{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Bayesian Statistics Using Sampling Methods\n", "\n", "This workbook adds more detail on the theoretical underpinnings of Metropolis Hastings MCMC and slightly tweaks and expounds on some examples from Thomas Wiecki's excellent [blogpost on this topic](http://twiecki.github.io/blog/2015/11/10/mcmc-sampling/). Of course, all errors are mine. \n", "\n", "In order to understand our parameters $\\theta$, MH MCMC trys to sample from our parameters' distribution, which is unknown. This distribution is the posterior, or $Pr(\\theta|y)$. If we can somehow construct this distribution we can extract information about $\\theta$. \n", "\n", "To see this, consider the following trivial example. Suppose we have one parameter and are somehow able to construct the posterior distribution. For simplicity, suppose the posterior is distributed $N(0,1)$ (the standard normal distribution). If we take draws from this distribution, we can learn alot about its shape:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "from scipy.stats import norm,uniform,lognorm\n", "import matplotlib.pyplot as plt\n", "import seaborn as sbn\n", "import warnings\n", "warnings.filterwarnings('ignore')\n", "\n", "sbn.set_style('white')\n", "sbn.set_context('talk')" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# define the posterior distribution\n", "def posterior(mean,std,N):\n", " return norm.rvs(mean,std,N)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The mean of theta is 0.002552528159105638\n", "The standard deviation of theta is 1.0027777454917184\n", "95% CI of theta is [-1.95823558 1.97137704]\n" ] }, { "data": { "image/png": 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\n", 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