\n",
"\n",
"This bound is useful once\n",
"$$ \n",
" 2 \\exp \\left(-\\frac{2n^2 a^2}{\\tau^2} \\right ) < 1,\n",
"$$\n",
"i.e.,\n",
"$$\n",
" a > \\sqrt{-2\\ln 1/2} \\cdot \\frac{\\tau}{2 n}.\n",
"$$\n",
"Since $\\sqrt{-2 \\ln 1/2} \\approx 1.18$, Hoeffding's inequality requires $a$ to be about 18% larger to\n",
"be useful compared to Chebychev's inequality, but it is exponentially better as $n$ grows."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Truncation\n",
"\n",
"If we know _a priori_ that $\\mu \\in [\\ell, u]$, we can take the intersection of any confidence interval $\\mathcal I$ with $[\\ell, u]$ without losing any coverage probability: if $\\mathcal I \\ni \\mu$, then $\\mathcal I \\cap [\\ell, u] \\ni \\mu$.\n",
"Hence, we can _truncate_ any confidence interval by taking its intersection with $[\\ell, u]$ without\n",
"reducing the confidence level.\n",
"\n",
"For instance, suppose $\\mathbb P X_j \\in [\\ell_j, u_j]) = 1, \\forall j = 1, \\ldots, n$. \n",
"Define $\\ell \\equiv \\frac{1}{n} \\sum_{j=1}^n \\ell_j$ and $u \\equiv \\frac{1}{n} \\sum_{j=1}^n u_j$.\n",
"Then\n",
"\n",
"$$\\mathbb P \\{ \\bar{X} \\in [\\ell, u] \\} = 1,$$\n",
"\n",
"and thus $\\mu \\equiv \\mathbb E \\bar{X} \\in [\\ell, u]$. Since we know with certainty that $\\mu \\in [\\ell, u]$,\n",
"any portion of a confidence interval $\\mathcal I$ that lies outside $[\\ell, u]$ cannot help it cover $\\mu$, and we might as well use $\\mathcal I \\cap [\\ell, u]$ if it is shorter than $\\mathcal I$: we gain precision without losing coverage.\n",
"\n",
"Basing a confidence interval on Chebychev's inequality or Hoeffding's inequality sometimes produces an interval that can be improved by truncation, decreasing the length (and expected length) without sacrificing coverage."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Comparing Hoeffding's Inequality and Chebychev's Inequality for the sample mean from finite bounded populations\n",
"\n",
"\n",
"Both inequalities bound $\\mathbb P_\\mu (|\\bar{X} - \\mu | \\ge a)$ in terms of $\\frac{\\sum_{j=1}^n (u_j - \\ell_j)^2}{n^2 a^2}$,\n",
"but Hoeffding's bound is exponentially better.\n",
"\n",
"Here is a comparison, normalized by taking $\\tau^2 \\equiv \\sum_{j=1}^n (u_j - \\ell_j)^2 = n$ so that $\\tau^2/n = 1$, as would be the case if $\\ell_j = 0$ and $u_j = 1$, $\\forall j$:\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# This is the first cell with code: set up the Python environment\n",
"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"import math\n",
"import numpy as np\n",
"import scipy as sp\n",
"import scipy.stats\n",
"from scipy.stats import binom\n",
"import pandas as pd\n",
"from ipywidgets.widgets import interact, interactive, fixed\n",
"import ipywidgets as widgets\n",
"from IPython.display import clear_output, display, HTML"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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PlizJdBQikmH1/WD+/PPP2bFjBwcccEDlY8uXL2fQoEH1DS1n5X6C6NMHVqwIy36LSKO3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\nEoSIiMSkBCEiIjH9fyRp98A1FwipAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def chebychev(n, ssByn, a):\n",
" return ssByn/(4.0 * n * np.square(a))\n",
"\n",
"def hoeffding(n, ssByn, a):\n",
" return 2*np.exp(-2.0 * n * np.square(a) / ssByn)\n",
"\n",
"def chebychevHoeffdingBoundPlot(n, ssByn = 1.0, aMax = 1.0, pts = 1000):\n",
" ''' \n",
" Plots the bounds on P( |\\bar{X} - \\mu | \\ge a) for Chebychev's inequality and \n",
" Hoeffding's inequality.\n",
" \\bar{X} is the sample mean of n independent bounded random variables.\n",
" ssBy is the mean sum of squares of the differences between the upper and lower bounds on \n",
" each of those n variables.\n",
" Plots the curves as pts points\n",
" '''\n",
" fig, ax = plt.subplots(nrows=1, ncols=1)\n",
" aMin = math.sqrt(-math.log(0.5)*ssByn/(2.0*n)) # smallest a for which Hoeffding bound is nontrivial\n",
" x = np.linspace(aMin, aMax, pts+1)\n",
" cheb, = plt.plot(x, np.fmin(1,chebychev(n, ssByn, x)), color='b', label='Chebychev')\n",
" hoef, = plt.plot(x, hoeffding(n, ssByn, x), color='r', label='Hoeffding')\n",
" plt.title(r'Bound on $\\mathbf{P} (|\\bar{X} - \\mu| > a)$')\n",
" plt.xlabel('a')\n",
" plt.ylabel('Probability')\n",
" plt.legend(loc = 'best')\n",
"\n",
"interact(chebychevHoeffdingBoundPlot, n=widgets.IntSlider(min=5, max=300, step=1, value=30),\\\n",
" ssByn = widgets.FloatSlider(min=0.5, max=10, step=0.1, value=1.0),\\\n",
" aMax = widgets.fixed(1), pts = widgets.fixed(1000)\n",
" )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Using Hoeffding's inequality for confidence intervals\n",
"\n",
"Suppose that $\\{X_j \\}_{j=1}^n$ are iid with support $[0, 1]$. \n",
"Then $\\tau = \\sqrt{n}$.\n",
"\n",
"
\n",
"It follows that\n",
"$$\n",
" A_\\alpha(\\theta) = \\left [0, \\theta + \\sqrt{\\frac{-\\ln \\alpha}{2n}} \\right ]\n",
"$$\n",
"is the (one-sided) acceptance region for a level-$\\alpha$ test of the hypothesis $\\mu = \\theta$;\n",
"$$\n",
" \\mathbb P_\\mu \\left (\\bar{X} - \\sqrt{\\frac{-\\ln \\alpha}{2n}} \\ge \\mu \\right ) \\le \\alpha;\n",
"$$\n",
"and \n",
"$$\\left [\\bar{X} - \\sqrt{\\frac{-\\ln \\alpha}{2n}}, 1 \\right ]$$ \n",
"is a one-sided\n",
"confidence interval for $\\mu$, with confidence level $1-\\alpha$.\n",
"
\n",
"\n",
"
\n",
"Similarly,\n",
"$$\n",
" A_\\alpha(\\theta) = \n",
" \\left [\\theta - \\sqrt{\\frac{-\\ln \\alpha/2}{2n}}, \\theta + \\sqrt{\\frac{-\\ln \\alpha/2}{2n}} \\right ]\n",
"$$\n",
"is the two-sided acceptance region for a level-$\\alpha$ test of the hypothesis $\\mu = \\theta$;\n",
"$$\n",
" \\mathbb P_\\mu \\left ( \\left | \\bar{X} - \\sqrt{\\frac{-\\ln \\alpha/2}{2n}} \\right | \\ge \\mu \\right ) \\le \\alpha;\n",
"$$\n",
"and \n",
"$$\\left [\\bar{X} - \\sqrt{\\frac{-\\ln \\alpha/2}{2n}}, \\bar{X} + \\sqrt{\\frac{-\\ln \\alpha/2}{2n}} \\right ]$$\n",
"is a two-sided\n",
"confidence interval for $\\mu$, with confidence level $1-\\alpha$.\n",
"
\n",
"\n",
"
\n",
"Let's compare Hoeffding confidence intervals and intervals based on the normal approximation by simulation, in the cases we saw previously in [Confidence intervals based on the normal approximation](normApprox.ipynb). We will use two-sided bounds, even though in applications one-sided bounds are often more interesting.\n",
"
\n",
"\n",
"
\n",
"Of course, if we know a priori that $\\mu \\in [\\ell, u]$, the intersection of the Hoeffding\n",
"confidence interval with $[\\ell, u]$ is still a $1-\\alpha$ confidence interval.\n",
"
Simulated coverage probability of Student-t and truncated Hoeffding confidence intervals for ' +\\\n",
" 'mixture of U[0,1] and pointmass at 0 population
' +\\\n",
" 'Nominal coverage probability ' + str(100*(1-alpha)) +\\\n",
" '%. Estimated from ' + str(round(reps,0)) + ' replications.'\n",
"\n",
"display(HTML(ansStr))\n",
"display(simTable)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Remarkably, the truncated Hoeffding intervals are often _shorter_ on average than the Student-t intervals, but have far better coverage."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## More in this direction...\n",
"\n",
"We could also apply other inequalities, such as Serfling's Inequality (a version of Hoeffding's inequality for sampling without replacement) or Feige's Inequality.\n",
"\n",
"For sampling without replacement, the \"Empirical Bernstein-Serfling Inequality\" of Bardenet and Maillard (2013) seems particularly worth exploring.\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## What's next?\n",
"Now we will consider some conservative methods for constructing lower confidence intervals for the mean of nonnegative populations\n",
"\n",
"- [Next: Lower confidence bounds for the mean of a nonnegative population: Markov's Inequality and Methods based on the Empirical Distribution Function](markov.ipynb)\n",
"- [Previous: Confidence bounds for the mean of a bounded population: Binomial and Hypergeometric](binom.ipynb)\n",
"- [Index](index.ipynb)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
""
],
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""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%run talkTools.py"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.1"
}
},
"nbformat": 4,
"nbformat_minor": 1
}