ee046202_tutorial_02_classic_statistics_confidence_intervals.ipynb

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    "# <img src=\"https://img.icons8.com/dusk/64/000000/artificial-intelligence.png\" style=\"height:50px;display:inline\"> EE 046202 - Technion - Unsupervised Learning & Data Analysis\n",
    "---\n",
    "\n",
    "#### <a href=\"https://taldatech.github.io\">Tal Daniel</a>\n",
    "\n",
    "## Tutorial 02 - Classical Methods in Statistical Inference - Confidence Intervals\n",
    "---"
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    "### <img src=\"https://img.icons8.com/bubbles/50/000000/checklist.png\" style=\"height:50px;display:inline\"> Agenda\n",
    "---\n",
    "* [Interval Estimation (Confidence Intervals)](#-Confidence-Intervals-(Interval-Estimation))\n",
    "    * [Formulation](#-Formulation)\n",
    "    * [Finding Interval Estimators](#-Finding-Interval-Estimators)\n",
    "* [Bootstrap Approach](#-Bootstrap)\n",
    "    * [Boostrap for the Median Estimator]()\n",
    "* [Confidence Interval with Chebyshev Inequality](#-Confidence-Interval-with-Chebyshev-Inequality)\n",
    "* [Confidence Interval with DKW Inequality](#-Confidence-Interval-with-DKW-Inequality)\n",
    "* [Recommended Videos](#-Recommended-Videos)\n",
    "* [Credits](#-Credits)"
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    "# imports for the tutorial\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from helpers.cdf_bounds import plot_cdf_confidence\n",
    "from scipy.stats import norm\n",
    "import time\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib notebook"
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    "### <img src=\"https://img.icons8.com/doodle/96/000000/defensive-wood-wall--v2.png\" style=\"height:50px;display:inline\"> Confidence Intervals (Interval Estimation)\n",
    "---\n",
    "* What is the problem with the **point estimation** for $\\theta$, $\\hat{\\theta}$?\n",
    "    * $\\hat{\\theta}$ alone does not give much information about $\\theta$!\n",
    "    * Without any additional information, we cannot tell how close is $\\hat{\\theta}$ to the real $\\theta$."
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    "* **Interval Estimation** - an interval that is *likely* to include the true value of $\\theta$.\n",
    "    * Instead of saying $\\hat{\\theta} = 8.14$, we report: $ [\\hat{\\theta}_l, \\hat{\\theta}_h ] = [6.65, 9.87 ] $, which we hope includes the real value of $\\theta$.\n",
    "    * We actually produce 2 estimates for $\\theta$: *high* estimate $\\hat{\\theta}_h$ and *low* estimate $\\hat{\\theta}_l$.\n",
    "    * We define two important properties of a confidence interval:\n",
    "        * **Length** - the precision which we can estimate $\\theta$. Defined as $\\hat{\\theta}_h - \\hat{\\theta}_l$.\n",
    "        * **Confidence Level** - how confident are we in the interval. It is the *probability* that the real $\\theta$ lies in the interval. We wish to have high confidence (usually 90-95%).\n",
    "        * <img src='./assets/tut_01_conf_int_1.jpg' style=\"height:300px\"> <a href=\"https://www.quora.com/Why-is-a-99-confidence-level-wider-when-constructing-confidence-intervals\">Image Source</a>"
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    "#### <img src=\"https://img.icons8.com/nolan/64/000000/sigma.png\" style=\"height:30px;display:inline\"> Formulation\n",
    "---\n",
    "Let $X_1, X_2, .., X_n$ be a random sample from a distribution with a parameter $\\theta$ that is to be estimated.\n",
    "\n",
    "**The goal** is to find *two* (point) estimators for $\\theta$:\n",
    "1. The low estimator: $$\\hat{\\theta}_l = \\hat{\\theta}_l(X_1, X_2,..., X_n)$$\n",
    "2. The high estimator: $$\\hat{\\theta}_h = \\hat{\\theta}_h(X_1, X_2,..., X_n)$$"
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    "* The interval estimator is:  $ [\\hat{\\theta}_l, \\hat{\\theta}_h ]$.\n",
    "* **Confidence Level** - defined to be $1-\\alpha$.\n",
    "* The estimators  $ \\hat{\\theta}_l, \\hat{\\theta}_h $ are chosen such that the **probability** that the interval  $ [\\hat{\\theta}_l, \\hat{\\theta}_h ]$ includes $\\theta$ is larger than the confidence level $1-\\alpha$. Thus, we wish $\\alpha$ to be **small**.\n",
    "    * Common values for $\\alpha$ are: $0.1, 0.05, 0.01$ which correspond to confidence levels 90%, 95% and 99% respectively.\n",
    "* Formally: find  $\\hat{\\theta}_l, \\hat{\\theta}_h$ such that $$ P(\\hat{\\theta}_l \\leq \\theta \\leq \\hat{\\theta}_h) \\geq 1- \\alpha $$ or $$ P(\\theta \\in [\\hat{\\theta}_l, \\hat{\\theta}_h ]) \\geq 1-\\alpha $$"
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    "#### <img src=\"https://img.icons8.com/dusk/64/000000/find-and-replace.png\" style=\"height:30px;display:inline\"> Finding Interval Estimators\n",
    "---\n",
    "* Recall that for a continuous random variable $X$ with **CDF** (Cumulative Distribution Function): $F_X(x) = P(X \\leq x)$.\n",
    "* We wish to find probability between bounds such that: $$ P(x_l \\leq X \\leq x_h) = 1 - \\alpha $$ $$\\to 1 - P(x_l \\leq X \\leq x_h) = \\alpha $$ $$ P(X \\leq x_l \\cup X\\geq x_h) = \\alpha = P(X \\leq x_l) + P(X\\geq x_h) $$\n",
    "We can choose (notice the different inequality signs): $$ P(X \\leq x_l) = \\frac{\\alpha}{2} \\text{, and } P(X \\geq x_h) = \\frac{\\alpha}{2} $$ Or, equivalently $$ F_X(x_l) = \\frac{\\alpha}{2} \\text{, and }  F_X(x_h) = 1 -  \\frac{\\alpha}{2}$$ <br> $$ \\rightarrow x_l = F_X^{-1}(\\frac{\\alpha}{2}),  x_h = F_X^{-1}(1 - \\frac{\\alpha}{2})$$"
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    "We call the interval $[x_l, x_h]$ a $(1-\\alpha)$ interval for $ X$\n",
    "   * Note: why do we divide $\\alpha$ by 2 and not by 4 (and then pick $\\frac{\\alpha}{4}, \\frac{3\\alpha}{4}$ instead of $\\frac{\\alpha}{2}$)?\n",
    "        * Since we are calculating *both* upper and lower bounds. Had we been interested in a single sided interval, then we would not have needed this division. Moreover, we would like our confidence level to be symmetrical in relation to both upper and lower bounds, there is no reason to give more weight to one over the other.\n",
    "<img src=\"./assets/tut_01_conf_int_2.png\" style=\"height:250px\"> <a href=\"https://www.probabilitycourse.com/chapter8/8_3_2_finding_interval_estimators.php\">Image Source</a>"
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    "#### <img src=\"https://img.icons8.com/dusk/64/000000/task.png\" style=\"height:30px;display:inline\"> Exercise - Confidence Intervals for Normal Distribution (Using CLT)\n",
    "---\n",
    "1. (Probability Exercise) Let $Z \\sim \\mathcal{N}(0,1)$. Find $x_l, x_h$ such that $$ P(x_l \\leq Z \\leq x_h) = 0.95 $$\n",
    "    * <a href=\"https://www.math.arizona.edu/~jwatkins/normal-table.pdf\"> Normal Table </a>\n",
    "2. Let $X_1, X_2, ..., X_n$ be a random sample from a normal distribution $\\mathcal{N}(\\theta, 1)$ ($\\mathbb{E}[X] = \\theta$). Find a 95% confidence interval for $\\theta$."
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    "#### <img src=\"https://img.icons8.com/dusk/64/000000/idea.png\" style=\"height:30px;display:inline\"> Solution\n",
    "---\n",
    "#### Section 1\n",
    "---\n",
    "$\\alpha = 0.05$, and $\\phi$ denotes the CDF of $Z$. Thus, we can choose: $$ x_l = \\phi^{-1}(\\frac{\\alpha}{2}) = \\phi^{-1}(0.025) = -1.96 $$ $$ x_h = \\phi^{-1}(1 - \\frac{\\alpha}{2}) = \\phi^{-1}(1 - 0.025) = 1.96 $$ More generally, if we assume $Z \\sim \\mathcal{N}(0,1)$, for any $p \\in [0,1]$, we define $z_p$ as the real value for which $P(Z>z_p) = p$. Therfore, $\\phi(z_p) = 1-p \\rightarrow z_p = \\phi^{-1}(1-p)$. In our case $$ P(-z_{\\frac{\\alpha}{2}} \\leq Z \\leq z_{\\frac{\\alpha}{2}}) = 1-\\alpha $$ <img src=\"./assets/tut_01_conf_int_3.png\" style=\"height:250px\"> <a href=\"https://www.probabilitycourse.com/chapter8/8_3_2_finding_interval_estimators.php\">Image Source</a>"
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    "#### Section 2\n",
    "---\n",
    "\n",
    "We begin with finding the **point estimator** for $\\theta$, which is the *mean* of the distribution. Recall from the previous tutorial that: $\\hat{\\theta} = \\overline{X} = \\frac{X_1 + X_2 + ... +X_n}{n}$ and since $X_i \\sim \\mathcal{N}(\\theta,1)$ and the samples are i.i.d. then: $\\overline{X} \\sim \\mathcal{N}(\\theta, \\frac{1}{n})$. We now perform **standartization** (normalization in our case): $$ \\frac{\\overline{X} - \\theta}{\\frac{1}{\\sqrt{n}}} = \\sqrt{n}(\\overline{X} - \\theta) \\sim \\mathcal{N}(0,1) $$\n",
    "* Why do we perform standartization? Because it is easier to work with $\\mathcal{N}(0,1)$ and we are familiar with its CDF and can easily find the values with the table. \n",
    "* Standartization in the general case where $\\sigma^2 \\neq 1$: $$ \\frac{\\overline{X} - \\theta}{\\frac{\\sigma}{\\sqrt{n}}} $$\n",
    "\n",
    "Thus: $$ P\\big( -1.96 \\leq  \\sqrt{n}(\\overline{X} - \\theta) \\leq 1.96 \\big) = 0.95 $$ $$ \\rightarrow P\\big(\\overline{X} -\\frac{1.96}{\\sqrt{n}} \\leq  \\theta \\leq \\overline{X} + \\frac{1.96}{\\sqrt{n}} \\big) = 0.95 $$ Therefore, the confidence interval is: $$ [\\hat{\\theta}_l, \\hat{\\theta}_h] = [\\overline{X} -\\frac{1.96}{\\sqrt{n}}, \\overline{X} + \\frac{1.96}{\\sqrt{n}}] $$"
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    "### <img src=\"https://img.icons8.com/color/96/000000/rocket-boot.png\" style=\"height:50px;display:inline\"> Bootstrap\n",
    "---\n",
    "* Bootstrap is a **sampling** method which we use to do estimations (and even train ML algorithms, as in Boosting/Bagging method). \n",
    "* The bootstrap approach relies on the assumption that drawing random samples **with replacement** from the data, simulates well the actual distribution of the data. \n",
    "* In that case, several **statistics** (e.g. the variance) can be calculated using random samples from the data."
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    "* **Motivation** - in general, we cannot compute the confidence interval. Thus, we need a numerical scheme to estimate it. This is where bootstrapping helps us.\n",
    "* In sampling **with replacement**, each sample unit of the population can occur one or more times in the sample.\n",
    "    * In statistics, resampling with replacement is called *bootstrapping*."
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    "* **Bootstrap Algorithm**:\n",
    "    * Denote the original sample: $ L_N = (x_1, x_2, ..., x_N) $\n",
    "    * Repeat $M$ times:\n",
    "        * Generate a sample $L_k$ of size $k$ from $L_N$ by sampling *with replacement* (samples can contain multiple copies of the same point).\n",
    "        * Compute $h$ from $L_k$ (that is, calculate an estimation $h$ using $L_k$).\n",
    "    * Denote the bootstrap values $H=(h^1, h^2, ..., h^M)$\n",
    "        * Use these values for calculating all the quantities of interest."
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    "#### <img src=\"https://img.icons8.com/dusk/64/000000/task.png\" style=\"height:30px;display:inline\"> Exercise - Bootstrap for the Median Estimator\n",
    "---\n",
    "Given $n$ i.i.d. samples, $\\{X_i\\}_{i=1}^n$, from $\\mathcal{D}$ consider the median estimator denoted by $\\hat{\\mu}_M$ and calculate the **standard deviation** using the bootstrap approach. Write down the boostrap algorithm for that case."
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    "#### <img src=\"https://img.icons8.com/dusk/64/000000/idea.png\" style=\"height:30px;display:inline\"> Solution\n",
    "---\n",
    "The algorithm:\n",
    "* Denote the original sample: $ \\mathcal{D} = X_n = (x_1, x_2, ..., x_n) $\n",
    "* Calculate the median estimator $T$: $T \\leftarrow median(\\mathcal{D})$\n",
    "* Repeat $M$ times:\n",
    "    * Generate a sample $L_k$ of size $k$ from $L_N$ by sampling *with replacement*.\n",
    "    * Compute $T_{boot}^k$ from $L_k$ (that is, calculate an estimation for the median $T_{boot}$ using $L_k$).\n",
    "        * $T_{boot}^k = median(L_k)$\n",
    "* Denote the bootstrap values $\\hat{T}=(T_{boot}^1, T_{boot}^2, ..., T_{boot}^M)$\n",
    "    * $Median \\leftarrow T$\n",
    "    * $\\hat{\\sigma}^2 \\leftarrow Variance(\\hat{T})$"
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    "# let's see the magic\n",
    "def std_bootstrap(x, k, m, estimator=np.mean):\n",
    "    estimate_value = estimator(x)\n",
    "    estimations = []\n",
    "    for _ in range(m):\n",
    "        # sample K with replacement\n",
    "        x_k = np.random.choice(x, size=k, replace=True)\n",
    "        t_k = estimator(x_k)\n",
    "        estimations.append(t_k)\n",
    "    return estimate_value, np.std(estimations)\n",
    "\n",
    "# generate real data\n",
    "n = 300  # number of totals samples\n",
    "K = 100  # number of bootstrap samples\n",
    "mu , sigma = 1, 4\n",
    "x = np.random.normal(mu, sigma, size=n)\n",
    "\n",
    "M = np.arange(1, 101)  # number of boostrap iterations\n",
    "\n",
    "stds = []\n",
    "for m in M:\n",
    "    stds.append(std_bootstrap(x=x, k=K, m=m, estimator=np.median)[1]) \n",
    "    # [1] because we don't care about the estimate, just the std"
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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(8, 6))\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "ax.plot(M, stds)\n",
    "ax.grid()\n",
    "ax.set_xlabel(\"Bootstrap Iterations (M)\")\n",
    "ax.set_ylabel(\"STD of the median estimator\")\n",
    "ax.set_title(\"Bootstrap STD of the median estimator as a function of M\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "* Bootstrap is a general sampling method that can be used to estimate other values other than the STD.\n",
    "* It has no convergence guarantees (unlike the sample mean for example). Thus, one should consider that values estimated with Bootstrap may be misleading.\n",
    "* Usually, Bootstrap is used for non-trivial estimators like the median or the correlation estimators (estimators that cannot be easily defined as an expectancy $\\mathbb{E}$ of an R.V.)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### <img src=\"https://img.icons8.com/ultraviolet/80/000000/less-than-2.png\" style=\"height:50px;display:inline\"> Confidence Interval with Chebyshev Inequality\n",
    "---\n",
    "* Recall the **Chebyshev Inequality**: if $X$ is a random variable with finite mean and variance $\\sigma^2$, then $$ P(|X-\\mathbb{E}[X]|\\geq \\epsilon)\\leq \\frac{\\sigma^2}{\\epsilon^2} $$ for all $\\epsilon > 0$.\n",
    "    * Also, letting $\\epsilon = k\\sigma$: $$ P(|X-\\mathbb{E}[X]|\\geq k\\sigma)\\leq \\frac{1}{k^2} $$\n",
    "        * This is useful when we want to measure the distance in \"standard deviations\" (e.g., a confidence interval of 2 standard deviations).\n",
    "* Chebyshev inequality allows us to calculate CIs given the variance (which can be estimated, for instance, with bootstrapping) of a random variable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "* For the sample mean $\\overline{X}=\\frac{1}{N}\\sum_{k=1}^NX_k$, Chebyshev inequality tells us: $$ P(|\\overline{X} -\\mu| \\geq \\epsilon) \\leq \\frac{\\hat{\\sigma}^2}{\\epsilon^2} = \\frac{\\sigma^2}{N\\epsilon^2} $$\n",
    "    * Recall that $\\hat{\\sigma}$ is the normalized empirical variance for a sample of size $N$.\n",
    "    * For example, for a 95% CI, $\\frac{\\sigma^2}{N\\epsilon^2} = \\alpha= 0.05 \\rightarrow \\epsilon = \\frac{\\sigma}{\\sqrt{0.05N}}$, which leads to: $$ \\overline{X} - \\frac{\\sigma}{\\sqrt{0.05N}} \\leq \\mu  \\leq \\overline{X} + \\frac{\\sigma}{\\sqrt{0.05N}} $$ with probability at least 0.95.\n",
    "* In practice, if you are not given $\\sigma$, then you can't calculate the CI."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### <img src=\"https://img.icons8.com/officel/80/000000/less-than-2.png\" style=\"height:50px;display:inline\"> Confidence Interval with DKW Inequality\n",
    "---\n",
    "The **Dvoretzky–Kiefer–Wolfowitz** (DKW) inequality bounds how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn.\n",
    "\n",
    "Let $X_1, X_2, …, X_n$ be real-valued independent and identically distributed (i.i.d.) random variables with cumulative distribution function (CDF) $F(\\cdot)$ (that is, they are sampled from $F$: $X_1, X_2, …, X_n \\sim F$). Let $F_n$ denote the associated **empirical distribution** function defined by: $$ F_n(x) = \\frac{1}{n}\\sum_{i=1}^n \\mathbb{1}_{\\{X_i \\leq x\\}}, x \\in \\mathbb{R} $$\n",
    "\n",
    "* $F(x)$ - the probability that a *single* random variable $X$ is smaller than $x$\n",
    "* $F_n(x)$ - the *fraction* of random variables that are smaller than $x$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The DKW inequality bounds the probability that the random function $F_n$ differs from $F$ by more than a given constant $\\epsilon > 0$ anywhere on the real line ($\\mathbb{R}$): $$ Pr\\big( \\underset{x \\in \\mathbb{R}}{sup} |F_n(x) - F(x)| > \\epsilon \\big) \\leq 2e^{-2n\\epsilon^2}, \\forall \\epsilon >0 $$\n",
    "\n",
    "* This strengthens the **Glivenko–Cantelli** theorem by quantifying the *rate of convergence* as $n$ tends to infinity.\n",
    "    * But it holds for any $n$.\n",
    "* Reminder: **Supremum**-$sup$ - the least element in a (partially ordered) set $P$ that is greater or equal to all elements of a subset $S$, if such element exists."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Producing CDF-based Confidence Bounds\n",
    "---\n",
    "* The DKW inequality is one method for generating **CDF-based confidence bounds** and producing a confidence band. \n",
    "* The purpose of this confidence interval is to contain the entire CDF at the specified **confidence level** .\n",
    "    * Alternative approaches attempt to only achieve the confidence level on each *individual point* which can allow for a tighter bound. \n",
    "* The DKW bounds runs parallel to, and is equally above and below, the empirical CDF.\n",
    "\n",
    "The interval that contains the **true CDF** $F(x)$, with probability $1-\\alpha$: $$ F_n(x) - \\epsilon \\leq F(x) \\leq  F_n(x) +\\epsilon$$ $$ \\alpha = 2e^{-2n\\epsilon^2} \\rightarrow \\epsilon = \\sqrt{\\frac{ln \\frac{2}{\\alpha}}{2n}} $$\n",
    "\n",
    "* What is so special about the DKW condifence bounds?\n",
    "    * It is **distribution free** - therefore, it is very loose in comparison to the bootstrap estimates, which are usually tighter.\n",
    "    * **Faster** to calculate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def compare_cdf_bounds():\n",
    "    # generate random data\n",
    "    x = np.random.randn(300)\n",
    "    # plot\n",
    "    fig = plt.figure(figsize=(12,5))\n",
    "    ax1 = fig.add_subplot(1,2,1)\n",
    "    ax1.grid()\n",
    "    s_time = time.time()\n",
    "    # Pointwise estimate (for each point!) confidence bounds using Beta distribution\n",
    "    plot_cdf_confidence(data=x, confidence=0.9, color='violet', label='Empirical CDF (90% confidence)', ax=ax1)\n",
    "    beta_time = time.time() - s_time\n",
    "    x_sorted = np.sort(x)\n",
    "    ax1.plot(x_sorted, norm.cdf(x_sorted), ':', label='True CDF')\n",
    "    ax1.legend()\n",
    "\n",
    "    ax2 = fig.add_subplot(1,2,2)\n",
    "    ax2.grid()\n",
    "    s_time = time.time()\n",
    "    # DKW (not pointwise!) confidence bounds\n",
    "    plot_cdf_confidence(data=x, confidence=0.9, color='violet', label='Empirical CDF (90% confidence)-DKW',\n",
    "                        estimator_name='DKW', ax=ax2)\n",
    "    dkw_time = time.time() - s_time\n",
    "    x_sorted = np.sort(x)\n",
    "    ax2.plot(x_sorted, norm.cdf(x_sorted), ':', label='True CDF')\n",
    "    ax2.legend()\n",
    "    print(\"pointwise: {:.3} sec, DKW: {:.3} sec\".format(beta_time, dkw_time))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pointwise: 0.121 sec, DKW: 0.00598 sec\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
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    }
   ],
   "source": [
    "compare_cdf_bounds()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### <img src=\"https://img.icons8.com/dusk/64/000000/task.png\" style=\"height:30px;display:inline\"> Exercise - Confidence Intervals Calculation\n",
    "---\n",
    "* Reminder: **Hoeffding Inequality**: Let $X_1, ..., X_n$ be i.i.d. random variables, bounded by the intervals $a_i \\leq X_i \\leq b_i$. Let the empirical mean be defined according to $\\overline{X} = \\frac{1}{n} \\sum_{i=1}^n X_i$. It holds that: $$ P(|\\overline{X} - \\mathbb{E}[\\overline{X}]| \\geq \\epsilon) \\leq 2e^{-\\frac{2n^2\\epsilon^2}{\\sum_{i=1}^n(a_i - b_i)^2}} $$\n",
    "\n",
    "Derive a 95% confidence band for the mean ($\\mathbb{E}[X]=\\mu$) using **DKW** (through the CDF bound) and **Hoeffding** inequalities.\n",
    "\n",
    "\n",
    "Assume:\n",
    "* True CDF: $F(x)$, estimated CDF: $F_n(x)$\n",
    "* Estimated mean: $\\overline{X}$\n",
    "* Number of samples $n=60$\n",
    "* Prior knowledge: the range of the random variables is $X \\in [0,45]$\n",
    "    * No specific distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### <img src=\"https://img.icons8.com/dusk/64/000000/idea.png\" style=\"height:30px;display:inline\"> Solution\n",
    "---\n",
    "#### Section 1\n",
    "---\n",
    "\n",
    "Using DKW: $$ P(\\underset{x \\in \\mathbb{R}}{sup} |F_n(x) - F(x)| > \\epsilon) \\leq 2e^{-2n\\epsilon^2} = 0.05 $$ $$ \\rightarrow \\epsilon = \\sqrt{\\frac{-\\ln(0.025)}{2*60}} = 0.175 $$ $$\\rightarrow F(x) \\in [F_n(x) -0.175, F_n(x) +0.175] $$\n",
    "   * Deriving a CI for the mean: we know that $\\mathbb{E}[X] = \\int_{0}^{45}(1-F(x))dx$ (from the Tail Sum formula), therefore: $$ \\int_{0}^{45}(1-F_n(x)\\pm0.175)dx = \\overline{X} \\pm 7.875 $$\n",
    "        * Why is $\\mathbb{E}[X] = \\int(1-F(x))dx$? Recall the Tail Sum formula from previous tutorial ($\\mathbb{E}[X] = \\sum_{i=1}^n P(X \\geq i)$). In the continuous space it is completely analogous! $$ \\mathbb{E}[X] = \\int_x P(X \\geq x)dx = \\int_x (1 - P(X \\leq x)) dx= \\int_x (1 - F(x))dx $$\n",
    "            * <a href=\"https://en.wikipedia.org/wiki/Expected_value#General_case_2\"> Proof </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Section 2\n",
    "---\n",
    "\n",
    "Using Hoeffding: $$ P(|\\overline{X} - \\mathbb{E}[\\overline{X}]| \\geq \\epsilon) \\leq 2e^{-\\frac{2n^2\\epsilon^2}{\\sum_{i=1}^n(a_i - b_i)^2}} = 2e^{-\\frac{2\\cdot 60^2\\epsilon^2}{\\sum_{i=1}^{60}(45 - 0)^2}} = 0.05 $$ $$ \\rightarrow \\epsilon = 7.89 \\rightarrow \\mathbb{E}[X] \\in [\\overline{X} -7.89, \\overline{X} +7.89] $$\n",
    "\n",
    "\n",
    "* We see that both approaches yield similar CI (same range) for the mean. In this sense, the estimators are equivalent.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### <img src=\"https://img.icons8.com/bubbles/50/000000/video-playlist.png\" style=\"height:50px;display:inline\"> Recommended Videos\n",
    "---\n",
    "#### <img src=\"https://img.icons8.com/cute-clipart/64/000000/warning-shield.png\" style=\"height:30px;display:inline\"> Warning!\n",
    "* These videos do not replace the lectures and tutorials.\n",
    "* Please use these to get a better understanding of the material, and not as an alternative to the written material.\n",
    "\n",
    "#### Video By Subject\n",
    "\n",
    "* Confidence Intervals - <a href=\"https://www.youtube.com/watch?v=yDEvXB6ApWc&t=600s\"> CrashCourse </a>\n",
    "* Bootstrap - <a href=\"https://www.youtube.com/watch?v=655X9eZGxls\">Confidence Intervals: Intro to bootstrapping proportions </a>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## <img src=\"https://img.icons8.com/dusk/64/000000/prize.png\" style=\"height:50px;display:inline\"> Credits\n",
    "---\n",
    "* Examples, exercises and definitions from <a href=\"https://probabilitycourse.com/\">Introduction to Probability, Statistics and Random Processes</a> - https://probabilitycourse.com\n",
    "* Icons from <a href=\"https://icons8.com/\">Icon8.com</a> - https://icons8.com\n",
    "* Datasets from <a href=\"https://www.kaggle.com/\">Kaggle</a> - https://www.kaggle.com/"
   ]
  }
 ],
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