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@tmarthal
Created October 16, 2013 21:30
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{
"metadata": {
"name": ""
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Hoeffding Inequality\n",
"\n",
"Run a computer simulation for flipping 1,000 virtual fair coins. Flip each coin independently 10 times. Focus on 3 coins as follows: c1 is the first coin flipped, crand is a coin chosen randomly from the 1,000, and cmin is the coin which had the minimum frequency of heads (pick the earlier one in case of a tie). Let \u03bd1, \u03bdrand, and \u03bdmin be the fraction of heads obtained for the 3 respective coins out of the 10 tosses. Run the experiment 100,000 times in order to reduce variability in your results (note that crand and cmin will change from run to run) and compute average values for the \u03bd\u2019s.\n",
"\n",
" 1. \u03bdmin is closest to:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import numpy as np\n",
"import random\n"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 1
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"num_samples = 100"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 11
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"v1,vrand,vmin = 0,0,0\n",
"values = []\n",
"for _ in range(num_samples):\n",
" # heads=1,Tails corresponds to 0, such that sum of the array is the sum of the heads\n",
" data = [[1 if x >= 0.5 else 0 for x in rand(10)] for i in range(1000)]\n",
" \n",
" c1 = data[0]\n",
" crand = data[random.randint(0, 999)]\n",
" cmin = data[0]\n",
" min_heads = sum(data[0])\n",
" for i in range(1,1000):\n",
" isum = sum(data[i])\n",
" if sum(data[i]) < min_heads:\n",
" cmin = data[i]\n",
" min_heads = sum(data[i])\n",
" #print i, vmin\n",
" #print c1,crand,cmin, min_heads\n",
" \n",
" v1 += sum(c1)\n",
" vrand += sum(crand)\n",
" vmin += min_heads\n",
" values.append(vmin)\n",
"v1,vrand,vmin"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 9,
"text": [
"(506, 509, 38)"
]
}
],
"prompt_number": 9
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# so the average vmin\n",
"1.0*vmin / num_samples"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 13,
"text": [
"0.38"
]
}
],
"prompt_number": 13
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# This is vmin aggregating -- ends up at 38 with num_samples==100\n",
"plt.plot(range(num_samples),values)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 15,
"text": [
"[<matplotlib.lines.Line2D at 0x10d94f3d0>]"
]
},
{
"metadata": {},
"output_type": "display_data",
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flpZGTk4OADk5OXXJPpzNmTOH4uJiCgsLWbp0KTfccANvvPGGLeciIiKCHj16\nsHPnTgDWrFlDbGwsqamptpuL6OhovvjiC6qqqjAMgzVr1hATE2PLuTihob8TaWlpLF26lOrqagoL\nC/nuu+8YOnRo44P5uwFgGIaxcuVKIyoqyujdu7cxZ86cQHxE0Prb3/5mOBwOY9CgQUZcXJwRFxdn\nfPTRR8b+/fuNxMREo0+fPkZSUpJx4MABq0NtVW6320hNTTUMw7DtXOTn5xtDhgwxBg4caIwePdqo\nqKiw7VzMnTvXiImJMfr372/cddddRnV1tW3m4rbbbjO6d+9utGvXzoiMjDReffXVRn/3p556yujd\nu7fRt29fIy8vz+v4uvhIRCSMBN2Np0VExHdK6iIiYURJXUQkjCipi4iEESV1EZEwoqQuIhJGlNRF\nRMKIkrqISBj5f0ylJd71JxZ+AAAAAElFTkSuQmCC\n",
"text": [
"<matplotlib.figure.Figure at 0x10d918150>"
]
}
],
"prompt_number": 15
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "pyout",
"prompt_number": 9,
"text": [
"0.38"
]
}
],
"prompt_number": 9
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# This is what the data looks like : 10 flips of a coin, 1000 times\n",
"[[1 if x >= 0.5 else 0 for x in rand(10)] for i in range(1000)]"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 2,
"text": [
"[[0, 0, 1, 1, 1, 0, 1, 1, 0, 1],\n",
" [1, 1, 1, 0, 0, 1, 1, 0, 1, 1],\n",
" [1, 0, 0, 1, 0, 1, 0, 1, 1, 0],\n",
" [1, 0, 1, 0, 0, 0, 0, 0, 0, 1],\n",
" [0, 1, 1, 0, 1, 1, 1, 0, 1, 1],\n",
" [1, 1, 1, 1, 1, 0, 1, 0, 1, 0],\n",
" [0, 1, 1, 1, 0, 1, 1, 1, 0, 1],\n",
" [0, 1, 1, 1, 0, 1, 0, 0, 0, 1],\n",
" [1, 1, 0, 1, 0, 1, 1, 1, 1, 1],\n",
" [0, 1, 1, 1, 0, 0, 0, 1, 1, 0],\n",
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" [0, 0, 1, 1, 0, 0, 0, 1, 1, 1],\n",
" [0, 1, 1, 0, 1, 0, 1, 0, 0, 1],\n",
" [0, 1, 1, 0, 0, 0, 0, 0, 0, 0],\n",
" [1, 1, 0, 1, 1, 1, 0, 1, 1, 1],\n",
" [0, 0, 0, 1, 0, 1, 0, 0, 0, 1],\n",
" [0, 1, 1, 1, 0, 0, 1, 0, 1, 1],\n",
" [1, 1, 1, 1, 1, 0, 1, 0, 0, 0],\n",
" [0, 1, 0, 0, 0, 0, 1, 1, 1, 0],\n",
" [1, 0, 0, 1, 0, 0, 0, 1, 0, 0],\n",
" [1, 0, 1, 0, 0, 1, 1, 0, 0, 0],\n",
" [1, 1, 1, 0, 1, 1, 0, 1, 1, 1],\n",
" [0, 0, 0, 1, 1, 1, 0, 0, 1, 0],\n",
" [0, 0, 1, 0, 1, 0, 1, 1, 0, 1],\n",
" [0, 1, 1, 0, 0, 0, 0, 0, 1, 1],\n",
" [0, 0, 1, 1, 1, 1, 1, 0, 1, 1],\n",
" [1, 1, 1, 0, 0, 1, 0, 1, 0, 0],\n",
" [0, 0, 0, 0, 0, 1, 0, 0, 0, 1],\n",
" [0, 0, 1, 1, 1, 0, 1, 1, 0, 1],\n",
" [1, 1, 0, 1, 0, 0, 1, 1, 0, 0],\n",
" [1, 1, 0, 0, 1, 0, 0, 1, 0, 1],\n",
" [1, 1, 0, 0, 0, 1, 0, 0, 0, 1],\n",
" [1, 1, 0, 0, 1, 1, 1, 0, 0, 1],\n",
" [0, 1, 0, 0, 0, 1, 0, 1, 1, 1],\n",
" [1, 1, 1, 0, 1, 1, 0, 0, 0, 1],\n",
" [1, 1, 0, 1, 1, 1, 1, 0, 1, 1],\n",
" [0, 0, 0, 1, 0, 1, 1, 0, 0, 1],\n",
" [1, 1, 1, 1, 1, 1, 0, 1, 1, 0],\n",
" [1, 0, 0, 0, 0, 0, 0, 1, 1, 0],\n",
" [0, 1, 1, 1, 0, 1, 1, 0, 0, 1],\n",
" [1, 1, 0, 1, 1, 0, 1, 0, 0, 0],\n",
" [1, 0, 1, 0, 1, 0, 1, 1, 1, 1],\n",
" [0, 1, 1, 1, 0, 0, 1, 1, 1, 0],\n",
" [0, 0, 1, 1, 0, 1, 1, 1, 0, 1],\n",
" [0, 0, 0, 0, 1, 0, 1, 1, 0, 1],\n",
" [0, 0, 0, 0, 1, 1, 1, 1, 0, 0],\n",
" [1, 1, 1, 0, 1, 0, 0, 0, 1, 0],\n",
" [1, 1, 1, 1, 0, 1, 1, 0, 1, 1],\n",
" [0, 0, 1, 0, 0, 0, 1, 0, 1, 0],\n",
" [1, 1, 1, 0, 0, 1, 1, 1, 0, 1],\n",
" [0, 1, 1, 1, 0, 1, 1, 0, 1, 1],\n",
" [0, 1, 0, 1, 1, 1, 0, 0, 0, 1],\n",
" [1, 1, 1, 1, 0, 0, 0, 0, 0, 1],\n",
" [1, 1, 1, 0, 0, 0, 0, 0, 0, 0],\n",
" [0, 0, 0, 1, 0, 0, 0, 1, 1, 0],\n",
" [1, 1, 1, 1, 0, 1, 0, 0, 0, 1],\n",
" [1, 1, 1, 1, 1, 1, 1, 1, 0, 0],\n",
" [1, 0, 1, 1, 0, 1, 0, 0, 0, 1],\n",
" [0, 0, 1, 1, 0, 1, 1, 1, 1, 1],\n",
" [1, 0, 0, 0, 1, 0, 1, 1, 0, 0],\n",
" [1, 0, 1, 1, 1, 0, 0, 1, 0, 1],\n",
" [1, 0, 0, 0, 1, 1, 1, 0, 1, 0],\n",
" [1, 1, 0, 1, 0, 0, 0, 0, 1, 0],\n",
" [1, 0, 1, 1, 1, 1, 1, 0, 1, 0]]"
]
}
],
"prompt_number": 2
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# How is vmin closest to 0.01 ? "
],
"language": "python",
"metadata": {},
"outputs": []
}
],
"metadata": {}
}
]
}
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