{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "import operator as op\n", "import functools as ft\n", "\n", "def summ(a_map):\n", " \"sum(map) does not do what one would expect it to do, but summ(map) does\"\n", " return ft.reduce(op.add, a_map)\n", "\n", "def integr(acc, new):\n", " \"reduction function to compute an integral\"\n", " if not acc: \n", " acc = [0]\n", " last = 0\n", " else:\n", " last = acc[-1]\n", " \n", " acc.append(last+new)\n", " return acc\n", " \n", "def pf(Proba):\n", " \"converting uniform draw into binary draw (factory function)\"\n", " def f(x):\n", " return 0 if x>P else 1\n", " f.__doc__ = \"converts uniform draw into binary draw with proba %.4f\" % Proba\n", " return f\n", "\n", "from scipy.misc import comb\n", "def pdf(n,p,k):\n", " \"the binomial density function B(n,p,k)\"\n", " return comb(n, k) * p**k * (1-p)**(n-k)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#Frequentist vs Bayesian statistics - Part I" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the first scenario we consider we are looking at something like coin-flips: we have a series of events, each of them being either _heads_ or _tails_. There are a number of questions we could ask\n", "\n", "- we might have a view on the probabilities; for example, we might want to ask whether or not this coin is _fair_, ie whether heads and tails appear with equal probability\n", " \n", "- we might have no view on probabilities, so we might ask what the respective probabilities of heads and tails are\n", "\n", "- we might want to go a step back and ask whether the series is random in the first place, and - if yes - if the variables are independent\n", "\n", "The first of those points is what one would generally call **hypothesis testing**: we have one hypothesis (and the emphasis is on _one_) and we want to know whether or not it is true. In the second case we are in the realm of **parameter estimation**. Note that this could also be framed as hypothesis testing: the hypothesis $H_p$ is that the parameter has the value $p$. The difference here is that we are testing an infinite number of hypothesis against a given that of data, in which case there is a chance that one will _stick_ even though it is not true (I have [written about hypothesis shopping before][shopping]). The last case is what one could term a **meta-hypothesis**, in the sense that if that one is false the other questions no longer make sense.\n", "\n", "So let's now set up our random sample: We will draw _N_ independent digital variables (1=heads, 0=tails) with a probability of heads of _P_. The number of heads will be called _K_, and the _realised_ probability, ie the ratio of _K_ and _N_, will be called _p_." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "N = 20 (200)\n", "P = 0.60\n", "K = 12 (115)\n", "p = 0.60 (0.57)\n" ] } ], "source": [ "N = 20\n", "P = 0.6\n", "l = list(map (pf(P), np.random.uniform(size=N)))\n", "ls = ft.reduce(lambda acc, n: acc + str(n), l, \"\")\n", "K = sum(l)\n", "l2 = list(map (pf(P), np.random.uniform(size=N*10)))\n", "ls2 = ft.reduce(lambda acc, n: acc + str(n), l, \"\")\n", "K2 = sum(l2)\n", "print (\"N = %i (%i)\" % (N,10*N))\n", "print (\"P = %.2f\" % P)\n", "print (\"K = %i (%i)\" % (K,K2))\n", "#print (\"K = %i ('%s')\" % (K,ls))\n", "print (\"p = %.2f (%.2f)\" % (K/N, K2/(10*N)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's look at our first task from above, _hypothesis testing_: we want to know whether the result we have obtained is compatible with our hypothesis that the probability equals _Ph_. For this we look at the problem from the other end: if our hypothesis was true and we ran this experiment, what result would we get? We can draw the graph of the probability density function (it is a binomial distribution, therefore it is a bar chart, not a continous function) and we can look at the corresponding value for the _K_ we found. It turns that the probability value itself is not that meaningful: the more samples we take - the bigger our _N_ - the smaller the chance of obtaining any given _K_. \n", "\n", "One common decision rule is to define an inner region and an outer region, for example symmetric, with 90% probability (approximately; in a binomial distribution we only have a discrete set of choices) in the inner region. We can then say that if our observation falls in the outer region our hypothesis should be rejected. Note that the opposite does not hold actually: we can't accept our hypothesis, we can only \"fail to reject it\"." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Ph = 0.5" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Probability for K=12 is 0.12\n", "Probability to be within K=7...13 for N=20: 0.88\n" ] } ], "source": [ "x = range(N)\n", "lo = 7\n", "hi = N-lo\n", "mdf = [pdf(N,Ph,k) for k in x]\n", "plt.bar(x,mdf)\n", "plt.title('scenario probabilities for N=%i and p=%0.2f' % (N,Ph))\n", "plt.xlabel('number of heads')\n", "plt.ylabel('probability')\n", "plt.show()\n", "print(\"Probability for K=%i is %.2f\" % (K,pdf(N,Ph,K)) )\n", "print(\"Probability to be within K=%i...%i for N=%i: %0.2f\" % (lo,hi,N,sum([(pp if (i>=lo and i<=hi) else 0) for i,pp in enumerate(mdf)])))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Probability for K=115 is 0.0060\n", "Probability to be within 90...110 for N=200: 0.86\n" ] } ], "source": [ "x = range(N*10)\n", "lo2 = 90\n", "hi2 = N*10-lo2\n", "mdf = [pdf(N*10,Ph,k) for k in x]\n", "plt.bar(x,mdf)\n", "plt.title('scenario probabilities for N=%i and p=%0.2f' % (N*10,Ph))\n", "plt.xlabel('number of heads')\n", "plt.ylabel('probability')\n", "plt.show()\n", "print(\"Probability for K=%i is %.4f\" % (K2,pdf(N*10,Ph,K2)) )\n", "print(\"Probability to be within %i...%i for N=%i: %0.2f\" % (lo2,hi2,N*10,sum([(pp if (i>=lo2 and i<=hi2) else 0) for i,pp in enumerate(mdf)])))" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Probability for K=12 is 0.00\n", "Probability to be within 0...3: 0.87\n" ] } ], "source": [ "x = range(N)\n", "lo3 = 0\n", "hi3 = 3\n", "mdf = [pdf(N,0.1,k) for k in x]\n", "plt.bar(x,mdf)\n", "plt.title('scenario probabilities for N=%i and p=%0.2f' % (N,0.1))\n", "plt.xlabel('number of heads')\n", "plt.ylabel('probability')\n", "plt.show()\n", "print(\"Probability for K=%i is %.2f\" % (K,pdf(N,0.1,K)) )\n", "print(\"Probability to be within %i...%i: %0.2f\" % (lo3,hi3,sum([(pp if (i>=lo3 and i<=hi3) else 0) for i,pp in enumerate(mdf)])))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P=0.60, Ph=0.50\n", "rejected 241 out of 1000 (N=20, acceptance range 7...13)\n" ] } ], "source": [ "NN = 1000\n", "kk = [summ(map (pf(P), np.random.uniform(size=N))) for _ in range(NN)]\n", "accepted = summ(map(lambda x: 1 if (x>=lo and x<=hi) else 0, kk))\n", "print (\"P=%.2f, Ph=%.2f\" % (P,Ph))\n", "print (\"rejected %i out of %i (N=%i, acceptance range %i...%i)\" % (NN-accepted, NN, N, lo, hi))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P=0.60, Ph=0.50\n", "rejected 920 out of 1000 (N=200, acceptance range 90...110)\n" ] } ], "source": [ "NN = 1000\n", "kk = [summ(map (pf(P), np.random.uniform(size=10*N))) for _ in range(NN)]\n", "accepted = summ(map(lambda x: 1 if (x>=lo2 and x<=hi2) else 0, kk))\n", "print (\"P=%.2f, Ph=%.2f\" % (P,Ph))\n", "print (\"rejected %i out of %i (N=%i, acceptance range %i...%i)\" % (NN-accepted, NN, N*10, lo2, hi2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What have we seen here:\n", "\n", "- When we test hypothesis, we define a set of outcomes that - conditional on the hypothesis being true - are so unlikely so that when they come up we reject it; this set of outcomes depends strongly on the number of points we are observing: we have seen that for N=20 the range 7..13 corresponds to 88% of the mass, whilst for N=200 the range 90..110 (which is correspondingly smaller) corresponds to 86% of the mass\n", "\n", "- We have then tested our hypothesis 'Ph=0.5' against a 1000 samples that were drawn from an underlying distribution 'P=0.5' with N=20 / N=200 (one sample is one draw of N items). In both cases we reject slightly more than 10% of our sample, which is consistent with our choice of the rejection region. To say it differently: our underlying hypothesis was correct, but we have rejected it in about 10% of cases\n", "\n", "- We have then changed the underlying distribution to 'P=0.6', still testing against the same hypothesis 'Ph=0.5'. We find that now for N=20 we reject about 25% of the samples, whilst for N=200 we reject about 90% of the samples. To rephrase this: our hypothesis was wrong, but we have not rejected it in 75% / 10% of the cases respectively\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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AxVnsl21v9EYdL5J2B34GdAbaA20k+bKwCXjlFejRw6+/yIUddoCuXeHFF5OO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\nnjCcc85lxROGc865rHjCcM45l5X/B0ZUCcmOCv14AAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "K=8\n", "x = np.linspace(0., 1., 100)\n", "mdf = [pdf(N,p,K) for p in x]\n", "plt.plot(x,mdf, '-')\n", "plt.title('probability for N=%i and K/N=%0.2f' % (N,K/N))\n", "plt.xlabel('probability parameter $p$')\n", "plt.ylabel('density')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "total mass = 4.7143\n" ] }, { "data": { "image/png": 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2IZzlL21oPXdfaWZnA/8C2gJ3uvt0MxscLR9abLBS+UaOhH33VVIQSVMhN7jd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\nmW2j138GTsrxffKVqxNPbJ3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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ms = sum(mdf)\n", "print (\"total mass = %.4f\" % ms)\n", "cdf = ft.reduce(integr, mdf)\n", "plt.title('cumulative probability for N=%i and K/N=%0.2f' % (N,K/N))\n", "plt.xlabel('probability parameter $p$')\n", "plt.ylabel('cumulative density')\n", "plot(x,cdf/ms)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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RE8ZeezlhWHtwwrCG9tpr6bEgEyaUHUltrmFYu3DCsIb2xBPp/osttyw7ktqc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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "K2=85\n", "x = np.linspace(0., 1., 500)\n", "mdf2 = [pdf(N*10,p,K2) for p in x]\n", "plt.plot(x,mdf2, '-')\n", "plt.title('probability for N=%i and K/N=%0.2f' % (10*N,K2/(N*10)))\n", "plt.xlabel('probability parameter $p$')\n", "plt.ylabel('density')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "total mass = 2.4826\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "ms2 = sum(mdf2)\n", "print (\"total mass = %.4f\" % ms2)\n", "cdf2 = ft.reduce(integr, mdf2)\n", "plt.title('cumulative probability for N=%i and K/N=%0.2f' % (10*N,K2/(10*N)))\n", "plt.xlabel('probability parameter $p$')\n", "plt.ylabel('cumulative density')\n", "plot(x,cdf2/ms2)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Zxy56zMPSGgy+oSw3ek8QCCVa89m3bx8//fQT7777bsG5Bg0asG7dOrp06cLR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\noDCaWtzXUpyOWFF6Uxb1+TnwqU8lChwtloWIpGFWFi96DnXUATtvnov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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(0., 1., 100)\n", "for K in (1,3,7,10,15):\n", " mdf = [pdf(N,p,K) for p in x]\n", " plt.plot(x,mdf, '-', label=\"K/N=%.2f\"%(K/N))\n", "plt.title('probability for N=%i' % (N))\n", "plt.xlabel('probability parameter $p$')\n", "plt.ylabel('density')\n", "plt.legend()\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.4.3" } }, "nbformat": 4, "nbformat_minor": 0 }