{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt, seaborn as sn, numpy as np, pandas as pd\n", "sn.set_context('notebook')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Distributions\n", "\n", "This notebook introduces some fundamental concepts that we’ll use extensively later on: **joint**, **marginal** and **conditional** distributions; the **sum** and **product** rules; and **Bayes’ Theorem**. \n", "\n", "The aim is to avoid lots of formal statistical notation and instead develop some intuition about what these things actually mean. Most of the examples use just two variables to make things easier to visualise, but everything here can be extended in a straightforward way to much larger parameter spaces. \n", "\n", "## 1.1. One dimensional probability densities\n", "\n", "Suppose we have a continuous random variable, $x$, with a **Gaussian** (or **Normal**) distribution with mean, $\\mu$, and standard deviation, $\\sigma$. This is often written as $x\\sim\\mathcal{N}(\\mu, \\sigma)$.\n", "\n", "The equation for the probability density, $P(x)$, is:\n", "\n", "$$P(x)=\\frac{1}{{\\sigma \\sqrt {2\\pi } }}e^{{{ - ( {x - \\mu } )^2 /{2\\sigma ^2 }}}}$$\n", "\n", "which we can plot like this:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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z+R1ELuP2UqUpVqAAn839lKNHj7iOI0HqagrCJGPMYGNM1uRfTFoK+lbOKxAi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\ngDNOI9QuX5bmVaowd+1qNm7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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def gaussian_function(x, mu=0, sigma=1):\n", " \"\"\" Simple example function to return the probability density of a Gaussian with mean, mu, \n", " and standard deviation, sigma, evaluated at values in x.\n", " \n", " Note that a better function is available in scipy.stats.norm - this version is just \n", " for illustration.\n", " \"\"\"\n", " return (1/(sigma*(2*np.pi)**0.5))*np.exp(-((x - mu)**2)/(2*sigma**2))\n", "\n", "# Pick some example values for mu and sigma\n", "mu = 5\n", "sigma = 3\n", "\n", "# Calculate probability densities, P_x, at a range of values of x\n", "x = np.arange(-5, 15, 0.1)\n", "P_x = gaussian_function(x, mu=mu, sigma=sigma)\n", "\n", "# Plot\n", "plt.plot(x, P_x, 'k-')\n", "plt.fill_between(x, P_x, where=((8<=x)&(x<=10)), alpha=0.3, color='red') # Explained below\n", "plt.xlabel('$x$', fontsize=20)\n", "plt.ylabel('$P(x)$', fontsize=20)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the values on the $P(x)$ axis are probability **densities** rather than probabilities: because $x$ is a continuous variable, the probability of it taking any exact value is zero. Instead, we can calculate the probability of $x$ lying *within a particular range* by integrating over that range. For example, the probability of $x$ lying between $a$ and $b$ is given by:\n", "\n", "$$P(a < x \\leq b)={\\int_{a}^{b} P(x) dx}$$\n", "\n", "which, for the example above where $a=8$ and $b=10$, is equal to the area shaded in red on the plot.\n", "\n", "Also, because $x$ must take some real value, the total area under the curve is equal to **1** i.e.\n", "\n", "$${\\int_{-\\infty}^{\\infty} P(x) dx} = 1$$\n", "\n", "## 1.2. More than one dimension: the joint distribution\n", "\n", "Suppose we have two independent random variables, $P(x)$ and $P(y)$, both normally distributed with means $\\mu_x$ and $\\mu_y$ and standard deviations $\\sigma_x$ and $\\sigma_y$ respectively." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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cK0kQPMjv7yUrK4vc3Ny4jy1TDEKIaIR3UXR3BQFkqeNcuW4nxdBJaA8D64FR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\n6Y5CJMjExARf/epXCQQC3H//Q2RnZ5sOKeV98pO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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Pick some example values for mu and sigma\n", "mu_x, mu_y = 5, 8\n", "sigma_x, sigma_y = 3, 5\n", "\n", "# Calculate densities\n", "x = np.arange(-10, 30, 0.1)\n", "P_x = gaussian_function(x, mu=mu_x, sigma=sigma_x)\n", "P_y = gaussian_function(x, mu=mu_y, sigma=sigma_y)\n", "\n", "# Plot\n", "fig, axes = plt.subplots(nrows=1, ncols=2)\n", "data = {'x':P_x, 'y':P_y}\n", "\n", "for idx, item in enumerate(['x', 'y']):\n", " axes[idx].plot(x, data[item], 'k-')\n", " axes[idx].set_xlabel('$%s$' % item, fontsize=20)\n", " axes[idx].set_ylabel('$P(%s)$' % item, fontsize=20)\n", "\n", "fig.subplots_adjust(wspace=0.8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we pick values at random from $x$ and $y$, we can calculate the **joint probability distribution**, $P(x,y)$:\n", "\n", "$$P(x,y)=P(x)P(y)$$\n", "\n", "We can represent this joint distribution as a surface in three dimensions, where the \"hills\" in the landscape represent areas with higher probability density." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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MaAQrGnWe536saLRoWf4nGs3/GPE4Rtxpcus+96539zHi8ZLHzr+OOq9t1Rlr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\nzm08F03z8MvLES/+EHHNiWnw2YFNaEtuJrHx3ymd85szX8MpoI8MEsiw0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7r8QnPvFHpFLpU8q1nnzyyRfItf6zePjhH/ODH9w3adk99/wvbrjhJp5/fmNzWbVaxXEmSlcd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\n3wWbzURRVhEIePH7HdhsPpKTj0NRTgFAVUFVy9m7910SEtqSmOjG5/uySnW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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import scipy.stats as stats\n", "from mpl_toolkits.mplot3d import Axes3D\n", "\n", "x = y = np.arange(-10, 30, 0.1)\n", "\n", "# Get a \"mesh\" for the x and y values\n", "X, Y = np.meshgrid(x, y)\n", "\n", "# This time we'll use the \"norm\" function from scipy.stats, although our gaussian_function \n", "# from above would work just as well\n", "norm_x = stats.norm.pdf(x, loc=mu_x, scale=sigma_x)\n", "norm_y = stats.norm.pdf(y, loc=mu_y, scale=sigma_y)\n", "M = np.dot(norm_y[:, None], norm_x[None, :])\n", "\n", "# Plot\n", "fig = plt.figure(figsize=(15,5))\n", "\n", "# First subplot in 3D\n", "ax1 = fig.add_subplot(121, projection='3d')\n", "ax1.plot_surface(X, Y, M, cmap=plt.cm.terrain)\n", "ax1.view_init(azim=390)\n", "ax1.set_xlabel('$x$', fontsize=16)\n", "ax1.set_ylabel('$y$', fontsize=16) \n", "ax1.set_zlabel('$P(x,y)$', fontsize=16)\n", "\n", "# Second subplot as a contour plot\n", "ax2 = fig.add_subplot(122)\n", "ax2.contour(X, Y, M)\n", "ax2.imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, \n", " extent=(-10, 30, -10, 30))\n", "ax2.set_xlabel('$x$', fontsize=16)\n", "ax2.set_ylabel('$y$', fontsize=16) " ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "In the 1D example above, the total area under the curve was equal to 1 and to find the probabiltiy of $x$ lying between $a$ and $b$ we integrated over that interval. Here the situation is exactly the same, except now we have an extra dimension. This means that the total **volume** under the surface must be equal to 1, because the values for $x$ and $y$ must lie somewhere in the $x-y$ plane.\n", "\n", "In a similar way, the probability of $x$ lying between $a$ and $b$ **and** $y$ lying between $c$ and $d$ is given by a double integral:\n", "\n", "$$P(a < x \\leq b, c < y \\leq d) = {\\int_{c}^{d} \\int_{a}^{b} P(x,y) dx dy}$$\n", "\n", "This integral corresponds to the **volume** beneath the surface element bounded by $a$, $b$, $c$ and $d$, which is illustrated by the red rectangle on the plot below." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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VkvQWGq/10z/WR6gN52qGoWnD8QnNodGQ/SMRQ9MCmS/MSb3pVJCzicqGkEN8uCCKbRMJ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\n20zduFPqUlJVipqUTA3OohxJeL6HSqioKElF6r/5osstt4yxfl2Rs+f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SBK7QUb8y5B3G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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from matplotlib.patches import Rectangle\n", "\n", "# Pick some example values for a, b, c and d\n", "a, b, c, d = 5, 10, 0, 5\n", "\n", "# Contour plot\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.contour(X, Y, M)\n", "ax.imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, alpha=0.6,\n", " extent=(-10, 30, -10, 30))\n", "ax.add_patch(Rectangle((a, c), b-a, d-c, facecolor='red', alpha=0.5))\n", "ax.set_xlabel('$x$', fontsize=16)\n", "ax.set_ylabel('$y$', fontsize=16) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1.3. Independent and dependent variables\n", "\n", "In the example above, $x$ and $y$ were **independent** variables: we first drew a value from $x$, then independently drew one from $y$, and then multiplied the probability densities together to get the density for the joint distribution. Because we used different values for the standard deviations of our two Gaussians ($\\sigma_x = 3$ and $\\sigma_y = 5$ in the example above), the contours of the joint distribution are ellipses, which are elongated parallel to the $y$ axis (reflecting the fact that the distribution has greater variability in this direction). More specifically, the **major axis** of each elipse is parallel to the $y$ axis, while the **minor axis** is parallel to the $x$ axis. It is important to understand that this is *only* the case because $x$ and $y$ are **independent**.\n", "\n", "The major and minor axes are related to the **eigenvectors** of the **covariance matrix**. This sounds complicated, but you can think of the covariance matrix as a table showing how each parameter (in this case $x$ and $y$) varies with the others. For the two-variable example here, it looks something like this:\n", "\n", "| |$x$|$y$|\n", "|:-:|:-:|:-:|\n", "|$x$|$\\sigma_x^2$|0|\n", "|$y$|0|$\\sigma_y^2$|\n", "\n", "Note that **variance** is simply the square of the standard deviation (i.e. $\\sigma_x^2$ and $\\sigma_y^2$). The two zeros in the table indicate that $x$ does not vary with $y$ and vice versa - in other words, these variables are independent.\n", "\n", "On the other hand, suppose we're considering a dataset where $x$ and $y$ depend on one another. For example, we may be working with a dataset where $x$ corresponds to peoples' weight and $y$ to their height. We would not expect these two variables to be independent, because we know that in general tall people tend to be heavier. Of course, we wouldn't use Gaussian distributions to represent height and weight (not least because the Gaussians assign non-zero probabilities to negative values). However, for the sake of keeping things simple we'll stick with the Gaussian example for now. If $x$ and $y$ co-vary, the covariance matrix looks like this:\n", "\n", "| |$x$|$y$|\n", "|:-:|:-:|:-:|\n", "|$x$|$\\sigma_x^2$|$\\sigma_{yx}^2$|\n", "|$y$|$\\sigma_{xy}^2$|$\\sigma_y^2$|\n", "\n", "where $\\sigma_{xy}^2 = \\sigma_{yx}^2 \\neq 0$.\n", "\n", "The result is that the major and minor axes (the eigenvectors) of the probability density contour elipses are rotated and stretched with respect to the $x$ and $y$ axes (see plot below). This occurs because the two variables are **dependent** on one another." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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3bUhx944UZ6YMXzg8XGlg5eXjSDwtBLtUNz/uXYfB0NPTw8SFCRtxKxULd5MQ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from scipy.stats import multivariate_normal\n", "\n", "# Use the same mu and sigma values from above. We need new values for sigma_xy and sigma_yx\n", "sigma_xy = sigma_yx = 3\n", "mus = [mu_x, mu_y]\n", "cov = [[sigma_x**2, sigma_yx**2], [sigma_xy**2, sigma_y**2]] # Covariance matrix\n", "\n", "# Grid of x and y values\n", "x = y = np.arange(-10, 30, 0.1)\n", "X, Y = np.meshgrid(x, y)\n", "pos = np.empty(X.shape + (2,))\n", "pos[:, :, 0] = X\n", "pos[:, :, 1] = Y\n", "\n", "# Create a \"frozen\" distribution\n", "rv = multivariate_normal(mus, cov)\n", "\n", "# Sample from the distribution\n", "M = rv.pdf(pos)\n", "\n", "# Contour plot\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.contour(X, Y, M)\n", "ax.imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, alpha=0.6,\n", " extent=(-10, 30, -10, 30))\n", "ax.set_xlabel('$x$', fontsize=16)\n", "ax.set_ylabel('$y$', fontsize=16) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1.4. Marginal and conditional distributions\n", "\n", "In section 1.2 we introduced the idea of a joint distribution and showed how joint probabilities can be interpreted as volumes beneath particular surface elements:\n", "\n", "$$P(a < x \\leq b, c < y \\leq d) = {\\int_{c}^{d} \\int_{a}^{b} P(x,y) dx dy}$$\n", "\n", "However, suppose we're only interested in the distribution over $x$, independent of the value of $y$. If we return to the height and weight example from above, we may wish to look at the joint distribution of height and weight and answer questions such as \"*What is the probability of weighing between 70 and 90 kgs, regardless of height?*\"\n", "\n", "To answer this, we can simply expand the limits in one of the integrals - the probability of $x$ lying between $a$ and $b$ regardless of $y$ is given by:\n", "\n", "$$P(a < x \\leq b) = P(a < x \\leq b, -\\infty < y \\leq \\infty) = {\\int_{-\\infty}^{\\infty} \\int_{a}^{b} P(x,y) dx dy}$$\n", "\n", "which is equivalent to the volume under the long strip shaded red on the left-hand plot below." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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BfaXETedJDWdJj+QI65K1Kw0qqgXpAdD7NMKaRlATDDBCTASJlow5RWGJoFTa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\nHr4e7NJ5mp9kE2W8V1zKslQ3mqZYrkVZVhTNFI1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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Pick some example values for a, b, c and d\n", "a, b, c, d = 5, 10, 0, 5\n", "\n", "# Set up for plotting\n", "fig, axes = plt.subplots(nrows=1, ncols=2)\n", "\n", "# Integrate out y\n", "axes[0].contour(X, Y, M)\n", "axes[0].imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, alpha=0.6,\n", " extent=(-10, 30, -10, 30))\n", "axes[0].axvspan(a, b, alpha=0.5, color='red')\n", "axes[0].set_xlabel('$x$', fontsize=16)\n", "axes[0].set_ylabel('$y$', fontsize=16)\n", "axes[0].set_title('Integrating out $y$')\n", "\n", "# Integrate out x\n", "axes[1].contour(X, Y, M)\n", "axes[1].imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, alpha=0.6,\n", " extent=(-10, 30, -10, 30))\n", "axes[1].axhspan(c, d, alpha=0.5, color='red')\n", "axes[1].set_xlabel('$x$', fontsize=16)\n", "axes[1].set_ylabel('$y$', fontsize=16)\n", "axes[1].set_title('Integrating out $x$')\n", "\n", "plt.subplots_adjust(wspace=0.8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we're interested in deriving an equation for the probability density distribution over $x$, rather than the actual probability of $x$ lying within a particular interval, this integral becomes:\n", "\n", "$$P(x) = \\int_y P(x,y) dy$$\n", "\n", "which is known as the **sum rule**. The process itself is called **marginalisation** and it's often said that we are \"*integrating out*\" the dependence on $y$. Integrating $y$ over all possible values essentially takes all the probability along the $y$ axis, sums it, and then stacks it up along the $x$ axis - hence the term *marginalisation*. A nice way to visualise this is to plot the **marginal distributions** for $x$ and $y$ along the edges of the plot:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# This time we'll use the handy jointplot function in Seaborn to plot \n", "# the same multivariate normal distribution as above\n", "# Rather than evaluating the densities over a regular grid, for simplicity\n", "# we'll just choose 1000 values at random\n", "data = np.random.multivariate_normal(mus, cov, size=1000)\n", "data = pd.DataFrame(data, columns=['$x$', '$y$'])\n", "sn.jointplot('$x$', '$y$', data, kind='kde', stat_func=None)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another question we might want to ask is, \"*Given that a person weighs between 70 and 90 kgs, what is the probability that their height is between 1.6 and 1.8 m?*\" This is a **conditional probability**, because we're interested in the probability that height lies in a certain range, *conditioned on the fact* that weight is between 70 and 90 kgs. \n", "\n", "More generally, we might ask, \"*What is the probability that $y$ lies between $c$ and $d$, **given** that $x$ lies between $a$ and $b$?*\" This is usually written as\n", "\n", "$$P(c < y \\leq d \\mid a < x \\leq b)$$\n", "\n", "where the $\\mid$ symbol is read as \"given\".\n", "\n", "To evaluate this conditional probability, we need to redistribute the volume beneath our probability density surface. We **know** that $a < x \\leq b$, so any probability density outside this range is now zero. Densities within the range therefore need to be re-weighted (\"re-normalised\"), so that the total probability of being anywhere within the strip bounded by $a < x \\leq b$ is equal to 1.\n", "\n", "From the discussion above, we know that the joint distribution is given by:\n", "\n", "$$P(a < x \\leq b, c < y \\leq d) = {\\int_{c}^{d} \\int_{a}^{b} P(x,y) dx dy}$$\n", "\n", "and the marginal distribution of $x$ by:\n", "\n", "$$P(a < x \\leq b, -\\infty < y \\leq \\infty) = {\\int_{-\\infty}^{\\infty} \\int_{a}^{b} P(x,y) dx dy}$$\n", "\n", "On the plot below, the volume under the surface representing the marginal is shaded red while the volume for the joint probability we want to work out is shaded black." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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/uUi8M6kMEweBFFDONtG/YhMDXWsoNLVNer+NI/2s6NlL29kj2Km2VFF3FJ2W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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Pick some example values for a, b, c and d\n", "a, b, c, d = 5, 10, 0, 5\n", "\n", "# Contour plot\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "ax.contour(X, Y, M)\n", "ax.imshow(M, interpolation='none', origin='lower',\n", " cmap=plt.cm.terrain, alpha=0.6,\n", " extent=(-10, 30, -10, 30))\n", "ax.axvspan(a, b, alpha=0.5, color='red')\n", "ax.add_patch(Rectangle((a, c), b-a, d-c, color='black'))\n", "ax.set_xlabel('$x$', fontsize=16)\n", "ax.set_ylabel('$y$', fontsize=16) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hopefully you can see from this illustration that the probability of $y$ lying between $c$ and $d$ **given** that $x$ lies between $a$ and $b$ is equal to the ratio of the volume beneath the black square to the volume beneath the red strip. We don't need to consider the area outside of the red strip because our calculation is *conditioned* on the assumption that the probability density in this region is zero.\n", "\n", "So, the volume beneath the black square is the **joint probability**; the volume beneath the red strip is the **marginal probability**; and the ratio of the two is the **conditional probability**:\n", "\n", "$$P(c < y \\leq d \\mid a < x \\leq b) = \\frac{P(a < x \\leq b, c < y \\leq d)}{P(a < x \\leq b)}$$\n", "\n", "In cleaner notation, we can write:\n", "\n", "$$P(y \\mid x) = \\frac{P(x,y)}{P(x)}$$\n", "\n", "This is known as the **product rule** and we can combine it with the **sum rule** to derive Bayes' famous equation. First, note that\n", "\n", "$$P(x,y) = P(y,x)$$ \n", "\n", "and therefore\n", "\n", "$$P(x,y) = P(y \\mid x)P(x) = P(y,x) = P(x \\mid y)P(y)$$\n", "\n", "This can be rearranged to give one form of Bayes' equation\n", "\n", "$$P(y \\mid x) = \\frac{P(x \\mid y)P(y)}{P(x)}$$\n", "\n", "We can now use the **sum rule** to re-write the denominator\n", "\n", "$$P(y \\mid x) = \\frac{P(x \\mid y)P(y)}{\\int_y P(x,y) dy}$$\n", "\n", "and finally use the **product rule** to re-write it again\n", "\n", "$$P(y \\mid x) = \\frac{P(x \\mid y)P(y)}{\\int_y P(x \\mid y)P(y) dy}$$\n", "\n", "At first glance this version might look more complicated than the others, but note that in this form **the denominator is the integral of the numerator**, which is a useful rearrangement.\n", "\n", "Bayes' equation is one of the fundamental building blocks for everything that follows. Hopefully this notebook has helped to provide some basic intuition as to where it comes from and how it relates to the properties of some simple distributions." ] } ], "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 }