVBLuXY6tYNU3bvWnd3Nyn7DHIpx7a61vesyN0URso6g1zKsX5d61bkUvYZ5FKOrW5V\n5Ht1rXcq8roVuZRVBrmUY3vd+SyxdeMUJ7tJmWWQSzmWTHZb6LH8DKzIpSwzyKUcW93sPWu9WrYi\nl7LOIJdybK2R3It8j4q82A5yt2mVsssgl3IsmbV+qN/yM7vWpcwyyKUcW61vUCoUqRRLux6fd7Kb\nlHkGuZRja/VNFsuVV9yLPGFFLmWfQS7lWK9bmALMlx0jl7Ku3OtgCKEI3AFcDtSAm2OMp7uO3wh8\nEqgDPwA+DhR6nSMpO9Yam5xfXdzzeLXYrsjtWpeyq19FfgMwF2O8GrgVuD05EEJYAD4DvC/G+DPA\nMeADnXOqu50jKTs2mw02m43BKnK71qXM6hfk1wD3AcQYHwKu6Dq2Drw7xrjeeVzuPHcN8NU9zpGU\nEat9dnUDtibC1bxpipRZ/YL8KLDU9bjR6W4nxtiKMZ4BCCF8AjgUY/xar3MkZcfa1g1Tdt8MJlEt\nlr1pipRhPcfIaQfyka7HxRhjM3nQCejPAm8GPjTIObsJIZwEbhuwzZJGYLXPLUwT8+Wyk92kDOsX\n5KeA64G7QwhXAQ/vOP4F2t3pH4wxtgY85xVijCeBk93PhRBOAI/3O1fS/gzStQ7tCW8v1lcm0SRJ\n+9AvyO8BrgshnOo8vqkzU/0w8B3go8CDwDdCCAC/t9s5I2+1pAPbuvPZHjdMScyXy6yv1mm1Wnuu\nN5eUnp5B3qmyb9nx9CNd/959O6hXniMpY1a39lnvM0ZeqtCkxWazwVyp33d/SZPmJDQppwbtWk9u\nnOKENymbDHIpp1YHnbXeOe6ENymbDHIpp9YGnbVeSrZpdS25lEUGuZRTA89aL9m1LmWZQS7l1Nqg\nY+TJHdCsyKVMMsilnFptbFLucS/yRNXJblKmGeRSTq3WN/pW4wDzZStyKcsMcimn1vrcizwx7xi5\nlGkGuZRDrVaL1fomi32WnsH2ZDcrcimbDHIphzabDRqt5oAVeTvsrcilbDLIpRzaWnpW6l+RO2td\nyjaDXMqhQW9hCq4jl7LOIJdyaK0x2Bpy6N7ZzSCXssggl3Joe5/1QSpyu9alLDPIpRza3p61/xh5\nuVCkWCjYtS5llEEu5VAyRj5IRV4oFJgvVazIpYwyyKUcSu58tjDArHVoT3izIpeyySCXcqjWbAd5\nMpGtn3ZFbpBLWWSQSzmUVNfVASvy+VKZml3rUiYZ5FIObQf5YBV5tVSm3mpSbzbG2SxJ+2CQSzk0\nbJC7TauUXQa5lEPDB7mbwkhZZZBLOZQE+Vxx0K71pCJ3nFzKGoNcyqFac5NqsUyxUBjo9VUrcimz\nDHIph2qN+sDd6tDdtW5FLmWNQS7l0LBBXnWym5RZBrmUQ+tDV+TeOEXKKoNcyplWq9WpyAfbDAYc\nI5eyzCCXcmaz2aBFa19j5HatS9ljkEs5k4Tx/IBLz8CudSnLDHIpZ2rN4TaD6X6t68il7DHIpZwZ\ndlc3cItWKcsMcilnku5xJ7tJs8Egl3LmYBW5XetS1hjkUs7sJ8jniiUKWJFLWWSQSzmzNWt9iCAv\nFApUS2VnrUsZZJBLObNVkQ+x/Aza3etOdpOyxyCXcma9Ofxkt+T1dq1L2WOQSzmznzHy5PV2rUvZ\nY5BLObPfIJ8vldlsNmi2muNolqR9MsilnNnPZDfo3t3N7nUpSwxyKWdq+9gQBrr3WzfIpSwxyKWc\n2X/XupvCSFlkkEs5s77P5Wdu0yplU8/f5BBCEbgDuByoATfHGE/veM0i8DXgozHG2Hnue8DZzkse\nizF+bNQNl7Q/tWadcqFIqTjc9/j5rSC3IpeypN9X8huAuRjj1SGEdwG3d54DIIRwBfBHwCVAq/Pc\nPECM8dqxtFjSgdQa9aHHx8E7oElZ1e8r+TXAfQAxxoeAK3Ycn6Md7LHruXcCiyGE+0MID3S+AEjK\niFpjc+gZ69DdtW5FLmVJv9/mo8BS1+NGCKEYY2wCxBi/BRBC6D5nBfhcjPGuEMJbgK+GEN6anLOb\nEMJJ4LZ9tF/SkGqNOsfmFoY+z4pcyqZ+Qb4EHOl6XOwVyB2PAI8CxBh/FEJ4HrgYeGavE2KMJ4GT\n3c+FEE4Aj/f5LElDanetH6QiN8ilLOnXtX4K+AWAEMJVwMMDvOdNtMfSCSFcQruqf/YAbZQ0IvVm\ng3qrua8g315Hbte6lCX9fpvvAa4LIZzqPL4phHAjcDjGeOce59wFfDmE8GByzgBVvKQJqDUawPCb\nwbTPcWc3KYt6BnmMsQXcsuPpR3Z53bVd/64DHx5J6ySNVC2589mQa8jB5WdSVrkhjJQj+91nHbar\neCtyKVsMcilH9rs9KzhGLmWVQS7lyEGCvFoqvew9JGWDQS7lyPo+73wGUCwUmSuWrMiljDHIpRw5\nSEUO7e5115FL2WKQSzly8CAvs163IpeyxCCXcqTW7Mxa38fyM4D5csUxciljDHIpR2oHGCOHdtd6\nrVmn2XKPJykrDHIpR9ZHMEYOzlyXssQgl3JkFGPkAGvOXJcywyCXcmQUs9YB1utW5FJWGORSjhxk\ni1ZoT3YDd3eTssQgl3Lk4JPdvHGKlDUGuZQjtWadIgXKhf396rvfupQ9BrmUI7VGnWqpTKFQ2Nf5\n20HuGLmUFQa5lCPrnSDfr+3JblbkUlYY5FKOtCvy/Y2PA8yXHSOXssYgl3Kk1tgcTUVu17qUGQa5\nlBPNVpONZmPfS8/AyW5SFhnkUk5sNBrA/jeDAVgwyKXMMcilnEjufFbd553PYHv9uZPdpOwwyKWc\nWD/gZjDQPdnNMXIpKwxyKScOus86QKlQpFIsedMUKUMMciknRhHk0LknuUEuZYZBLuXEQW+Yklgo\nle1alzLEIJdyYlQVebVUcbKblCEGuZQTo5jsBrBQrlBr1mm2mqNolqQDMsilnNiqyA+w/Ay2K/qa\n3etSJhjkUk5srSM/8Bi527RKWWKQSzkxqsluyTata46TS5lgkEs5URvRGPl8ufKy95OULoNcyonR\nrSNvn++mMFI2GORSTqyPcPlZ9/tJSpdBLuXEqCryZLKbXetSNhjkUk4kQT53wOVnTnaTssUgl3Ki\n1tykWixTLBQO9D7eAU3KFoNcyolao37gbnXYrsjX7VqXMsEgl3Ki1qgfeA05GORS1hjkUk6sN+oH\nXkMOBrmUNQa5lAOtVmsMXeuOkUtZYJBLObDZbNCiNZogTya7OWtdygSDXMqBUa0hBygVilSKJbvW\npYwwyKUc2L7z2cHHyKHdvW7XupQNPb+ehxCKwB3A5UANuDnGeHrHaxaBrwEfjTHGQc6RNFmjuhd5\nYr5UtiKXMqJfRX4DMBdjvBq4Fbi9+2AI4QrgQeANQGuQcyRNXhK6o1h+1n6fikEuZUS/IL8GuA8g\nxvgQcMWO43O0gzsOcY6kCRvlGDm0g7zWqNNstfq/WNJY9Qvyo8BS1+NGp+scgBjjt2KMTw9zjqTJ\nG3mQd2au1xwnl1LX77d6CTjS9bgYY2yO+pwQwkngtj7vK2mftoN8dJPdoN1lv1AezXtK2p9+QX4K\nuB64O4RwFfDwAO859DkxxpPAye7nQggngMcH+DxJfYy6Il9wdzcpM/r9Vt8DXBdCONV5fFMI4Ubg\ncIzxzkHPGUE7JR3AenO0k92Syt5NYaT09fytjjG2gFt2PP3ILq+7ts85klI0juVn4DatUhY4CU3K\ngZGPkZftWpeywiCXcmAcy8/AIJeywCCXcqDWCVwnu0mzxyCXciCpyEc32a39Pmt1x8iltBnkUg4k\nk9LmRlWRd8bIa1bkUuoMcikHas06lWKJUmE0v/LJGPmaQS6lziCXcqDWqI9s6RlsB7kVuZQ+g1zK\ngVpjc2QT3WB7rH3NdeRS6gxyKQdqjfqIg9yd3aSsMMilGddqtVgfcZCXikUqxZJd61IGGOTSjKs1\n6jRaTQ6VqyN93/lS2a51KQMMcmnGLddrAByuzI30fedLFStyKQMMcmnGLW+2g/xQZdQVecXlZ1IG\nGOTSjFtJKvKRd61XqDXqNFutkb6vpOEY5NKMW9ncAMZQkZfbk+dqjpNLqTLIpRmXdK2PoyIHb5wi\npc0gl2Zc0rV+aAyT3cAgl9JmkEszbjnpWh/D8jMwyKW0GeTSjNua7DaGWesA697KVEqVQS7NuJVk\n+Vl5xF3rZbvWpSwwyKUZt1yvMVcsjexe5AnHyKVsMMilGbeyuTHypWfgGLmUFQa5NOOW67WRLz2D\n7orcMXIpTQa5NMPqzQa1Rn3kS8/AW5lKWWGQSzNspT6epWcAC052kzLBIJdm2NaubmMYI686Ri5l\ngkEuzbBxLT0DWHCMXMoEg1yaYcv18dzCFFx+JmWFQS7NsGR71nF0rZeKRcqFopPdpJQZ5NIMG9e9\nyBML5Ypd61LKDHJphm2NkY9h+RlAtVSxa11KmUEuzbDlzvKzsVXkBrmUOoNcmmHbFfl4grxaKrPe\nqNNstcby/pL6M8ilGbZSr1GksLVUbNSSTWE2HCeXUmOQSzNseXODQ5U5CoXCWN7fJWhS+gxyaYYt\nb9bGsj1rwiCX0meQSzOq2WqxWt8YyxryxPatTO1al9JikEszaq2+QYvW2Ca6QXv5WfuzrMiltBjk\n0oxa3toMZjxryME7oElZYJBLM2qlsz3rOCvypNt+aXN9bJ8hqTeDXJpRyfas45zsdnxuEYCXaqtj\n+wxJvRnk0ozavhf5+LrWz5tbAODsxtrYPkNSbwa5NKOWN8dfkR+rtoP8RYNcSk2518EQQhG4A7gc\nqAE3xxhPdx2/Hvg0UAe+FGP8Yuf57wFnOy97LMb4sTG0XVIPK/Xxj5HPlyrMlyp2rUsp6hnkwA3A\nXIzx6hDCu4DbO88RQqgAnweuAFaBUyGE/wacA4gxXju2VkvqK9lnfVw3TEkcn1vgJStyKTX9utav\nAe4DiDE+RDu0E28HHo0xno0xbgLfBN4LvBNYDCHcH0J4oPMFQNKEbS0/G+MYOcB51UVW6xvuty6l\npF+QHwWWuh43Ot3tybGzXcfOAceAFeBzMcafB34d+ErXOZImZGv52Zgrcie8Senq17W+BBzpelyM\nMTY7/z6749gR4EXgEeBRgBjjj0IIzwMXA8/s9SEhhJPAbUO1XFJPK/Ua86UKpeJ4v0efV20vQXtx\nY42LFo70ebWkUesX5KeA64G7QwhXAQ93Hfsh8JYQwnHaVfjPAp8DbqI9Oe43QgiX0K7cn+31ITHG\nk8DJ7udCCCeAxwf8/5C0w/Jmbezd6rBdkTvhTUpHv6/q9wDrIYRTtCe6fSqEcGMI4Vc74+K/CdwP\nfAu4K8b4LHAXcDSE8CDw58BNXVW8pAlotVqs1DfG3q0OXUFu17qUip4VeYyxBdyy4+lHuo7fC9y7\n45w68OFRNVDS8DaaDTabjbEuPUskXetW5FI6nIQmzaDtpWcT7Fq3IpdSYZBLM2h76dn4K/Kjc/MU\nKRjkUkoMcmkGTWrpGUCxUOTo3Lxd61JKDHJpBiUV+STGyKHdvX52Y41WqzWRz5O0zSCXZtAkx8ih\nPeGt3mpu3ahF0uQY5NIMWkmhIgcnvElpMMilGbR9L/IJBXmyBG3DcXJp0gxyaQYtT3CyG3Tv7mZF\nLk2aQS7NoJUJLj8DOG/OilxKi0EuzaCVzRrlQpG5Ymkin3de1TFyKS0GuTSDlusbHK5UKRQKE/m8\nrYrcrnVp4gxyaQatbNYmNj4OsFCuUC2V7VqXUmCQSzOm0Wqy1tjk0ARuYdrtvLlFK3IpBQa5NGOS\n7VkPT7Aih/bM9eV6jc1mY6KfK+WdQS7NmGQN+aQ2g0kkE97OOuFNmiiDXJox27u6Tb5rHZzwJk2a\nQS7NmO191ifftQ6uJZcmzSCXZsxyvTNGPvGu9WRTGCtyaZIMcmnGLG+uA5PbnjVxfGubVityaZIM\ncmnGPLPyEgCvWTw60c895u5uUioMcmnGPLn8AgulChfNH57o5x6rLFDAyW7SpBnk0gxZq2/wD2vn\nuPTI+RPbnjVRKhY5Upl3sps0YQa5NEOeXH4BgEsPX5DK559XXeSljTVarVYqny/lkUEuzZAnzyVB\nfn4qn398boHNZoPVzsx5SeNnkEszZKsiP5JOkB+bc8KbNGkGuTRDnlx+nsPlKhdUD6Xy+cla8hdd\ngiZNjEEuzYjlzRrPra+kMtEtkezu5n7r0uQY5NKMeGo53fFxgONbu7tZkUuTYpBLM+KJZKLbkXRm\nrEPXfuuuJZcmxiCXZsSTy88D6Vbkxzp3QHvRilyaGINcmhFPnnuBo5X5rao4DYvlCpViyclu0gQZ\n5NIMOLuxxosbq6lOdAMoFAqcOHwBz6y8ZJhLE2KQSzPgqZR3dOt25UWX0gK+c+bJtJsi5YJBLs2A\nZKLbiZQ2gun2Ty78KYoUDHJpQgxyaQZkYaJb4sjcPG87/hqeWH6BM2vn0m6ONPMMcmnKtVotnjz3\nAserixxNcaJbtysvuhSAb595KuWWSLPPIJem3EsbayxtrmeiGk/8owteR7lQ5Ntnnki7KdLMM8il\nKZf2rUt3s1ie47LzL+HHq2d5ZuWltJsjzTSDXJpyT5xrj49nYaJbtys63etOepPGyyCXplxSkb8+\nQ13rAJef/1rmiiW+feZJWq1W2s2RZpZBLk2x5c0aT5x7jgvnD3G4Uk27OS9TLZV55wWv48z68taX\nDUmjZ5BLU2qjUeeOv/trVuubXPWqN6bdnF1dafe6NHYGuTSFmq0md8VvcXrpOa686FL++esvS7tJ\nu3rH8YtZKFX4zpmnaNq9Lo1FudfBEEIRuAO4HKgBN8cYT3cdvx74NFAHvhRj/GK/cyQdTKvV4s9P\nf5fvP/804dir+ZW3XkUxxf3Ve6kUS/zjC3+Kb/3DY9z71A/4Z697B3Olnn92JA2pX0V+AzAXY7wa\nuBW4PTkQQqgAnweuA94L/FoI4VWdc6q7nSPp4O5/+u/462d/xGsXz+OWd7yHSrGUdpN6+rnXvo1D\n5Sr//am/5bbv3stDP3nc6lwaoX5fja8B7gOIMT4UQrii69jbgUdjjGcBQgjfBH4WeDfw1T3OGdpq\nfYPlzdpB3kLKqFeGWasFTVo0Wk2arRaNZpMXaqs8u3qWZ1fP8uPVszy6dIbj1UU+cdn7WCjPpdDu\n4bz20Hn8xyuv56v/7//yjWciX4p/w9efibz34jdzuDLPYnmOQ+U5FsoVihQoFAoUgAIFdu9oyGbv\ngzQK+5m02i/IjwJLXY8bIYRijLHZOXa269g54Fifc4ZRAvj0A/+V6vlHhzxVmk0F4DWLR/kXrwms\nnHmBFaZnNvi7Khfx1osP8bWn/57vP/EY8YnH0m6SlDkP3fr7J4CnY4z1Qc/pF+RLwJGux92BfHbH\nsSPAS33O2VUI4SRw227Hvv/ZL/dpopQ/f5l2AySNy+PAG4AnBj2hX5CfAq4H7g4hXAU83HXsh8Bb\nQgjHgRXa3eqfo91fuNc5u4oxngROdj8XQqgC68CbgcYA/y/an+SHRuPldR4/r/H4eY3H73Hg6WFO\nKPTacSmEUGB7BjrATcBPA4djjHeGED4A/Hvak+buijH+4W7nxBgfGep/Y/vzWzFGB8TGyGs8GV7n\n8fMaj5/XePz2c417VuQxxhZwy46nH+k6fi9w7wDnSJKkMXBDGEmSpphBLknSFMt6kP+HtBuQA17j\nyfA6j5/XePy8xuM39DXuOdlNkiRlW9YrckmS1INBLknSFDPIJUmaYga5JElTzCCXJGmKGeSSJE2x\nfjdNGbsQQpHtvdlrwM0xxtNdx68HPg3UgS/FGL+YSkOn3ADX+Ubgk7Sv8w+Aj3e229WA+l3jrtf9\nMfB8jPHfTbiJU2+An+Mrgdtp3/H1GeAjMcaNNNo6zQa4zh8Efpv2TbK+FGP8o1QaOuVCCO8CfjfG\neO2O54fKvSxU5DcAczHGq4Fbaf8SAhBCqACfB64D3gv8WgjhVam0cvr1us4LwGeA98UYf4b2feU/\nkEorp9ue1zgRQvhXwGW0/wBqeL1+jgvAHwP/Msb4HuABvFPXfvX7WU7+Ll8D/JsQwrEJt2/qhRD+\nLXAnUN3x/NC5l4Ugvwa4DyDG+BBwRdextwOPxhjPxhg3gW/Svl2qhtfrOq8D744xrncel4G1yTZv\nJvS6xoQQrgb+KfAF2hWjhtfrGr8VeB74zRDC/wLOizHGibdwNvT8WQY2gfOABdo/y34xHd6jwC/y\nyr8FQ+deFoL8KLDU9bjR6dZJjp3tOnaOdrWo4e15nWOMrRjjGYAQwieAQzHGr6fQxmm35zUOIVxM\n+5a//xpD/CB6/b24ELga+APg54D3hxCuRfvR6zpDu0L/LvC3wF/FGLtfqwHEGP+Cdtf5TkPnXhaC\nfAk40vW4GGNsdv59dsexI8CLk2rYjOl1nQkhFEMI/wl4P/ChSTduRvS6xr9EO2j+B/BbwC+HED4y\n4fbNgl7X+HnalUyMMdZpV5Q7K0kNZs/rHEJ4Pe0vpJcCJ4BXhxB+aeItnF1D514WgvwU8AsAIYSr\ngIe7jv35B1k0AAABCUlEQVQQeEsI4XgIYY5298LfTL6JM6HXdYZ2d28V+GBXF7uGs+c1jjH+QYzx\nis6klt8F/izG+F/SaeZU6/Vz/BhwOITwps7j99CuGDW8Xtd5HmgAtU64/4R2N7tGY+jcS/2mKZ0J\nKsnsSICbgJ8GDscY7wwhfIB2l2QRuCvG+IfptHS69brOwHc6/z3Ydcrvxxj/cqKNnHL9fpa7Xvcr\nQIgx/vbkWzndBvh7kXxRKgCnYoyfSqel022A6/wp4Jdpz695FPjVTi+IhhBCOEH7S/3VnZVD+8q9\n1INckiTtXxa61iVJ0j4Z5JIkTTGDXJKkKWaQS5I0xQxySZKmmEEuSdIUM8glSZpi/x9+HqY/FGBe\n5gAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fair_flips = bernoulli_flips = np.random.binomial(n=1, p=.5, size=1000)\n", "p_fair = np.array([np.product(st.bernoulli.pmf(fair_flips, p)) for p in params])\n", "p_fair = p_fair / np.sum(p_fair)\n", "plt.plot(params, p_fair)\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Basically we created 1,000 fair coin flips and then generated the sampling distribution like we did before (except we divided by the sum of the sampling distribution to make the values sum to 1). Now we have a \"fair coin\" prior on our parameters. This basically means that before we saw any data we thought coin flips were fair. And we can see that assumption in our prior distribution by the fact that our prior distribution peaks at .5 and is almost all between .4 and .6.\n", "\n", "I know what you are thinking - this is pretty boring. The sampling and prior distributions look exactly the same. So lets mix things up. Lets keep our fair prior but change our data to be an unfair coin:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjgAAAGECAYAAAA7lVplAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nJEladf5/njpOK65uYM4AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "unfair_flips = bernoulli_flips = np.random.binomial(n=1, p=.8, size=1000)\n", "p_unfair = np.array([np.product(st.bernoulli.pmf(unfair_flips, p)) for p in params])\n", "fig, axes = plt.subplots(2, 1, sharex=True)\n", "axes[0].plot(params, p_unfair)\n", "axes[0].set_title(\"Sampling Distribution\")\n", "axes[1].plot(params, p_fair)\n", "axes[1].set_title(\"Prior Distribution\")\n", "sns.despine()\n", "plt.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ah - now this is interesting. We have strong data evidence of an unfair coin (since we generated the data we know it is unfair with p=.8), but our prior beliefs are telling us that coins are fair. How do we deal with this?\n", "\n", "#Bayes Theorem (Posterior Distribution)\n", "\n", "Bayes theorem is what allows us to go from our sampling and prior distributions to our posterior distribution. The **posterior distribution is the $P(\\theta|X)$**. Or in English, the probability of our parameters given our data. And if you think about it that is what we really want. We are typically given our data - from maybe a survey or web traffic - and we want to figure out what parameters are most likely given our data. So how do we get to this posterior distribution? Here comes some math (don't worry it is not too bad):\n", "\n", "By definition, we know that (If you don't believe me, check out this [page](https://people.richland.edu/james/lecture/m170/ch05-cnd.html) for a refresher): \n", "* $P(A|B) = \\dfrac{P(A,B)}{P(B)}$. Or in English, the probability of seeing A given B is the probability of seeing them both divided by the probability of B.\n", "* $P(B|A) = \\dfrac{P(A,B)}{P(A)}$. Or in English, the probability of seeing B given A is the probability of seeing them both divided by the probability of A.\n", "\n", "You will notice that both of these values share the same numerator, so:\n", "* $P(A,B) = P(A|B)*P(B)$\n", "* $P(A,B) = P(B|A)*P(A)$\n", "\n", "Thus:\n", "\n", "$P(A|B)*P(B) = P(B|A)*P(A)$\n", "\n", "Which implies:\n", "\n", "$P(A|B) = \\dfrac{P(B|A)*P(A)}{P(B)}$\n", "\n", "And plug in $\\theta$ for $A$ and $X$ for $B$:\n", "\n", "$P(\\theta|X) = \\dfrac{P(X|\\theta)*P(\\theta)}{P(X)}$\n", "\n", "Nice! Now we can plug in some terminology we know:\n", "\n", "$Posterior = \\dfrac{likelihood * prior}{P(X)}$\n", "\n", "But what is the $P(X)$? Or in English, the probability of our data? That sounds wierd... Let's go back to some math and use $B$ and $A$ again:\n", "\n", "We know that $P(B) = \\sum_{A} P(A,B)$ (check out this [page](http://en.wikipedia.org/wiki/Marginal_distribution) for a refresher)\n", "\n", "And from our definitions above, we know that:\n", "\n", "$P(A,B) = P(B|A)*P(A)$\n", "\n", "Thus:\n", "\n", "$P(B) = \\sum_{A} P(B|A)*P(A)$\n", "\n", "Plug in our $\\theta$ and $X$:\n", "\n", "$P(X) = \\sum_{\\theta} P(X|\\theta)*P(\\theta)$\n", "\n", "Plug in our terminology:\n", "\n", "$P(X) = \\sum_{\\theta} likelihood * prior$\n", "\n", "Wow! Isn't that awesome! But what do we mean by $\\sum_{\\theta}$. This means to sum over all the values of our parameters. In our coin flip example, we defined 100 values for our parameter p, so we would have to calculated the likelihood * prior for each of these values and sum all those anwers. That is our denominator for Bayes Theorem. Thus our final answer for Bayes is:\n", "\n", "$Posterior = \\dfrac{likelihood * prior}{\\sum_{\\theta} likelihood * prior}$\n", "\n", "That was a lot of text. Let's do some more coding and put everything together." ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def bern_post(n_params=100, n_sample=100, true_p=.8, prior_p=.5, n_prior=100):\n", " params = np.linspace(0, 1, n_params)\n", " sample = np.random.binomial(n=1, p=true_p, size=n_sample)\n", " likelihood = np.array([np.product(st.bernoulli.pmf(sample, p)) for p in params])\n", " #likelihood = likelihood / np.sum(likelihood)\n", " prior_sample = np.random.binomial(n=1, p=prior_p, size=n_prior)\n", " prior = np.array([np.product(st.bernoulli.pmf(prior_sample, p)) for p in params])\n", " prior = prior / np.sum(prior)\n", " posterior = [prior[i] * likelihood[i] for i in range(prior.shape[0])]\n", " posterior = posterior / np.sum(posterior)\n", " \n", " fig, axes = plt.subplots(3, 1, sharex=True, figsize=(8,8))\n", " axes[0].plot(params, likelihood)\n", " axes[0].set_title(\"Sampling Distribution\")\n", " axes[1].plot(params, prior)\n", " axes[1].set_title(\"Prior Distribution\")\n", " axes[2].plot(params, posterior)\n", " axes[2].set_title(\"Posterior Distribution\")\n", " sns.despine()\n", " plt.tight_layout()\n", " \n", " return posterior" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjgAAAI4CAYAAABndZP2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "example_post = bern_post()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You will notice that I set 100 as the number of observations for the prior and likelihood. This increases the variance of our distributions. More data typically decreases the spread of a distribution. Also, as you get more data to estimate your likelihood, the prior distribution matters less. " ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjgAAAI4CAYAAABndZP2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nBRwRERFpOwo4IiIi0nb+P71Mo44icH1fAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "moredata_post = bern_post(n_sample=1000)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can see that effect in the graphs above. Because we have more data to help us estimate our likelihood our posterior distribution is closer to our likelihood. Pretty cool.\n", "\n", "##Conclusion\n", "\n", "There you have it. An introduction to Bayes Theorem. Now if you ever are doubting the fairness of a coin you know how to investigate that problem! Or maybe the probability of a population voting yes for a law? Or any other yes/no outcome. I know what you are thinking - not that impressive. You want to predict things or run a fancy algorithm... in time my friends. I actually would like to write a series of posts leading up to Bayesian Linear Regression. Hopefully this is the first post in that series :)\n", "\n", "###Some Side Notes\n", "\n", "1. You will notice that the denominator for Bayes Theorem is just a constant. So if you only want to get the maximum posterior value, you don't even need to calculate that constant. For this reason you will often see the posterior shown as proportional (or $\\propto$ in math) to the likelihood * prior.\n", "2. Frequentist statistics is focused on the likelihood. Or you could say that frequentists are bayesians with a non-informative prior (like a uniform distribution). But don't hate on frequentists too much; most of bayesian inference in applied settings relies of frequentists statistics.\n", "3. Now that you know frequentist statistics focuses on the likelihood it is much clearer why people often misinterpret frequentist confidence intervals. The likelihood is $P(X|\\theta)$ - or the probability of our data given our parameters. That is a bit wierd because we are given our data, not our parameters. What most frequentists models do is take the maximum of the likelihood distribution (or Maximum Likelihood Estimation (MLE)). Basically find what parameters maximize the probability of seeing our data. The important point here is that they are treating the data as random, so what the frequentist confidence interval is saying is that if you were to keep getting new data (maybe some more surveys) and calculate confidence intervals on each of these new samples, 95% of these samples would have confidence intervals that contain the true parameter you are trying to estimate." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "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": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }