{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": false }, "source": [ "# [Introduction to Data Science: A Comp-Math-Stat Approach](https://lamastex.github.io/scalable-data-science/as/2019/)\n", "## YOIYUI001, Summer 2019 \n", "©2019 Raazesh Sainudiin. [Attribution 4.0 International (CC BY 4.0)](https://creativecommons.org/licenses/by/4.0/)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 10. Convergence of Limits of Random Variables, Confidence Set Estimation and Testing\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Inference and Estimation: The Big Picture\n", "\n", "- Limits\n", " - Limits of Sequences of Real Numbers\n", " - Limits of Functions\n", " - Limit of a Sequence of Random Variables\n", "- Convergence in Distribution\n", "- Convergence in Probability\n", "- Some Basic Limit Laws in Statistics\n", "- Weak Law of Large Numbers\n", "- Central Limit Theorem \n", "- Asymptotic Normality of the Maximum Likelihood Estimator\n", "- Set Estimators - Confidence Intervals and Sets from Maximum Likelihood Estimators\n", "- Parametric Hypothesis Test - From Confidence Interval to Wald test\n", " \n", "\n", "### Inference and Estimation: The Big Picture\n", "\n", "The Models and their maximum likelihood estimators we discussed earlier fit into our Big Picture, which is about inference and estimation and especially inference and estimation problems where computational techniques are helpful. \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
 Point estimationSet estimationHypothesis Testing
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Parametric

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MLE of finitely many parameters
done

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Asymptotically Normal Confidence Intervals
about to see ...

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Wald Test from Confidence Interval
about to see ...

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Non-parametric
(infinite-dimensional parameter space)

\n", "		coming up ... coming up ... coming up ...
\n", "\n", "But before we move on we have to discuss what makes it all work: the idea of limits - where do you get to if you just keep going?\n", "\n", "## Limits\n", "\n", "We talked about the likelihood function and maximum likelihood estimators for making point estimates of model parameters. For example for the $Bernoulli(\\theta^*)$ RV (a $Bernoulli$ RV with true but possibly unknown parameter $\\theta^*$, we found that the likelihood function was $L_n(\\theta) = \\theta^{t_n}(1-\\theta)^{(n-t_n)}$ where $t_n = \\displaystyle\\sum_{i=1}^n x_i$. We also found the maxmimum likelihood estimator (MLE) for the $Bernoulli$ model, $\\widehat{\\theta}_n = \\frac{1}{n}\\displaystyle\\sum_{i=1}^n x_i$. \n", "\n", "We demonstrated these ideas using samples simulated from a $Bernoulli$ process with a secret $\\theta^*$. We had an interactive plot of the likelihood function where we could increase $n$, the number of simulated samples or the amount of data we had to base our estimate on, and see the effect on the shape of the likelihood function. The animation belows shows the changing likelihood function for the Bernoulli process with unknown $\\theta^*$ as $n$ (the amount of data) increases.\n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", " \n", " \n", "
Likelihood function for Bernoulli process, as $n$ goes from 1 to 1000 in a continuous loop.
\n", "\n", "For large $n$, you can probably make your own guess about the true value of $\\theta^*$ even without knowing $t_n$. As the animation progresses, we can see the likelihood function 'homing in' on $\\theta = 0.3$. \n", "\n", "We can see this in another way, by just looking at the sample mean as $n$ increases. An easy way to do this is with running means: generate a very large sample and then calculate the mean first over just the first observation in the sample, then the first two, first three, etc etc (running means were discussed in an earlier worksheet if you want to go back and review them in detail in your own time). Here we just define a function so that we can easily generate sequences of running means for our $Bernoulli$ process with the unknown $\\theta^*$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Preparation: Let's just evaluate the next cel and focus on concepts.\n", "\n", "You can see what they are as you need to." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "def likelihoodBernoulli(theta, n, tStatistic):\n", " '''Bernoulli likelihood function.\n", " theta in [0,1] is the theta to evaluate the likelihood at.\n", " n is the number of observations.\n", " tStatistic is the sum of the n Bernoulli observations.\n", " return a value for the likelihood of theta given the n observations and tStatistic.'''\n", " retValue = 0 # default return value\n", " if (theta >= 0 and theta <= 1): # check on theta\n", " mpfrTheta = RR(theta) # make sure we use a Sage mpfr \n", " retValue = (mpfrTheta^tStatistic)*(1-mpfrTheta)^(n-tStatistic)\n", " return retValue\n", " \n", "def bernoulliFInverse(u, theta):\n", " '''A function to evaluate the inverse CDF of a bernoulli.\n", " \n", " Param u is the value to evaluate the inverse CDF at.\n", " Param theta is the distribution parameters.\n", " Returns inverse CDF under theta evaluated at u'''\n", " \n", " return floor(u + theta)\n", " \n", "def bernoulliSample(n, theta, simSeed=None):\n", " '''A function to simulate samples from a bernoulli distribution.\n", " \n", " Param n is the number of samples to simulate.\n", " Param theta is the bernoulli distribution parameter.\n", " Param simSeed is a seed for the random number generator, defaulting to 30.\n", " Returns a simulated Bernoulli sample as a list.'''\n", " \n", " set_random_seed(simSeed)\n", " us = [random() for i in range(n)]\n", " set_random_seed(None)\n", " return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n", " \n", "def bernoulliSampleSecretTheta(n, theta=0.30, simSeed=30):\n", " '''A function to simulate samples from a bernoulli distribution.\n", " \n", " Param n is the number of samples to simulate.\n", " Param theta is the bernoulli distribution parameter.\n", " Param simSeed is a seed for the random number generator, defaulting to 30.\n", " Returns a simulated Bernoulli sample as a list.'''\n", " \n", " set_random_seed(simSeed)\n", " us = [random() for i in range(n)]\n", " set_random_seed(None)\n", " return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n", "\n", "def bernoulliRunningMeans(n, myTheta, mySeed = None):\n", " '''Function to give a list of n running means from bernoulli with specified theta.\n", " \n", " Param n is the number of running means to generate.\n", " Param myTheta is the theta for the Bernoulli distribution\n", " Param mySeed is a value for the seed of the random number generator, defaulting to None.'''\n", " \n", " sample = bernoulliSample(n, theta=myTheta, simSeed = mySeed)\n", " from pylab import cumsum # we can import in the middle of code\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n", " \n", "#return a plot object for BernoulliLikelihood using the secret theta bernoulli generator\n", "def plotBernoulliLikelihoodSecretTheta(n):\n", " '''Return a plot object for BernoulliLikelihood using the secret theta bernoulli generator.\n", " \n", " Param n is the number of simulated samples to generate and do likelihood plot for.'''\n", " \n", " thisBSample = bernoulliSampleSecretTheta(n) # make sample\n", " tn = sum(thisBSample) # summary statistic\n", " from pylab import arange\n", " ths = arange(0,1,0.01) # get some values to plot against\n", " liks = [likelihoodBernoulli(t,n,tn) for t in ths] # use the likelihood function to generate likelihoods\n", " redshade = 1*n/1000 # fancy colours\n", " blueshade = 1 - redshade\n", " return line(zip(ths, liks), rgbcolor = (redshade, 0, blueshade))\n", " \n", "def cauchyFInverse(u):\n", " '''A function to evaluate the inverse CDF of a standard Cauchy distribution.\n", " \n", " Param u is the value to evaluate the inverse CDF at.'''\n", " \n", " return RR(tan(pi*(u-0.5)))\n", " \n", "def cauchySample(n):\n", " '''A function to simulate samples from a standard Cauchy distribution.\n", " \n", " Param n is the number of samples to simulate.'''\n", " \n", " us = [random() for i in range(n)]\n", " return [cauchyFInverse(u) for u in us]\n", "\n", "def cauchyRunningMeans(n):\n", " '''Function to give a list of n running means from standardCauchy.\n", " \n", " Param n is the number of running means to generate.'''\n", " \n", " sample = cauchySample(n)\n", " from pylab import cumsum\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n", "\n", "def twoRunningMeansPlot(nToPlot, iters):\n", " '''Function to return a graphics array containing plots of running means for Bernoulli and Standard Cauchy.\n", " \n", " Param nToPlot is the number of running means to simulate for each iteration.\n", " Param iters is the number of iterations or sequences of running means or lines on each plot to draw.\n", " Returns a graphics array object containing both plots with titles.'''\n", " xvalues = range(1, nToPlot+1,1)\n", " for i in range(iters):\n", " shade = 0.5*(iters - 1 - i)/iters # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToPlot)\n", " cRunningMeans = cauchyRunningMeans(nToPlot)\n", " bPts = zip(xvalues, bRunningMeans)\n", " cPts = zip(xvalues, cRunningMeans)\n", " if (i < 1):\n", " p1 = line(bPts, rgbcolor = (shade, 0, 1))\n", " p2 = line(cPts, rgbcolor = (1-shade, 0, shade))\n", " cauchyTitleMax = max(cRunningMeans) # for placement of cauchy title\n", " else:\n", " p1 += line(bPts, rgbcolor = (shade, 0, 1))\n", " p2 += line(cPts, rgbcolor = (1-shade, 0, shade))\n", " if max(cRunningMeans) > cauchyTitleMax: cauchyTitleMax = max(cRunningMeans)\n", " titleText1 = \"Bernoulli running means\" # make title text\n", " t1 = text(titleText1, (nToGenerate/2,1), rgbcolor='blue',fontsize=10) \n", " titleText2 = \"Standard Cauchy running means\" # make title text\n", " t2 = text(titleText2, (nToGenerate/2,ceil(cauchyTitleMax)+1), rgbcolor='red',fontsize=10)\n", " return graphics_array((p1+t1,p2+t2))\n", "\n", "def pmfPointMassPlot(theta):\n", " '''Returns a pmf plot for a point mass function with parameter theta.'''\n", " \n", " ptsize = 10\n", " linethick = 2\n", " fudgefactor = 0.07 # to fudge the bottom line drawing\n", " pmf = points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " pmf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n", " pmf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n", " pmf += line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick)\n", " pmf += line([(theta+.05,0),(theta+2,0)], rgbcolor=\"blue\",thickness=linethick)\n", " pmf+= text(\"Point mass f\", (theta,1.1), rgbcolor='blue',fontsize=10)\n", " pmf.axes_color('grey') \n", " return pmf\n", " \n", "def cdfPointMassPlot(theta):\n", " '''Returns a cdf plot for a point mass function with parameter theta.'''\n", " \n", " ptsize = 10\n", " linethick = 2\n", " fudgefactor = 0.07 # to fudge the bottom line drawing\n", " cdf = line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick) # padding\n", " cdf += points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " cdf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n", " cdf += line([(theta,1),(theta+2,1)], rgbcolor=\"blue\", thickness=linethick) # padding\n", " cdf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n", " cdf+= text(\"Point mass F\", (theta,1.1), rgbcolor='blue',fontsize=10)\n", " cdf.axes_color('grey') \n", " return cdf\n", " \n", "def uniformFInverse(u, theta1, theta2):\n", " '''A function to evaluate the inverse CDF of a uniform(theta1, theta2) distribution.\n", " \n", " u, u should be 0 <= u <= 1, is the value to evaluate the inverse CDF at.\n", " theta1, theta2, theta2 > theta1, are the uniform distribution parameters.'''\n", " \n", " return theta1 + (theta2 - theta1)*u\n", "\n", "def uniformSample(n, theta1, theta2):\n", " '''A function to simulate samples from a uniform distribution.\n", " \n", " n > 0 is the number of samples to simulate.\n", " theta1, theta2 (theta2 > theta1) are the uniform distribution parameters.'''\n", " \n", " us = [random() for i in range(n)]\n", " \n", " return [uniformFInverse(u, theta1, theta2) for u in us]\n", "\n", "def exponentialFInverse(u, lam):\n", " '''A function to evaluate the inverse CDF of a exponential distribution.\n", " \n", " u is the value to evaluate the inverse CDF at.\n", " lam is the exponential distribution parameter.'''\n", " \n", " # log without a base is the natural logarithm\n", " return (-1.0/lam)*log(1 - u)\n", " \n", "def exponentialSample(n, lam):\n", " '''A function to simulate samples from an exponential distribution.\n", " \n", " n is the number of samples to simulate.\n", " lam is the exponential distribution parameter.'''\n", " \n", " us = [random() for i in range(n)]\n", " \n", " return [exponentialFInverse(u, lam) for u in us]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get back to our running means of Bernoullin RVs:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def bernoulliSecretThetaRunningMeans(n, mySeed = None):\n", " '''Function to give a list of n running means from Bernoulli with unknown theta.\n", " \n", " Param n is the number of running means to generate.\n", " Param mySeed is a value for the seed of the random number generator, defaulting to None\n", " Note: the unknown theta parameter for the Bernoulli process is defined in bernoulliSampleSecretTheta\n", " Return a list of n running means.'''\n", " \n", " sample = bernoulliSampleSecretTheta(n, simSeed = mySeed)\n", " from pylab import cumsum # we can import in the middle of code\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can use this function to look at say 5 different sequences of running means (they will be different, because for each iteration, we will simulate a different sample of $Bernoulli$ observations). " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 1500\n", "iterations = 5\n", "xvalues = range(1, nToGenerate+1,1)\n", "for i in range(iterations):\n", " redshade = 0.5*(iterations - 1 - i)/iterations # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToGenerate)\n", " pts = zip(xvalues,bRunningMeans)\n", " if (i == 0):\n", " p = line(pts, rgbcolor = (redshade,0,1))\n", " else:\n", " p += line(pts, rgbcolor = (redshade,0,1))\n", "show(p, figsize=[5,3], axes_labels=['n','sample mean'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What we notice is how the different lines **converge** on a sample mean of close to 0.3. \n", "\n", "Is life always this easy? Unfortunately no. In the plot below we show the well-behaved running means for the $Bernoulli$ and beside them the running means for simulated standard $Cauchy$ random variables. They are all over the place, and each time you re-evaluate the cell you'll get different all-over-the-place behaviour. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 1500\n", "iterations = 5\n", "g = twoRunningMeansPlot(nToGenerate, iterations) # uses above function to make plot\n", "show(g,figsize=[10,5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We talked about the Cauchy in more detail in an earlier notebook. If you cannot recall the detail and are interested, go back to that in your own time. The message here is that although with the Bernoulli process, the sample means converge as the number of observations increases, with the Cauchy they do not. \n", "\n", "\n", "\n", "# Limits of a Sequence of Real Numbers\n", "\n", "A sequence of real numbers $x_1, x_2, x_3, \\ldots $ (which we can also write as $\\{ x_i\\}_{i=1}^\\infty$) is said to converge to a limit $a \\in \\mathbb{R}$,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} x_i = a$$\n", "\n", "if for every natural number $m \\in \\mathbb{N}$, a natural number $N_m \\in \\mathbb{N}$ exists such that for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$\n", "\n", "What is this saying? $\\left|x_j - a\\right|$ is measuring the closeness of the $j$th value in the sequence to $a$. If we pick bigger and bigger $m$, $\\frac{1}{m}$ will get smaller and smaller. The definition of the limit is saying that if $a$ is the limit of the sequence then we can get the sequence to become as close as we want ('arbitrarily close') to $a$, and to stay that close, by going far enough into the sequence ('for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$')\n", "\n", "($\\mathbb{N}$, the natural numbers, are just the 'counting numbers' $\\{1, 2, 3, \\ldots\\}$.)\n", "\n", " \n", "\n", "Take a trivial example, the sequence $\\{x_i\\}_{i=1}^\\infty = 17, 17, 17, \\ldots$\n", "\n", "Clearly, $\\underset{i \\rightarrow \\infty}{\\lim} x_i = 17$, but let's do this formally:\n", "\n", "For every $m \\in \\mathbb{N}$, take $N_m =1$, then\n", "\n", "$\\forall$ $j \\geq N_m=1, \\left|x_j -17\\right| = \\left|17 - 17\\right| = 0 \\leq \\frac{1}{m}$, as required.\n", "\n", "($\\forall$ is mathspeak for 'for all' or 'for every')\n", "\n", "\n", "\n", "What about $\\{x_i\\}_{i=1}^\\infty = \\displaystyle\\frac{1}{1}, \\frac{1}{2}, \\frac{1}{3}, \\ldots$, i.e., $x_i = \\frac{1}{i}$?\n", "\n", "$\\underset{i \\rightarrow \\infty}{\\lim} x_i = \\underset{i \\rightarrow \\infty}{\\lim}\\frac{1}{i} = 0$\n", "\n", "For every $m \\in \\mathbb{N}$, take $N_m = m$, then $\\forall$ $j \\geq m$, $\\left|x_j - 0\\right| \\leq \\left |\\frac{1}{m} - 0\\right| = \\frac{1}{m}$\n", "\n", "### YouTry\n", "\n", "Think about $\\{x_i\\}_{i=1}^\\infty = \\frac{1}{1^p}, \\frac{1}{2^p}, \\frac{1}{3^p}, \\ldots$ with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} \\displaystyle\\frac{1}{i^p} = 0$, provided $p > 0$.\n", "\n", "You can draw the plot of this very easily using the Sage symbolic expressions we have already met (`f.subs(...)` allows us to substitute a particular value for one of the symbolic variables in the symbolic function `f`, in this case a value to use for $p$)." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('i, p')\n", "f = 1/(i^p)\n", "# make and show plot, note we can use f in the label\n", "plot(f.subs(p=1), (x, 0.1, 3), axes_labels=('i',f)).show(figsize=[6,3]) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What about $\\{x_i\\}_{i=1}^\\infty = 1^{\\frac{1}{1}}, 2^{\\frac{1}{2}}, 3^{\\frac{1}{3}}, \\ldots$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} i^{\\frac{1}{i}} = 1$.\n", "\n", "This one is not as easy to see intuitively, but again we can plot it with SageMath." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('i')\n", "f = i^(1/i)\n", "n=500\n", "p=plot(f.subs(p=1), (x, 0, n), axes_labels=('i',f)) # main plot\n", "p+=line([(0,1),(n,1)],linestyle=':') # add a dotted line at height 1\n", "p.show(figsize=[6,3]) # show the plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, $\\{x_i\\}_{i=1}^\\infty = p^{\\frac{1}{1}}, p^{\\frac{1}{2}}, p^{\\frac{1}{3}}, \\ldots$, with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} p^{\\frac{1}{i}} = 1$ provided $p > 0$.\n", "\n", "You can cut and paste (with suitable adaptations) to try to plot this one as well ..." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "x" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(end of You Try)\n", "\n", "---\n", "\n", "*back to the real stuff ...*\n", "\n", "# Limits of Functions\n", "\n", "We say that a function $f(x): \\mathbb{R} \\rightarrow \\mathbb{R}$ has a limit $L \\in \\mathbb{R}$ as $x$ approaches $a$:\n", "\n", "$$\\underset{x \\rightarrow a}{\\lim} f(x) = L$$\n", "\n", "provided $f(x)$ is arbitrarily close to $L$ for all ($\\forall$) values of $x$ that are sufficiently close to but not equal to $a$.\n", "\n", "For example\n", "\n", "Consider the function $f(x) = (1+x)^{\\frac{1}{x}}$\n", "\n", "$\\underset{x \\rightarrow 0}{\\lim} f(x) = \\underset{x \\rightarrow 0}{\\lim} (1+x)^{\\frac{1}{x}} = e \\approx 2.71828\\cdots$\n", "\n", "even though $f(0) = (1+0)^{\\frac{1}{0}}$ is undefined!" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "# x is defined as a symbolic variable by default by Sage so we do not need var('x')\n", "f = (1+x)^(1/x)\n", "# uncomment and try evaluating next line\n", "#f.subs(x=0) # this will give you an error message" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "BUT: If you are intersted in the \"Art of dividing by zero\" talk to Professor Warwick Tucker in Maths Department!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can get some idea of what is going on with two plots on different scales" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f = (1+x)^(1/x)\n", "n1=5\n", "p1=plot(f.subs(p=1), (x, 0.001, n1), axes_labels=('x',f)) # main plot\n", "t1 = text(\"Large scale plot\", (n1/2,e), rgbcolor='blue',fontsize=10) \n", "n2=0.1\n", "p2=plot(f.subs(p=1), (x, 0.0000001, n2), axes_labels=('x',f)) # main plot\n", "p2+=line([(0,e),(n2,e)],linestyle=':') # add a dotted line at height e\n", "t2 = text(\"Small scale plot\", (n2/2,e+.01), rgbcolor='blue',fontsize=10) \n", "show(graphics_array((p1+t1,p2+t2)),figsize=[6,3]) # show the plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "all this has been laying the groundwork for the topic of real interest to us ...\n", "\n", "# Limit of a Sequence of Random Variables\n", "\n", "We want to be able to say things like $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$ in some sensible way. $X_i$ are some random variables, $X$ is some 'limiting random variable', but what do we mean by 'limiting random variable'?\n", "\n", "To help us, lets introduce a very very simple random variable, one that puts all its mass in one place. " ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "theta = 2.0\n", "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),\\\n", " figsize=[8,2]) # show the plots" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is known as the $Point\\,Mass(\\theta)$ random variable, $\\theta \\in \\mathbb(R)$: the density $f(x)$ is 1 if $x=\\theta$ and 0 everywhere else\n", "\n", "$$\n", "f(x;\\theta) =\n", "\\begin{cases}\n", "0 & \\text{ if } x \\neq \\theta \\\\\n", "1 & \\text{ if } x = \\theta\n", "\\end{cases}\n", "$$\n", "\n", "$$\n", "F(x;\\theta) =\n", "\\begin{cases}\n", "0 & \\text{ if } x < \\theta \\\\\n", "1 & \\text{ if } x \\geq \\theta\n", "\\end{cases}\n", "$$\n", "\n", "So, if we had some sequence $\\{\\theta_i\\}_{i=1}^\\infty$ and $\\underset{i \\rightarrow \\infty}{\\lim} \\theta_i = \\theta$\n", "\n", "and we had a sequence of random variables $X_i \\sim Point\\,Mass(\\theta_i)$, $i = 1, 2, 3, \\ldots$\n", "\n", "then we could talk about a limiting random variable as $X \\sim Point\\,Mass(\\theta)$:\n", "\n", "i.e., we could talk about $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 202 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of point mass rvs converging on theta = 0\n", "ptsize = 20\n", "i = 1\n", "theta_i = 1/i\n", "p = points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n", "p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n", "while theta_i > 0.01:\n", " i+=1\n", " theta_i = 1/i\n", " p += points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n", "p += points((0,1), rgbcolor=\"red\", pointsize=ptsize)\n", "p += line([(0,0),(0,1)], rgbcolor=\"red\", linestyle=':')\n", "p.show(xmin=-1, xmax = 2, ymin=0, ymax = 1.1, axes=false, gridlines=[None,[0]], \\\n", " figsize=[7,2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we want to generalise this notion of a limit to other random variables (that are not necessarily $Point\\,Mass(\\theta_i)$ RVs)\n", "\n", "What about one many of you will be familiar with - the 'bell-shaped curve' \n", "\n", "## The $Gaussian(\\mu, \\sigma^2)$ or $Normal(\\mu, \\sigma^2)$ RV?\n", "\n", "The probability density function (PDF) $f(x)$ is given by\n", "\n", "$$\n", "f(x ;\\mu, \\sigma) = \\displaystyle\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left(\\frac{-1}{2\\sigma^2}(x-\\mu)^2\\right)\n", "$$\n", "\n", "The two parameters, $\\mu \\in \\mathbb{R} := (-\\infty,\\infty)$ and $\\sigma \\in (0,\\infty)$, are sometimes referred to as the location and scale parameters.\n", "\n", "To see why this is, use the interactive plot below to have a look at what happens to the shape of the density function $f(x)$ when you change $\\mu$ or increase or decrease $\\sigma$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "f9cdfb72fc0f45de899199dca263d861", "version_major": 2, "version_minor": 0 }, "text/plain": [ "SW50ZXJhY3RpdmUgZnVuY3Rpb24gPGZ1bmN0aW9uIF8gYXQgMHg3Zjg1Y2QyNmQ3ZDA+IHdpdGggMiB3aWRnZXRzCiAgbXlfbXU6IEV2YWxUZXh0KHZhbHVlPXUnMCcsIGRlc2NyaXB0aW9uPXXigKY=\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@interact\n", "def _(my_mu=input_box(0, label='mu') ,my_sigma=input_box(1,label='sigma')):\n", " '''Interactive function to plot the normal pdf and ecdf.'''\n", " \n", " if my_sigma > 0:\n", " html('

Normal('+str(my_mu)+','+str(my_sigma)+'2)

')\n", " var('mu sigma')\n", " f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", " p1=plot(f.subs(mu=my_mu,sigma=my_sigma), \\\n", " (x, my_mu - 3*my_sigma - 2, my_mu + 3*my_sigma + 2),\\\n", " axes_labels=('x','f(x)'))\n", " show(p1,figsize=[8,3])\n", " else:\n", " print \"sigma must be greater than 0\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the sequence of random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "We can use the animation below to see how the PDF $f_{i}(x)$ looks as we move through the sequence of $X_i$ (the animation only goes to $i = 25$, $\\sigma = 0.04$ but you get the picture ...)\n", "\n", "\n", " \n", " \n", " \n", "\n", " \n", " \n", "
Normal curve animation, looping through $\\sigma = \\frac{1}{i}$ for $i = 1, \\dots, 25$
\n", "\n", "We can see that the probability mass of $X_i \\sim Normal(0, \\frac{1}{i})$ increasingly concentrates about 0 as $i \\rightarrow \\infty$ and $\\frac{1}{i} \\rightarrow 0$\n", "\n", "Does this mean that $\\underset{i \\rightarrow \\infty}{\\lim} X_i = Point\\,Mass(0)$?\n", "\n", "No, because for any $i$, however large, $P(X_i = 0) = 0$ because $X_i$ is a continuous RV (for any continous RV $X$, for any $x \\in \\mathbb{R}$, $P(X=x) = 0$).\n", "\n", "So, we need to refine our notions of convergence when we are dealing with random variables\n", "\n", "# Convergence in Distribution\n", "\n", "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable. Let $F_i$ denote the distribution function (DF) of $X_i$ and let $F$ denote the distribution function of $X$.\n", "\n", "Now, if for any real number $t$ at which $F$ is continuous,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n", "\n", "(in the sense of the convergence or limits of functions we talked about earlier)\n", "\n", "Then we can say that the sequence or RVs $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in distribution** and write $X_i \\overset{d}{\\rightarrow} X$.\n", "\n", "An equivalent way of defining convergence in distribution is to go right back to the meaning of the probabilty space 'under the hood' of a random variable, a random variable $X$ as a mapping from the sample space $\\Omega$ to the real line ($X: \\Omega \\rightarrow \\mathbb{R}$), and the sample points or outcomes in the sample space, the $\\omega \\in \\Omega$. For $\\omega \\in \\Omega$, $X(\\omega)$ is the mapping of $\\omega$ to the real line $\\mathbb{R}$. We could look at the set of $\\omega$ such that $X(\\omega) \\leq t$, i.e. the set of $\\omega$ that map to some value on the real line less than or equal to $t$, $\\{\\omega: X(\\omega) \\leq t \\}$. \n", "\n", "Saying that for any $t \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ is the equivalent of saying that for any $t \\in \\mathbb{R}$, \n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:X_i(\\omega) \\leq t \\}\\right) = P\\left(\\{\\omega: X(\\omega) \\leq t\\right)$$\n", "\n", "Armed with this, we can go back to our sequence of $Normal$ random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "and let $X \\sim Point\\,Mass(0)$,\n", "\n", "and say that the $X_i$ **converge in distribution** to the $x \\sim Point\\,Mass$ RV $X$,\n", "\n", "$$X_i \\overset{d}{\\rightarrow} X$$\n", "\n", "What we are saying with convergence in distribution, informally, is that as $i$ increases, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by the limiting random variable. In this case, as $i$ increases, the $Point\\,Mass(0)$ is becoming a better and better model for the next outcome of a random experiment with outcomes $\\sim Normal(0,\\frac{1}{i})$." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 14 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of converging normal distributions\n", "my_mu = 0\n", "upper = my_mu + 5; lower = -upper; # limits for plot\n", "var('mu sigma')\n", "stop_i = 12\n", "html('

N(0,1) to N(0, 1/'+str(stop_i)+')

')\n", "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", "p=plot(f.subs(mu=my_mu,sigma=1.0), (x, lower, upper), rgbcolor = (0,0,1))\n", "for i in range(2, stop_i, 1): # just do a few of them\n", " shade = 1-11/i # make them different colours\n", " p+=plot(f.subs(mu=my_mu,sigma=1/i), (x, lower, upper), rgbcolor = (1-shade, 0, shade))\n", "textOffset = -0.2 # offset for placement of text - may need adjusting \n", "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n", "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### There is an interesting point to note about this convergence: \n", "\n", "We have said that the $X_i \\sim Normal(0,\\frac{1}{i})$ with distribution functions $F_i$ converge in distribution to $X \\sim Point\\,Mass(0)$ with distribution function $F$, which means that we must be able to show that for any real number $t$ at which $F$ is continuous,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n", "\n", "Note that for any of the $X_i \\sim Normal(0, \\frac{1}{i})$, $F_i(0) = \\frac{1}{2}$, and also note that for $X \\sim Point,Mass(0)$, $F(0) = 1$, so clearly $F_i(0) \\neq F(0)$. \n", "\n", "What has gone wrong? \n", "\n", "Nothing: we said that we had to be able to show that $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ for any $t \\in \\mathbb{R}$ at which $F$ is continuous, but the $Point\\,Mass(0)$ distribution function $F$ is not continous at 0!" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "theta = 0.0\n", "# show the plots\n", "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),figsize=[8,2]) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Convergence in Probability\n", "\n", "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable. Let $F_i$ denote the distribution function (DF) of$X_i$ and let $F$ denote the distribution function of $X$.\n", "\n", "Now, if for any real number $\\varepsilon > 0$,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$$\n", "\n", "Then we can say that the sequence $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in probability** and write $X_i \\overset{P}{\\rightarrow} X$.\n", "\n", "Or, going back again to the probability space 'under the hood' of a random variable, we could look the way the $X_i$ maps each outcome $\\omega \\in \\Omega$, $X_i(\\omega)$, which is some point on the real line, and compare this to mapping $X(\\omega)$. \n", "\n", "Saying that for any $\\varepsilon \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$ is the equivalent of saying that for any $\\varepsilon \\in \\mathbb{R}$, \n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:|X_i(\\omega) - X(\\omega)| > \\varepsilon \\}\\right) = 0$$\n", "\n", "Informally, we are saying $X$ is a limit in probabilty if, by going far enough into the sequence $X_i$, we can ensure that the mappings $X_i(\\omega)$ and $X(\\omega)$ will be arbitrarily close to each other on the real line for all $\\omega \\in \\Omega$.\n", "\n", "**Note** that convergence in distribution is implied by convergence in probability: convergence in distribution is the weakest form of convergence; any sequence of RV's that converges in probability to some RV $X$ also converges in distribution to $X$ (but not necessarily vice versa). " ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of converging normal distributions\n", "my_mu = 0\n", "var('mu sigma')\n", "upper = 0.2; lower = -upper\n", "i = 20 # start part way into the sequence\n", "lim = 100 # how far to go\n", "stop_i = 12\n", "html('

N(0,1/'+str(i)+') to N(0, 1/'+str(lim)+')

')\n", "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", "p=plot(f.subs(mu=my_mu,sigma=1.0/i), (x, lower, upper), rgbcolor = (0,0,1))\n", "for j in range(i, lim+1, 4): # just do a few of them\n", " shade = 1-(j-i)/(lim-i) # make them different colours\n", " p+=plot(f.subs(mu=my_mu,sigma=1/j), (x, lower,upper), rgbcolor = (1-shade, 0, shade))\n", "textOffset = -1.5 # offset for placement of text - may need adjusting \n", "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n", "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For our sequence of $Normal$ random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "and $X \\sim Point\\,Mass(0)$,\n", "\n", "It can be shown that the $X_i$ converge in probability to $X \\sim Point\\,Mass(0)$ RV $X$,\n", "\n", "$$X_i \\overset{P}{\\rightarrow} X$$\n", "\n", "(the formal proof of this involves Markov's Inequality, which is beyond the scope of this course). \n", "\n", "# Some Basic Limit Laws in Statistics\n", "\n", "Intuition behind Law of Large Numbers and Central Limit Theorem\n", "\n", "Take a look at the Khan academy videos on the Law of Large Numbers and the Central Limit Theorem. This will give you a working idea of these theorems. In the sequel, we will strive for a deeper understanding of these theorems on the basis of the two notions of convergence of sequences of random variables we just saw.\n", " \n", "\n", "## Weak Law of Large Numbers\n", "\n", "Remember that a statistic is a random variable, so a sample mean is a random variable. If we are given a sequence of independent and identically distributed RVs, $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$, then we can also think of a sequence of random variables $\\overline{X}_1, \\overline{X}_2, \\ldots, \\overline{X}_n, \\ldots$ ($n$ being the sample size). \n", "\n", "Since $X_1, X_2, \\ldots$ are $IID$, they all have the same expection, say $E(X_1)$ by convention.\n", "\n", "If $E(X_1)$ exists, then the sample mean $\\overline{X}_n$ converges in probability to $E(X_1)$ (i.e., to the expectatation of any one of the individual RVs):\n", "\n", "$$\n", "\\text{If} \\quad X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1 \\ \\text{and if } \\ E(X_1) \\ \\text{exists, then } \\ \\overline{X}_n \\overset{P}{\\rightarrow} E(X_1) \\ .\n", "$$\n", "\n", "Going back to our definition of convergence in probability, we see that this means that for any real number $\\varepsilon > 0$, $\\underset{n \\rightarrow \\infty}{\\lim} P\\left(|\\overline{X}_n - E(X_1)| > \\varepsilon\\right) = 0$\n", "\n", "Informally, this means that means that, by taking larger and larger samples we can make the probability that the average of the observations is more than $\\varepsilon$ away from the expected value get smaller and smaller.\n", "\n", "Proof of this is beyond the scope of this course, but we have already seen it in action when we looked at the $Bernoulli$ running means. Have another look, this time with only one sequence of running means. You can increase $n$, the sample size, and change $\\theta$. Note that the seed for the random number generator is also under your control. This means that you can get replicable samples: in particular, in this interact, when you increase the sample size it looks as though you are just adding more to an existing sample rather than starting from scratch with a new one. " ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "8899b48028ea42dd9843f9752b83f94d", "version_major": 2, "version_minor": 0 }, "text/plain": [ "SW50ZXJhY3RpdmUgZnVuY3Rpb24gPGZ1bmN0aW9uIF8gYXQgMHg3Zjg1Y2NiYmFjMDg+IHdpdGggMyB3aWRnZXRzCiAgblRvR2VuOiBUcmFuc2Zvcm1JbnRTbGlkZXIodmFsdWU9MTAwLCBkZXPigKY=\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@interact\n", "def _(nToGen=slider(1,1500,1,100,label='n'),my_theta=input_box(0.3,label='theta'),rSeed=input_box(1234,label='random seed')):\n", " '''Interactive function to plot running mean for a Bernoulli with specified n, theta and random number seed.'''\n", " \n", " if my_theta >= 0 and my_theta <= 1:\n", " html('

Bernoulli('+str(my_theta.n(digits=2))+')

')\n", " xvalues = range(1, nToGen+1,1)\n", " bRunningMeans = bernoulliRunningMeans(nToGen, myTheta=my_theta, mySeed=rSeed)\n", " pts = zip(xvalues, bRunningMeans)\n", " p = line(pts, rgbcolor = (0,0,1))\n", " p+=line([(0,my_theta),(nToGen,my_theta)],linestyle=':',rgbcolor='grey')\n", " show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print 'Theta must be between 0 and 1'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Central Limit Theorem\n", "\n", "You have probably all heard of the Central Limit Theorem before, but now we can relate it to our definition of convergence in distribution. \n", "\n", "Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist,\n", "\n", "then\n", "\n", "$$\n", "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n", "$$\n", "\n", "And remember $Z \\sim Normal(0,1)$?\n", "\n", "Consider $Z_n := \\displaystyle\\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}}$\n", "\n", "If $\\overline{X}_n = \\displaystyle\\frac{1}{n} \\displaystyle\\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, then $\\overline{X}_n -E(X_1) \\overset{d}{\\rightarrow} X-E(X_1) \\sim Normal \\left( 0,\\frac{V(X_1)}{n} \\right)$\n", "\n", "and $\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right) \\overset{d}{\\rightarrow} \\sqrt{n} \\left( X-E(X_1) \\right) \\sim Normal \\left( 0,V(X_1) \\right)$\n", "\n", "so $Z_n := \\displaystyle \\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}} \\overset{d}{\\rightarrow} Z \\sim Normal \\left( 0,1 \\right)$\n", "\n", "Thus, for sufficiently large $n$ (say $n>30$), probability statements about $\\overline{X}_n$ can be approximated using the $Normal$ distribution. \n", "\n", "The beauty of the CLT, as you have probably seen from other courses, is that $\\overline{X}_n \\overset{d}{\\rightarrow} Normal \\left( E(X_1), \\frac{V(X_1)}{n} \\right)$ does not require the $X_i$ to be normally distributed. \n", "\n", "We can try this with our $Bernoulli$ RV generator. First, a small number of samples:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[3/5, 1/2, 1/2, 3/10, 1/2]" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "theta, n, samples = 0.6, 10, 5 # concise way to set some variable values\n", "sampleMeans=[] # empty list\n", "for i in range(0, samples, 1): # loop \n", " thisMean = QQ(sum(bernoulliSample(n, theta)))/n # get a sample and find the mean\n", " sampleMeans.append(thisMean) # add mean to the list of means\n", "sampleMeans # disclose the sample means" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can use the interactive plot to increase the number of samples and make a histogram of the sample means. According to the CLT, for lots of reasonably-sized samples we should get a nice symmetric bell-curve-ish histogram centred on $\\theta$. You can adjust the number of bins in the histogram as well as the number of samples, sample size, and $\\theta$. " ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "5ce67616de144078ac0e7ae0c6215b78", "version_major": 2, "version_minor": 0 }, "text/plain": [ "SW50ZXJhY3RpdmUgZnVuY3Rpb24gPGZ1bmN0aW9uIF8gYXQgMHg3Zjg1Y2M1NjQwNTA+IHdpdGggNCB3aWRnZXRzCiAgcmVwbGljYXRlczogVHJhbnNmb3JtSW50U2xpZGVyKHZhbHVlPTEwMCzigKY=\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=slider(1,3000,1,100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_theta=input_box(0.3,label='theta'),Bins=5):\n", " '''Interactive function to plot distribution of replicates of sample means for n IID Bernoulli trials.'''\n", " \n", " if my_theta >= 0 and my_theta <= 1 and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1): \n", " thisMean = RR(sum(bernoulliSample(nToGen, my_theta)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Bernoulli sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print 'Theta must be between 0 and 1, and samples > 0'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Increase the sample size and the numbe rof bins in the above interact and see if the histograms of the sample means are looking more and more normal as the CLT would have us believe." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But although the $X_i$ do not have to be $\\sim Normal$ for $\\overline{X}_n = \\overset{d}{\\rightarrow} X \\sim Normal\\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, remember that we said \"Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist\", then,\n", "\n", "$$\n", "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n", "$$\n", "\n", "This is where is all goes horribly wrong for the standard $Cauchy$ distribution (any $Cauchy$ distribution in fact): neither the expectation nor the variance exist for this distribution. The Central Limit Theorem cannot be applied here. In fact, if $X_1,X_2,\\ldots \\overset{IID}{\\sim}$ standard $Cauchy$, then $\\overline{X}_n = \\displaystyle \\frac{1}{n} \\sum_{i=1}^n X_i \\sim$ standard $Cauchy$.\n", "\n", "### YouTry\n", "\n", "Try looking at samples from two other RVs where the expectation and variance do exist, the $Uniform$ and the $Exponential$:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "9338da0eb0ad4d959bfbefc11947853c", "version_major": 2, "version_minor": 0 }, "text/plain": [ "SW50ZXJhY3RpdmUgZnVuY3Rpb24gPGZ1bmN0aW9uIF8gYXQgMHg3Zjg1Y2M1NjQxNDA+IHdpdGggNSB3aWRnZXRzCiAgcmVwbGljYXRlczogRXZhbFRleHQodmFsdWU9dScxMDAnLCBkZXNjcmnigKY=\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=input_box(100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_theta1=input_box(2,label='theta1'),\\\n", " my_theta2=input_box(4,label='theta1'),Bins=5):\n", " '''Interactive function to plot distribution of \n", " sample means for n IID Uniform(theta1, theta2) trials.'''\n", " \n", " if (my_theta1 < my_theta2) and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1):\n", " \n", " thisMean = RR(sum(uniformSample(nToGen, my_theta1, my_theta2)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Uniform sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print 'theta1 must be less than theta2, and samples > 0'" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "13c3f2f51e1c4fb49955764c73e8ab87", "version_major": 2, "version_minor": 0 }, "text/plain": [ "SW50ZXJhY3RpdmUgZnVuY3Rpb24gPGZ1bmN0aW9uIF8gYXQgMHg3Zjg1Y2M4NWNiOTA+IHdpdGggNCB3aWRnZXRzCiAgcmVwbGljYXRlczogRXZhbFRleHQodmFsdWU9dScxMDAnLCBkZXNjcmnigKY=\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=input_box(100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_lambda=input_box(0.1,label='lambda'),Bins=5):\n", " '''Interactive function to plot distribution of \\\n", " sample means for an Exponential(lambda) process.'''\n", " \n", " if my_lambda > 0 and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1): \n", " thisMean = RR(sum(exponentialSample(nToGen, my_lambda)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Exponential sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print 'lambda must be greater than 0, and samples > 0'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Properties of the MLE\n", "\n", "The LLN (law of large numbers) and CLT (central limit theorem) are statements about the limiting distribution of the sample mean of IID random variables whose expectation and variance exists. How does this apply to the MLE (maximum likelihood estimator)?\n", "\n", "Consider the following generic parametric model for our data or observations:\n", "\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x; \\theta^*) \\ \\text{ or } \\ f(x; \\theta^*)\n", "$$\n", "\n", "We do not know the true parameter $\\theta^*$ under the model for our data. Our task is to estimate the unknown parameter $\\theta^*$ using the MLE:\n", "\n", "$$\\widehat{\\Theta}_n = argmax_{\\theta \\in \\mathbf{\\Theta}} l(\\theta)$$\n", "\n", "The amazing think about the MLE is its following properties:\n", "\n", "### 1. The MLE is *asymptotically consistent*\n", "\n", "$$\\boxed{\\widehat{\\Theta}_n \\overset{P}{\\rightarrow} \\theta^*}$$\n", "\n", "So when the number of observations (sample size $n$) goes to infinity, our MLE converges in probability to the true parameter $\\theta^* \\in \\mathbf{\\Theta}$.\n", "\n", "Interestingly, one can work out the details and find that the MLE $\\widehat{\\Theta}_n$, which is also a random variable based on $n$ IID samples that takes values in the parameter space $\\mathbf{\\Theta}$, is also normally distributed for large sample sizes.\n", "\n", "### 2. The MLE is *equivariant*\n", "\n", "$$\\boxed{\\text{If } \\ \\widehat{\\Theta}_n \\ \\text{ is the MLE of } \\ \\theta^* \\ \\text{ then } \\ g(\\widehat{\\Theta}_n) \\ \\text{ is the MLE of } \\ g(\\theta^*)}$$\n", "\n", "This is a very useful property, since any function $g : \\mathbf{\\Theta} \\to \\mathbb{R}$ of interest is at our disposal by merely applying $g$ to the the MLE. Often $g$ is some sort of reward that depends on the unknown parameter $\\theta^*$.\n", "\n", "### 3. The MLE is *asymptotically normal* \n", "\n", "$$\\boxed{ \\frac{\\left(\\widehat{\\Theta}_n - \\theta^*\\right)}{\\widehat{se}_n} \\overset{d}{\\rightarrow} Normal(0,1) }\n", "\\quad \\text{ or equivalently, } \\quad\n", "\\boxed{ \\widehat{\\Theta}_n \\overset{d}{\\rightarrow} Normal( \\theta^*, \\widehat{se}_n^2) }\n", "$$\n", "\n", "where, $\\widehat{se}_n$ is the *estimated standard error* of the MLE:\n", "\n", "$$\\boxed{ \\widehat{se}_n \\ \\text{ is an estimate of the } \\ \\sqrt{V\\left(\\widehat{\\Theta}_n \\right)}}$$\n", "\n", "We can compute $\\widehat{se}_n$ with the following formula:\n", "\n", "$$\\boxed{\\widehat{se}_n = \\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}$$\n", "\n", "where, the expectation is called the *Fisher information* of one sample or $I_1$:\n", "\n", "$$\\boxed{ I_1 := E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) = \n", "\\begin{cases}\n", "\\displaystyle{\\int{\\left(-\\frac{\\partial^2 \\log f(x;\\theta)}{\\partial \\theta^2} \\right) f(x; \\theta)} dx} & \\text{ for continuous RV } X\\\\\n", "\\displaystyle{\\sum_x{\\left(-\\frac{\\partial^2 \\log f(x;\\theta)}{\\partial \\theta^2} \\right) f(x; \\theta)}}& \\text{ for discrete RV } X\n", "\\end{cases}\n", "}\n", "$$\n", "\n", "Other two properties (not needed for this course) include:\n", "\n", "- *asymptotic efficiency*, i.e., among a class of well-behaved estimators, the MLE has the smallest variance at least for large samples, and\n", "- *approximately Bayes*, i.e., the MLE is approximately the *Bayes estimator* (some of you may see Bayesian methods of estimation in advanced courses in statistical machine learning or in latest AI methods)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Confidence Interval and Set Estimation from MLE\n", "\n", "An immediate implication of the asymptotic normality of the MLE, which informally states that the distribution of the MLE can be approximated by a Normal random variable, is to obtain confidence intervals for the unkown parameter $\\theta^*$.\n", "\n", "Recall that in set estimation, as opposed to point estimation, we estimate the unknown parameter using a random set based on the data (typically intervals in 1D) that \"traps\" the true parameter $\\theta^*$ with a very high probability, say $0.95$. We typically express such probality in terms of $1-\\alpha$, so the $95\\%$ confidence interval is seen as a $1-\\alpha$ confidence interval with $\\alpha=0.05$. From the the asymptotic normality of the MLE, we get the following confidence interval for the unknown $\\theta^*$:\n", "\n", "\n", "$$\n", "\\boxed{\\text{If } \\quad \n", "\\displaystyle{C_n := \\left( \\widehat{\\Theta}_n - z_{\\alpha/2} \\widehat{se}_n, \\, \\widehat{\\Theta}_n + z_{\\alpha/2} \\widehat{se}_n \\right)} \\quad \\text{ then } \\quad P \\left( \\{ \\theta^* \\in C_n \\} ; \\theta^* \\right) \\underset{n \\to \\infty}{\\longrightarrow} 1-\\alpha , \\quad \\text{ where } z_{\\alpha/2} = \\Phi^{[-1]}(1-\\alpha/2).\n", "}\n", "$$\n", "\n", "Recall that $P \\left( \\{ \\theta^* \\in C_n \\} ; \\theta^* \\right)$ is simply the probability of the event that $\\theta^*$ will be in $C_n$, the $1-\\alpha$ confidence interval, given the data is distributed according to the model with true parameter $\\theta^*$.\n", "\n", "NOTE: $\\Phi^{[-1]}(1-\\alpha/2)$ is merely the inverse distribution function (CDF) of the standard normal RV. \n", "\n", "$$\n", "\\text{For } \\alpha=0.05, z_{\\alpha/2}=1.96 \\approxeq 2, \\text{ so: } \\quad \\boxed{\\widehat{\\Theta}_n \\pm 2 \\widehat{se}_n} \\quad \\text{ is an approximate 95% confidence interval.}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example of Confidence Interval for IID $Bernoulli(\\theta)$ Trials\n", "\n", "We already know that the MLE for the model with $n$ IID $Bernoulli(\\theta)$ Trials is the sample mean, i.e.,\n", "\n", "$$X_1,X_2,\\ldots, X_n \\overset{IID}{\\sim} Bernoulli(\\theta^*) \\implies \\widehat{\\Theta}_n = \\overline{X}_n$$\n", "\n", "Our task now is to obtain the $1-\\alpha$ confidence interval based on this MLE.\n", "\n", "To get the confidence interval we need to obtain $\\widehat{se}_n$ by computing the following:\n", "\n", "$$\n", "\\begin{array}{cc}\n", "\\widehat{se}_n &=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\n", "\\end{array}\n", "$$\n", "$I_1 := E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right)$ is called the Fisher Information of one sample.\n", "Since our IID samples are from a discrete distribution with \n", "\n", "$$\n", "\\begin{array}{cc}\n", "f(x; \\theta) = \\theta^x (1-\\theta)^{1-x} \n", "&\\implies& \\displaystyle{\\log \\left( f(x;\\theta) \\right) = x \\log(\\theta) +(1-x) \\log(1-\\theta)}\\\\\n", "&\\implies& \\displaystyle{\\frac{\\partial}{\\partial \\theta} \\left(\\log \\left( f(x;\\theta) \\right)\\right)} \n", "= \\displaystyle{\\frac{x}{\\theta} -\\frac{1-x}{1-\\theta}} \\\\\n", "&\\implies& \\displaystyle{\\frac{\\partial^2}{\\partial \\theta^2} \\left(\\log \\left( f(x;\\theta) \\right)\\right)} \n", "= \\displaystyle{-\\frac{x}{\\theta^2} - \\frac{1-x}{(1-\\theta)^2}}\\\\\n", "&\\implies& \\displaystyle{E \\left( - \\frac{\\partial^2}{\\partial \\theta^2} \\left(\\log \\left( f(x;\\theta) \\right)\\right) \\right)} \n", "= \\displaystyle{\\sum_{x\\in\\{0,1\\}} \\left( \\frac{x}{\\theta^2} + \\frac{1-x}{(1-\\theta)^2} \\right) f(x; \\theta) = \\frac{\\theta}{\\theta^2} + \\frac{1-\\theta}{(1-\\theta)^2} = \\frac{1}{\\theta(1-\\theta)}}\n", "\\end{array}\n", "$$\n", "\n", "Note that we have implicitly assumed that the $x$ values are only $0$ or $1$ by ignoring the indicator term $\\mathbf{1}_{\\{0,1\\}}(x)$ in $f(x;\\theta)$. But this is okay as we are carefully doing the sums over just $x \\in \\{0,1\\}$.\n", "\n", "Now, by using the formula for $\\widehat{se}_n$, we can obtain:\n", "\n", "$$\n", "\\begin{array}{cc}\n", "\\widehat{se}_n \n", "&=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\\\\\n", "&=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n \\frac{1}{\\theta(1-\\theta)} \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\\\\\n", "&=& \\displaystyle{\\sqrt{\\frac{\\widehat{\\theta}_n(1-\\widehat{\\theta}_n)}{n}}}\n", "\\end{array}\n", "$$\n", "\n", "Finally, we can complete our task by obtaining the 95% confidence interval for $\\theta^*$ as follows:\n", "\n", "$$\n", "\\displaystyle{ \\widehat{\\theta}_n \\pm 2 \\widehat{se}_n = \\widehat{\\theta}_n \\pm 2 \\sqrt{\\frac{\\widehat{\\theta}_n(1-\\widehat{\\theta}_n)}{n}} = \\overline{x}_n \\pm 2 \\sqrt{\\frac{\\overline{x}_n(1-\\overline{x}_n)}{n}} }\n", "$$" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 42 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 100\n", "replicates = 20\n", "xvalues = range(1, nToGenerate+1,1)\n", "for i in range(replicates):\n", " redshade = 0.5*(replicates - 1 - i)/replicates # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToGenerate)\n", " pts = zip(xvalues,bRunningMeans)\n", " if (i == 0):\n", " p = line(pts, rgbcolor = (redshade,0,1))\n", " else:\n", " p += line(pts, rgbcolor = (redshade,0,1))\n", " mle=bRunningMeans[nToGenerate-1]\n", " se95Correction=2.0*sqrt(mle*(1-mle)/nToGenerate)\n", " lower95CI = mle-se95Correction\n", " upper95CI = mle+se95Correction\n", " p += line([(nToGenerate+i,lower95CI),(nToGenerate+i,upper95CI)], rgbcolor = (redshade,0,1), thickness=0.5)\n", "p += line([(1,0.3),(nToGenerate+replicates,0.3)], rgbcolor='black', thickness='2')\n", "p += text('sample mean up to n='+str(nToGenerate)+' and their 95% confidence intervals',(nToGenerate/1.5,1),fontsize=16)\n", "show(p, figsize=[10,6])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 5\n", "\n", "Obtain the 95% Confidence Interval for the $\\lambda^*$ from the experiment based on $n$ IID $Exponential(\\lambda)$ trials.\n", "\n", "Write down your answer by returning the right answer in the function `SampleExamProblem5` in the next cell.\n", "Your function call `SampleExamProblem5(sampleWaitingTimes)` on the Orbiter waiting times data should return the 95% confidence interval for the unknown parameter $\\lambda^*$." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Sample Exam Problem 5\n", "# Only replace the XXX below, do not change the function naemes or parameters\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "def SampleExamProblem5(exponentialSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown rate parameter lambda* \n", " from n IID Exponential(lambda*) trials in the input numpy array called exponentialSamples'''\n", " XXX\n", " XXX\n", " XXX\n", " lower95CI=XXX\n", " upper95CI=XXX\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = SampleExamProblem5(sampleWaitingTimes)\n", "print \"The 95% CI for lambda in the Orbiter Waiting time experiment = \"\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 5 Solution\n", "\n", "We can obtain the 95% Confidence Interval for the $\\lambda^*$ for the experiment based on $n$ IID $Exponential(\\lambda)$ trials, by hand or using SageMath symbolic computations (typically both).\n", "\n", "Let $X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} Exponential(\\lambda^*)$. \n", "\n", "We saw that the ML estimator of $\\lambda^* \\in (0,\\infty)$ is $\\widehat{\\Lambda}_n = 1/\\, \\overline{X}_n$ and its ML estimate is $\\widehat{\\lambda}_n=1/\\, \\overline{x}_n$, where $x_1,x_2,\\ldots,x_n$ are our observed data.\n", "\n", "Let us obtain $I_1$, the Fisher Information of one sample, for this experiment to find the standard error:\n", "\n", "$$\n", "\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n) = \\frac{1}{\\sqrt{n \\left. I_1 \\right\\vert_{\\lambda=\\widehat{\\lambda}_n}}}\n", "$$\n", "\n", "and construct an approximate $95\\%$ confidence interval for $\\lambda^*$ using the asymptotic normality of its ML estimator $\\widehat{\\Lambda}_n$.\n", "\n", "Since the probability density function $f(x;\\lambda)=\\lambda e^{-\\lambda x}$, for $x\\in [0,\\infty)$, we have,\n", "\n", "$$\n", "\\begin{align}\n", "I_1 &= - E \\left( \\frac{\\partial^2 \\log f(X;\\lambda)}{\\partial^2 \\lambda} \\right) = - \\int_{x \\in [0,\\infty)} \\left( \\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda} \\right) \\lambda e^{-\\lambda x} \\ dx\n", "\\end{align}\n", "$$\n", "\n", "Let us compute the above integrand next.\n", "$$\n", "\\begin{align}\n", "\\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda}\n", "&:=\n", "\\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log \\left( \\lambda e^{-\\lambda x} \\right) \\right) \\right)\n", "= \\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log(\\lambda) + \\log(e^{-\\lambda x} \\right) \\right) \\\\\n", "&= \\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log(\\lambda) -\\lambda x \\right) \\right)\n", "= \\frac{\\partial}{\\partial \\lambda} \\left( {\\lambda}^{-1} - x \\right) = - \\lambda^{-2} - 0 = -\\frac{1}{\\lambda^2}\n", "\\end{align}\n", "$$\n", "Now, let us evaluate the integral by recalling that the expectation of the constant $1$ is 1 for any RV $X$ governed by some parameter, say $\\theta$. For instance when $X$ is a continuous RV, $E_{\\theta}(1) = \\int_{x \\in \\mathbb{X}} 1 \\ f(x;\\theta) = \\int_{x \\in \\mathbb{X}} \\ f(x;\\theta) = 1$. Therefore, the Fisher Information of one sample is\n", "$$\n", "\\begin{align}\n", "I_1(\\theta) = - \\int_{x \\in \\mathbb{X} = [0,\\infty)} \\left( \\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda} \\right) \\lambda e^{-\\lambda x} \\ dx\n", " &= - \\int_{0}^{\\infty} \\left(-\\frac{1}{\\lambda^2} \\right) \\lambda e^{-\\lambda x} \\ dx \\\\\n", "& = - \\left(-\\frac{1}{\\lambda^2} \\right) \\int_{0}^{\\infty} \\lambda e^{-\\lambda x} \\ dx = \\frac{1}{\\lambda^2} \\ 1 = \\frac{1}{\\lambda^2}\n", "\\end{align}\n", "$$\n", "Now, we can compute the desired estimated standard error, by substituting in the ML estimate $\\widehat{\\lambda}_n = 1/(\\overline{x}_n) := 1 / \\left( \\sum_{i=1}^n x_i \\right)$ of $\\lambda^*$, as follows:\n", "$$\n", "\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n) \n", "= \\frac{1}{\\sqrt{n \\left. I_1 \\right\\vert_{\\lambda=\\widehat{\\lambda}_n}}}\n", "= \\frac{1}{\\sqrt{n \\frac{1}{\\widehat{\\lambda}_n^2} }}\n", "= \\frac{\\widehat{\\lambda}_n}{\\sqrt{n}}\n", "= \\frac{1}{\\sqrt{n} \\ \\overline{x}_n}\n", "$$\n", "Using $\\widehat{\\mathsf{se}}_n(\\widehat{\\lambda}_n)$ we can construct an approximate $95\\%$ confidence interval $C_n$ for $\\lambda^*$, due to the asymptotic normality of the ML estimator of $\\lambda^*$, as follows:\n", "$$\n", "C_n\n", "= \\widehat{\\lambda}_n \\pm 2 \\frac{\\widehat{\\lambda}_n}{\\sqrt{n}}\n", "= \\frac{1}{\\overline{x}_n} \\pm 2 \\frac{1}{\\sqrt{n} \\ \\overline{x}_n} .\n", "$$\n", "Let us compute the ML estimate and the $95\\%$ confidence interval for the rate parameter for the waiting times at the Orbiter bus-stop. The sample mean $\\overline{x}_{132}=9.0758$ and the ML estimate is:\n", "$$\\widehat{\\lambda}_{132}=1/\\,\\overline{x}_{132}=1/9.0758=0.1102 ,$$\n", "and the $95\\%$ confidence interval is:\n", "$$\n", "C_n\n", "= \\widehat{\\lambda}_{132} \\pm 2 \\frac{\\widehat{\\lambda}_{132}}{\\sqrt{132}}\n", "= \\frac{1}{\\overline{x}_{132}} \\pm 2 \\frac{1}{\\sqrt{132} \\, \\overline{x}_{132}} = 0.1102 \\pm 2 \\cdot 0.0096 = [0.091, 0.129] .\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "logfx = log(lam*e^(-lam*x))\n", "d2logfx = -1/lam^2\n", "FisherInformation1 = lam^(-2)\n", "StdErr = lam/sqrt(n)\n", "EstStdErr = 1/(sqrt(n)*sampMean)\n" ] }, { "data": { "text/plain": [ "(1/sampMean - 2/(sqrt(n)*sampMean), 1/sampMean + 2/(sqrt(n)*sampMean))" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Sample Exam Problem 5 Solution\n", "# solution is straightforward by following these steps symbolically\n", "# or you can do it by hand with pen/paper or do both to be safe\n", "\n", "## STEP 1 - define the variables you need\n", "lam,x,n = var('lam','x','n')\n", "\n", "## STEP 2 - get symbolic expression for the likelihood of one sample\n", "logfx = log(lam*exp(-lam*x)).full_simplify()\n", "print \"logfx = \", logfx\n", "\n", "## STEP 3 - find second derivate of expression from STEP 2 w.r.t. parameter\n", "d2logfx = logfx.diff(lam,2).full_simplify()\n", "print \"d2logfx = \", d2logfx\n", "\n", "## STEP 4 - to get Fisher Information of one sample\n", "## integrate d2logfx * f(x) over x in [0,Infinity), f(x) id PDF lam*exp(-lam*x)\n", "assume(lam>0) # usually you need make such assume's for integrate to work - see suggestions in error messages\n", "FisherInformation1 = -integrate(d2logfx*lam*exp(-lam*x),x,0,Infinity)\n", "print \"FisherInformation1 = \",FisherInformation1\n", "\n", "## STEP 5 - get Standard Error from FisherInformation1\n", "StdErr = 1/sqrt(n*FisherInformation1)\n", "print \"StdErr = \",StdErr\n", "\n", "## STEP 6 - get Standard Error from Standard Error and MLE or lamHat\n", "# lamHat = 1/xBar = 1/sampleMean; know from before\n", "lamHat,sampMean = var('lamHat','sampMean')\n", "lamHat = 1/sampMean\n", "EstStdErr = StdErr.subs(lam=lamHat)\n", "print \"EstStdErr = \",EstStdErr\n", "\n", "## STEP 7 - Get lower and upper 95% CI\n", "(lamHat-2*EstStdErr, lamHat+2*EstStdErr)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The 95% CI for lambda in the Orbiter Waiting time experiment = \n", "(0.09100312972775282, 0.12936414907024382)\n" ] } ], "source": [ "# Sample Exam Problem 5 Solution\n", "# Only replace the XXX below, do not change the function naemes or parameters\n", "import numpy as np\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "def SampleExamProblem5(exponentialSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown rate parameter lambda* \n", " from n IID Exponential(lambda*) trials in the input numpy array called exponentialSamples'''\n", " sampleMean = exponentialSamples.mean()\n", " n=len(exponentialSamples)\n", " correction=RR(2/(sqrt(n)*sampleMean)) # you can also replace RR by float here or you get expressions\n", " lower95CI=1.0/sampleMean - correction\n", " upper95CI=1.0/sampleMean + correction\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = SampleExamProblem5(sampleWaitingTimes)\n", "print \"The 95% CI for lambda in the Orbiter Waiting time experiment = \"\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": { "lx_assignment_number": "3", "lx_problem_cell_type": "PROBLEM" }, "source": [ "---\n", "## Assignment 3, PROBLEM 5\n", "Maximum Points = 3" ] }, { "cell_type": "markdown", "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "5", "lx_problem_points": "3" }, "source": [ "\n", "Obtain the 95% CI based on the asymptotic normality of the MLE for the mean paramater $\\lambda$ based on $n$ IID $Poisson(\\lambda^*)$ trials.\n", "\n", "Recall that a random variable $X \\sim Poisson(\\lambda)$ if its probability mass function is:\n", "\n", "$$\n", "f(x; \\lambda) = \\exp{(-\\lambda)} \\frac{\\lambda^x}{x!}, \\quad \\lambda > 0, \\quad x \\in \\{0,1,2,\\ldots\\}\n", "$$\n", "\n", "The MLe $\\widehat{\\lambda}_n = \\overline{x}_n$, the sample mean.\n", "\n", "Work out your answer and express it in the next cell by replacing `XXX`s." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "5", "lx_problem_points": "3" }, "outputs": [], "source": [ "# Only replace the XXX below, do not change the function naemes or parameters\n", "import numpy as np\n", "samplePoissonCounts = np.array([0,5,11,5,6,8,9,0,1,14,2,4,4,11,2,12,10,5,6,1,7,9,8,0,5,7,11,6,0,1])\n", "\n", "def Assignment3Problem5(poissonSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown parameter lambda* \n", " from n IID Poisson(lambda*) trials in the input numpy array called samplePoissonCounts'''\n", " XXX\n", " XXX\n", " XXX\n", " lower95CI=XXX\n", " upper95CI=XXX\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = Assignment3Problem5(samplePoissonCounts)\n", "print \"The 95% CI for lambda based on IID Poisson(lambda) data in samplePoissonCounts = \"\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Hypothesis Testing\n", "\n", "The subset of *all posable hypotheses* that have the property of *[falsifiability](https://en.wikipedia.org/wiki/Falsifiability)* constitute the space of *scientific hypotheses*. \n", "Roughly, a falsifiable statistical hypothesis is one for which a statistical experiment can be designed to produce data or empirical observations that an experimenter can use to falsify or reject it. \n", "In the *statistical decision problem of hypothesis testing*, we are interested in empirically falsifying a scientific hypothesis, i.e. we attempt to reject a hypothesis on the basis of empirical observations or data. \n", "Thus, hypothesis testing has its roots in the *philosophy of science* and is based on *Karl Popper's falsifiability criterion for demarcating scientific hypotheses from the set of all posable hypotheses*.\n", "\n", "## Introduction\n", "Usually, the hypothesis we **attempt to reject or falsify** is called the **null hypothesis** or $H_0$ and its complement is called the **alternative hypothesis** or $H_1$. \n", "For example, consider the following two hypotheses:\n", "\n", "- $H_0$: The average waiting time at an Orbiter bus stop *is less than or equal to* $10$ minutes.\n", "- $H_1$: The average waiting time at an Orbiter bus stop *is more than* $10$ minutes.\n", "\n", "If the sample mean $\\overline{x}_n$ is much larger than $10$ minutes then we may be inclined to reject the null hypothesis that the average waiting time is less than or equal to $10$ minutes. \n", "\n", "Suppose we are interested in the following slightly different hypothesis test for the Orbiter bus stop problem:\n", "\n", "- $H_0$: The average waiting time at an Orbiter bus stop *is equal to* $10$ minutes.\n", "- $H_1$: The average waiting time at an Orbiter bus stop *is not* $10$ minutes.\n", "\n", "Once again we can use the sample mean as the test statistic, but this time we may be inclined to reject the null hypothesis if the sample mean $\\overline{x}_n$ is much larger than *or* much smaller than $10$ minutes. \n", "The procedure for rejecting such a null hypothesis is called the **Wald test** we are about to see.\n", "\n", "More generally, suppose we have the following parametric experiment based on $n$ IID trials:\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x_1;\\theta^*), \\quad \\text{ with an unknown (and fixed) } \\theta^* \\in \\mathbf{\\Theta} \\ .\n", "$$\n", "\n", "Let us partition the parameter space $\\mathbf{\\Theta}$ into $\\mathbf{\\Theta}_0$, the null parameter space, and $\\mathbf{\\Theta}_1$, the alternative parameter space, i.e.,\n", "$$\\mathbf{\\Theta}_0 \\cup \\mathbf{\\Theta}_1 = \\mathbf{\\Theta}, \\qquad \\text{and} \\qquad \\mathbf{\\Theta}_0 \\cap \\mathbf{\\Theta}_1 = \\emptyset \\ .$$\n", "\n", "Then, we can formalise testing the null hypothesis versus the alternative as follows:\n", "$$\n", "H_0 : \\theta^* \\in \\mathbf{\\Theta}_0 \\qquad \\text{versus} \\qquad H_1 : \\theta^* \\subset \\mathbf{\\Theta}_1 \\ .\n", "$$\n", "\n", "The basic idea involves finding an appropriate **rejection region** $\\mathbb{X}_R$ within the **data space** $\\mathbb{X}$ and rejecting $H_0$ if the observed data $x:=(x_1,x_2,\\ldots,x_n)$ falls inside the rejection region $\\mathbb{X}_R$,\n", "$$\n", "\\text{If $x:=(x_1,x_2,\\ldots,x_n) \\in \\mathbb{X}_R \\subset \\mathbb{X}$, then reject $H_0$, else do not reject $H_0$.}\n", "$$\n", "Typically, the rejection region $\\mathbb{X}_R$ is of the form:\n", "$$\n", "\\mathbb{X}_R := \\{ x:=(x_1,x_2,\\ldots,x_n) : T(x) > c \\}\n", "$$\n", "where, $T$ is the **test statistic** and $c$ is the **critical value**. Thus, the problem of finding $\\mathbb{X}_R$ boils down to that of finding $T$ and $c$ that are appropriate. Once the rejection region is defined, the possible outcomes of a hypothesis test are summarised in the following table.\n", "\n", "\n", "The outcomes of a hypothesis test, in general, are:\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
'true state of nature'Do not reject $H_0$
Reject $H_0$
\n", "

$H_0$ is true

\n", "

 

\n", "		\n", "

OK 

\n", "		\n", "

Type I error

\n", "
\n", "

$H_0$ is false

\n", "		Type II errorOK
\n", "\n", "So, intuitively speaking, we want a small probability that we reject $H_0$ when $H_0$ is true (minimise Type I error). Similarly, we want to minimise the probability that we fail to reject $H_0$ when $H_0$ is false (type II error). Let us formally see how to achieve these goals.\n", "\n", "## Power, Size and Level of a Test\n", "\n", "### Power Function \n", "\n", "The **power function** of a test with rejection region $\\mathbb{X}_R$ is\n", "$$\n", "\\boxed{\n", "\\beta(\\theta) := P_{\\theta}(x \\in \\mathbb{X}_R)\n", "}\n", "$$\n", "So $\\beta(\\theta)$ is the power of the test if the data were generated under the parameter value $\\theta$, i.e. the probability that the observed data $x$, sampled from the distribution specified by $\\theta$, falls in the rejection region $\\mathbb{X}_R$ and thereby leads to a rejection of the null hypothesis.\n", "\n", "### Size of a test\n", "The $\\mathsf{size}$ of a test with rejection region $\\mathbb{X}_R$ is the supreme power under the null hypothesis, i.e.~the supreme probability of rejecting the null hypothesis when the null hypothesis is true:\n", "$$\n", "\\boxed{\n", "\\mathsf{size} := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} \\beta(\\theta) := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} P{\\theta}(x \\in \\mathbb{X}_R) \\ .\n", "}\n", "$$\n", "The $\\mathsf{size}$ of a test is often denoted by $\\alpha$. A test is said to have $\\mathsf{level}$ $\\alpha$ if its $\\mathsf{size}$ is less than or equal to $\\alpha$.\n", "\n", "\n", "## Wald test\n", "\n", "The Wald test is based on a direct relationship between the $1-\\alpha$ confidence interval and a $\\mathsf{size}$ $\\alpha$ test. It can be used for testing simple hypotheses involving a scalar parameter.\n", "\n", "### Definition\n", "\n", "Let $\\widehat{\\Theta}_n$ be an asymptotically normal estimator of the fixed and possibly unknown parameter $\\theta^* \\in \\mathbf{\\Theta} \\subset \\mathbb{X}$ in the parametric IID experiment:\n", "\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x_1;\\theta^*) \\enspace .\n", "$$\n", "\n", "Consider testing:\n", "\n", "$$\n", "\\boxed{H_0: \\theta^* = \\theta_0 \\qquad \\text{versus} \\qquad H_1: \\theta^* \\neq \\theta_0 \\enspace .}\n", "$$\n", "\n", "Suppose that the null hypothesis is true and the estimator $\\widehat{\\Theta}_n$ of $\\theta^*=\\theta_0$ is asymptotically normal:\n", "\n", "$$\n", "\\boxed{\n", "\\theta^*=\\theta_0, \\qquad \\frac{\\widehat{\\Theta}_n - \\theta_0}{\\widehat{\\mathsf{se}}_n} \\overset{d}{\\to} Normal(0,1) \\enspace .}\n", "$$\n", "\n", "Then, **the Wald test based on the test statistic $W$** is:\n", "$$\n", "\\boxed{\n", "\\text{Reject $H_0$ when $|W|>z_{\\alpha/2}$, where $W:=W((X_1,\\ldots,X_n))=\\frac{\\widehat{\\Theta}_n ((X_1,\\ldots,X_n)) - \\theta_0}{\\widehat{\\mathsf{se}}_n}$.}\n", "}\n", "$$\n", "The rejection region for the Wald test is:\n", "$$\n", "\\boxed{\n", "\\mathbb{X}_R = \\{ x:=(x_1,\\ldots,x_n) : |W (x_1,\\ldots,x_n) | > z_{\\alpha/2} \\} \\enspace .\n", "}\n", "$$\n", "\n", "### Asymptotic $\\mathsf{size}$ of a Wald test\n", "\n", "As the sample size $n$ approaches infinity, the $\\mathsf{size}$ of the Wald test approaches $\\alpha$ :\n", "\n", "$$\n", "\\boxed{\n", "\\mathsf{size} = P_{\\theta_0} \\left( |W| > z_{\\alpha/2} \\right) \\to \\alpha \\enspace .}\n", "$$\n", "\n", "**Proof:** Let $Z \\sim Normal(0,1)$. The $\\mathsf{size}$ of the Wald test, i.e.~the supreme power under $H_0$ is:\n", "\n", "$$\n", "\\begin{align}\n", "\\mathsf{size}\n", "& := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} \\beta(\\theta) := \\sup_{\\theta \\in \\{\\theta_0\\}} P_{\\theta}(x \\in \\mathbb{X}_R) = P_{\\theta_0}(x \\in \\mathbb{X}_R) \\\\\n", "& = P_{\\theta_0} \\left( |W| > z_{\\alpha/2} \\right) = P_{\\theta_0} \\left( \\frac{|\\widehat{\\theta}_n - \\theta_0|}{\\widehat{\\mathsf{se}}_n} > z_{\\alpha/2} \\right) \\\\\n", "& \\to P \\left( |Z| > z_{\\alpha/2} \\right)\\\\\n", "& = \\alpha \\enspace .\n", "\\end{align}\n", "$$\n", "\n", "Next, let us look at the power of the Wald test when the null hypothesis is false.\n", "\n", "### Asymptotic power of a Wald test\n", "\n", "Suppose $\\theta^* \\neq \\theta_0$. The power $\\beta(\\theta^*)$, which is the probability of correctly rejecting the null hypothesis, is approximately equal to:\n", "\n", "$$\n", "\\boxed{\n", "\\Phi \\left( \\frac{\\theta_0-\\theta^*}{\\widehat{\\mathsf{se}}_n} - z_{\\alpha/2} \\right) +\n", "\\left( 1- \\Phi \\left( \\frac{\\theta_0-\\theta^*}{\\widehat{\\mathsf{se}}_n} + z_{\\alpha/2} \\right) \\right) \\enspace ,\n", "}\n", "$$\n", "where, $\\Phi$ is the DF of $Normal(0,1)$ RV. Since ${\\widehat{\\mathsf{se}}_n} \\to 0$ as $n \\to 0$ the power increase with sample $\\mathsf{size}$ $n$. Also, the power increases when $|\\theta_0-\\theta^*|$ is large.\n", "\n", "Now, let us make the connection between the $\\mathsf{size}$ $\\alpha$ Wald test and the $1-\\alpha$ confidence interval explicit.\n", "\n", "### The $\\mathsf{size}$ Wald test\n", "\n", "The $\\mathsf{size}$ $\\alpha$ Wald test rejects:\n", "\n", "$$\n", "\\boxed{\n", "\\text{ $H_0: \\theta^*=\\theta_0$ versus $H_1: \\theta^* \\neq \\theta_0$ if and only if $\\theta_0 \\notin C_n := (\\widehat{\\theta}_n-{\\widehat{\\mathsf{se}}_n} z_{\\alpha/2}, \\widehat{\\theta}_n+{\\widehat{\\mathsf{se}}_n} z_{\\alpha/2})$.\n", "}}\n", "$$\n", "\n", "$$\\boxed{\\text{Therefore, testing the hypothesis is equivalent to verifying whether the null value $\\theta_0$ is in the confidence interval.}}$$\n", "\n", "\n", "### Example: Wald test for the mean waiting times at our Orbiter bus-stop\n", "\n", "Let us use the Wald test to attempt to reject the null hypothesis that the mean waiting time at our Orbiter bus-stop is $10$ minutes under an IID $Exponential(\\lambda^*)$ model. Let $\\alpha=0.05$ for this test. We can formulate this test as follows:\n", "$$\n", "H_0: \\lambda^* = \\lambda_0= \\frac{1}{10} \\quad \\text{versus} \\quad H_1: \\lambda^* \\neq \\frac{1}{10}, \\quad \\text{where, } \\quad X_1\\ldots,X_{132} \\overset{IID}{\\sim} Exponential(\\lambda^*) \\enspace .\n", "$$\n", "We already obtained the $95\\%$ confidence interval based on its MLE's asymptotic normality property to be $[0.0914, 0.1290]$. \n", "\n", "$$\\boxed{\\text{Since our null value $\\lambda_0=0.1$ belongs to this confidence interval, we fail to reject the null hypothesis from a $\\mathsf{size}$ $\\alpha=0.05$ Wald test.}}$$\n", "\n", "We will revisit this example in a more computationally explicit fasion soon below." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A Live Example: Simulating Bernoulli Trials to understand Wald Tests\n", "\n", "Let's revisit the MLE for the $Bernoulli(\\theta^*)$ model with $n$ IID trails, we have already seen, and test the null hypothesis that the unknown $\\theta^* = \\theta_0 = 0.5$.\n", "\n", "Thus, we are interested in the null hypothesis $H_0$ versus the alternative hypothesis $H_1$:\n", "\n", "$$\\displaystyle{H_0: \\theta^*=\\theta_0 \\quad \\text{ versus } \\quad H_1: \\theta^* \\neq \\theta_0, \\qquad \\text{ with }\\theta_0=0.5}$$\n", "\n", "We can test this hypothesis with Type I error at $\\alpha$ using the **size-$\\alpha$ Wald Test** that builds on the asymptotic normality of the MLE, i.e., \n", "$$\\displaystyle{ \\frac{\\widehat{\\theta}_n - \\theta_0}{\\widehat{se}_n} \\overset{d}{\\to} Normal(0,1)}$$\n", "\n", "The size-$\\alpha$ Wald test is:\n", "\n", "$$\n", "\\boxed{\n", "\\text{Reject } \\ H_0 \\quad \\text{ when } |W| > z_{\\alpha/2}, \\quad \\text{ where, } \\quad W = \\frac{\\widehat{\\theta}_n - \\theta_0}{\\widehat{se}_n}\n", "}\n", "$$" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.9518001458970666\n" ] }, { "data": { "text/plain": [ "False" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "# do a live simulation ... to implement this test...\n", "# simulate from Bernoulli(theta0) n samples\n", "# make mle\n", "# construct Wald test\n", "# make a decision - i.e., decide if you will reject or fail to reject the H0: theta0=0.5\n", "trueTheta=0.45\n", "n=20\n", "myBernSamples=np.array([floor(random()+trueTheta) for i in range(0,n)])\n", "#myBernSamples\n", "mle=myBernSamples.mean() # 1/mean\n", "mle\n", "NullTheta=0.5\n", "se=sqrt(mle*(1.0-mle)/n)\n", "W=(mle-NullTheta)/se\n", "print abs(W)\n", "alpha = 0.05\n", "abs(W) > 2 # alpha=0.05, so z_{alpha/2} =1.96 approx=2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 6 \n", "\n", "Consider the following model for the parity (odd=1, even=0) of the first Lotto ball to pop out of the NZ Lotto machine. We had $n=1114$ IID trials:\n", "\n", "$$\\displaystyle{X_1,X_2,\\ldots,X_{1114} \\overset{IID}{\\sim} Bernoulli(\\theta^*)}$$\n", "\n", "and know from this dataset that the number of odd balls is $546=\\sum_{i=1}^{1114} x_i$.\n", "\n", "Your task is to perform a Wald Test of size $\\alpha=0.05$ to try to reject the null hypothesis that the chance of seeing an odd ball out of the NZ Lotto machine is exactly $1/2$, i.e.,\n", "\n", "$$\\displaystyle{H_0: \\theta^*=\\theta_0 \\quad \\text{ versus } \\quad H_1: \\theta^* \\neq \\theta_0, \\qquad \\text{ with }\\theta_0=0.5}$$\n", "\n", "Show you work by replacing `XXX`s with the right expressions in the next cell." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Sample Exam Problem 6 Problem\n", "\n", "## STEP 1: get the MLE thetaHat\n", "thetaHat=XXX \n", "print \"mle thetaHat = \",thetaHat\n", "\n", "## STEP 2: get the NullTheta or theta0\n", "NullTheta=XXX\n", "print \"Null value of theta under H0 = \", NullTheta\n", "\n", "## STEP 3: get estimated standard error\n", "seTheta=XXX # for Bernoulli trials from earleir in 10.ipynb\n", "print \"estimated standard error\",seTheta\n", "\n", "# STEP 4: get Wald Statistic\n", "W=XXX\n", "print \"Wald staatistic = \",W\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "# do NOT change anything below\n", "rejectNullSampleExamProblem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullSampleExamProblem6):\n", " print \"we reject the null hypothesis that theta_0=0.5\"\n", "else:\n", " print \"we fail to reject the null hypothesis that theta_0=0.5\"" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mle thetaHat = 273/557\n", "Null value of theta under H0 = 0.500000000000000\n", "estimated standard error 0.0149776163832414\n", "Wald staatistic = -0.659272243178650\n", "we fail to reject the null hypothesis that theta_0=0.5\n" ] } ], "source": [ "# Sample Exam Problem 6 Solution\n", "\n", "## STEP 1: get the MLE thetaHat\n", "n=1114 # sample size\n", "thetaHat=546/n # MLE is sample mean for IID Bernoulli trials\n", "print \"mle thetaHat = \",thetaHat\n", "\n", "## STEP 2: get the NullTheta or theta0\n", "NullTheta=0.5\n", "print \"Null value of theta under H0 = \", NullTheta\n", "\n", "## STEP 3: get estimated standard error\n", "seTheta=sqrt(thetaHat*(1.0-thetaHat)/n) # for Bernoulli trials from earleir in 10.ipynb\n", "print \"estimated standard error\",seTheta\n", "\n", "# STEP 4: get Wald Statistic\n", "W=(thetaHat-NullTheta)/seTheta\n", "print \"Wald staatistic = \",W\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "rejectNullSampleExamProblem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullSampleExamProblem6):\n", " print \"we reject the null hypothesis that theta_0=0.5\"\n", "else:\n", " print \"we fail to reject the null hypothesis that theta_0=0.5\"" ] }, { "cell_type": "markdown", "metadata": { "lx_assignment_number": "3", "lx_problem_cell_type": "PROBLEM" }, "source": [ "---\n", "## Assignment 3, PROBLEM 6\n", "Maximum Points = 3" ] }, { "cell_type": "markdown", "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "6", "lx_problem_points": "3" }, "source": [ "\n", "For the Orbiter waiting time problem, assuming IID trials as follows: \n", "\n", "$$\\displaystyle{X_1,X_2,\\ldots,X_{n} \\overset{IID}{\\sim} Exponential(\\lambda^*)}$$\n", "\n", "Your task is to perform a Wald Test of size $\\alpha=0.05$ to try to reject the null hypothesis that the waiting time at the Orbiter bus-stop, i.e., the inter-arrival time between buses, is exactly $10$ minutes:\n", "\n", "$$\\displaystyle{H_0: \\lambda^*=\\lambda_0 \\quad \\text{ versus } \\quad H_1: \\lambda^* \\neq \\lambda_0, \\qquad \\text{ with }\\lambda_0=0.1}$$\n", "\n", "Show you work by replacing `XXX`s with the right expressions in the next cell." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "6", "lx_problem_points": "3" }, "outputs": [], "source": [ "import numpy as np\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "#test H0: lambda=0.1\n", "## STEP 1: get the MLE thetaHat\n", "lambdaHat=XXX # you need to use sampleWaitingTimes here!\n", "print \"mle lambdaHat = \",lambdaHat\n", "\n", "## STEP 2: get the NullLambda or lambda0\n", "NullLambda=XXX\n", "print \"Null value of lambda under H0 = \", NullLambda\n", "\n", "## STEP 3: get estimated standard error\n", "seLambda=XXX # see Sample Exam Problem 5 in 10.ipynb\n", "print \"estimated standard error\",seLambda\n", "\n", "# STEP 4: get Wald Statistic\n", "W=XXX\n", "print \"Wald statistic = \",W\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "# do NOT change anything below\n", "rejectNullAssignment3Problem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullAssignment3Problem6):\n", " print \"we reject the null hypothesis that lambda0=0.1\"\n", "else:\n", " print \"we fail to reject the null hypothesis that lambda0=0.1\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## P-value\n", "\n", "It is desirable to have a more informative decision than simply reporting \"reject $H_0$\" or \"fail to reject $H_0$.\"\n", "\n", "For instance, we could ask whether the test rejects $H_0$ for each $\\mathsf{size}=\\alpha$. \n", "Typically, if the test rejects at $\\mathsf{size}$ $\\alpha$ it will also reject at a larger $\\mathsf{size}$ $\\alpha' > \\alpha$. \n", "Therefore, there is a smallest $\\mathsf{size}$ $\\alpha$ at which the test rejects $H_0$ and we call this $\\alpha$ the $\\text{p-value}$ of the test.\n", "\n", "$$\\boxed{\\text{The smallest $\\alpha$ at which a $\\mathsf{size}$ $\\alpha$ test rejects the null hypothesis $H_0$ is the $\\text{p-value}$.}}$$\n" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 11 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p=text('Reject $H_0$?',(12,12)); p+=text('No',(30,10)); p+=text('Yes',(30,15)); p+=text('p-value',(70,10))\n", "p+=text('size',(65,4)); p+=text('$0$',(40,4)); p+=text('$1$',(90,4)); p+=points((59,5),rgbcolor='red',size=50)\n", "p+=line([(40,17),(40,5),(95,5)]); p+=line([(40,10),(59,10),(59,15),(90,15)]);\n", "p+=line([(68,9.5),(59.5,5.5)],rgbcolor='red'); p.show(axes=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Definition of p-value\n", "Suppose that for every $\\alpha \\in (0,1)$ we have a $\\mathsf{size}$ $\\alpha$ test with rejection region $\\mathbb{X}_{R,\\alpha}$ and test statistic $T$. Then,\n", "$$\n", "\\text{p-value} := \\inf \\{ \\alpha: T(X) \\in \\mathbb{X}_{R,\\alpha} \\} \\enspace .\n", "$$\n", "That is, the p-value is the smallest $\\alpha$ at which a $\\mathsf{size}$ $\\alpha$ test rejects the null hypothesis.\n", "\n", "### Understanding p-value\n", "If the evidence against $H_0$ is strong then the p-value will be small. However, a large p-value is not strong evidence in favour of $H_0$. This is because a large p-value can occur for two reasons:\n", "\n", "- $H_0$ is true.\n", "- $H_0$ is false but the test has low power (i.e., high Type II error).\n", "\n", "Finally, it is important to realise that *p-value is not the probability that the null hypothesis is true*, i.e. $\\text{p-value} \\, \\neq P(H_0|x)$, where $x$ is the data. The following itemisation of implications for the evidence scale is useful.\n", "\n", "The scale of the evidence against the null hypothesis $H_0$ in terms of the range of the p-values has the following interpretation that is commonly used:\n", "\n", "- P-value $\\in (0.00, 0.01]$ $\\implies$ Very strong evidence against $H_0$\n", "- P-value $\\in (0.01, 0.05]$ $\\implies$ Strong evidence against $H_0$\n", "- P-value $\\in (0.05, 0.10]$ $\\implies$ Weak evidence against $H_0$\n", "- P-value $\\in (0.10, 1.00]$ $\\implies$ Little or no evidence against $H_0$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we will see a convenient expression for the p-value for certain tests.\n", "\n", "### The p-value of a hypothesis test\n", "\n", "Suppose that the $\\mathsf{size}$ $\\alpha$ test based on the test statistic $T$ and critical value $c_{\\alpha}$ is of the form:\n", "\n", "$$\n", "\\text{Reject $H_0$ if and only if $T:=T((X_1,\\ldots,X_n))> c_{\\alpha}$,}\n", "$$\n", "\n", "then\n", "\n", "$$\n", "\\boxed{\n", "\\text{p-value} \\, = \\sup_{\\theta \\in \\mathbf{\\Theta}_0} P_{\\theta}(T((X_1,\\ldots,X_n)) \\geq t:=T((x_1,\\ldots,x_n))) \\enspace ,}\n", "$$\n", "\n", "where, $(x_1,\\ldots,x_n)$ is the observed data and $t$ is the observed value of the test statistic $T$. \n", "\n", "In words, **the p-value is the supreme probability under $H_0$ of observing a value of the test statistic the same as or more extreme than what was actually observed.**\n", "\n", "\n", "Let us revisit the Orbiter waiting times example from the p-value perspective.\n", "\n", "### Example: p-value for the parametric Orbiter bus waiting times experiment\n", "\n", "Let the waiting times at our bus-stop be $X_1,X_2,\\ldots,X_{132} \\overset{IID}{\\sim} Exponential(\\lambda^*)$. Consider the following testing problem:\n", "\n", "$$\n", "H_0: \\lambda^*=\\lambda_0=\\frac{1}{10} \\quad \\text{versus} \\quad H_1: \\lambda^* \\neq \\lambda_0 \\enspace .\n", "$$\n", "\n", "We already saw that the Wald test statistic is:\n", "\n", "$$\n", "W:=W(X_1,\\ldots,X_n)= \\frac{\\widehat{\\Lambda}_n-\\lambda_0}{\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n)} = \\frac{\\frac{1}{\\overline{X}_n}-\\lambda_0}{\\frac{1}{\\sqrt{n}\\overline{X}_n}} \\enspace .\n", "$$\n", "\n", "The observed test statistic is:\n", "\n", "$$\n", "w=W(x_1,\\ldots,x_{132})=\n", "\\frac{\\frac{1}{\\overline{X}_{132}}-\\lambda_0}{\\frac{1}{\\sqrt{132}\\overline{X}_{132}}}\n", "= \\frac{\\frac{1}{9.0758}-\\frac{1}{10}}{\\frac{1}{\\sqrt{132} \\times 9.0758}} = 1.0618 \\enspace .\n", "$$\n", "Since, $W \\overset{d}{\\to} Z \\sim Normal(0,1)$, the p-value for this Wald test is:\n", "\n", "$$\n", "\\begin{align}\n", "\\text{p-value} \\, \n", "&= \\sup_{\\lambda \\in \\mathbf{\\Lambda}_0} P_{\\lambda} (|W|>|w|)= \\sup_{\\lambda \\in \\{\\lambda_0\\}} P_{\\lambda} (|W|>|w|) = P_{\\lambda_0} (|W|>|w|) \\\\\n", "& \\to P (|Z|>|w|)=2 \\Phi(-|w|)=2 \\Phi(-|1.0618|)=2 \\times 0.1442=0.2884 \\enspace .\n", "\\end{align}\n", "$$\n", "\n", "Therefore, there is little or no evidence against $H_0$ that the mean waiting time under an IID $Exponential$ model of inter-arrival times is exactly ten minutes.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Preparation for Nonparametric Estimation and Testing\n", "### YouTry Later\n", "\n", "Python's `random` for sampling and sequence manipulation\n", "\n", "The Python `random` module, available in SageMath, provides a useful way of taking samples if you have already generated a 'population' to sample from, or otherwise playing around with the elements in a sequence. See http://docs.python.org/library/random.html for more details. Here we will try a few of them.\n", "\n", "The aptly-named sample function allows us to take a sample of a specified size from a sequence. We will use a list as our sequence:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[70, 57, 80, 26, 99, 59, 98, 93, 13, 86]" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "popltn = range(1, 101, 1) # make a population\n", "sample(popltn, 10) # sample 10 elements from it at random" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Each call to sample will select unique elements in the list (note that 'unique' here means that it will not select the element at any particular position in the list more than once, but if there are duplicate elements in the list, such as with a list [1,2,4,2,5,3,1,3], then you may well get any of the repeated elements in your sample more than once). sample samples with replacement, which means that repeated calls to sample may give you samples with the same elements in." ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n" ] } ], "source": [ "popltnWithDuplicates = range(1, 11, 1)*4 # make a population with repeated elements\n", "print(popltnWithDuplicates)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[4, 10, 10, 9, 2, 5, 6, 9, 1, 1]\n", "[6, 8, 10, 9, 5, 8, 2, 6, 4, 1]\n", "[9, 7, 3, 2, 1, 2, 3, 8, 10, 2]\n", "[3, 6, 4, 7, 2, 8, 5, 8, 7, 10]\n", "[9, 4, 7, 9, 3, 8, 10, 3, 2, 7]\n" ] } ], "source": [ "for i in range (5):\n", " print sample(popltnWithDuplicates, 10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Try experimenting with choice, which allows you to select one element at random from a sequence, and shuffle, which shuffles the sequence in place (i.e, the ordering of the sequence itself is changed rather than you being given a re-ordered copy of the list). It is probably easiest to use lists for your sequences. See how `shuffle` is creating permutations of the list. You could use `sample` and `shuffle` to emulate *permuations of k objects out of n* ...\n", "\n", "You may need to check the documentation to see how use these functions." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?sample" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?shuffle" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?choice" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.7", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.15" }, "lx_course_instance": "Summer 2019", "lx_course_name": "Introduction to Data Science: A Comp-Math-Stat Approach", "lx_course_number": "YOIYUI001" }, "nbformat": 4, "nbformat_minor": 2 }