{ "metadata": { "name": "", "signature": "sha256:c582638faa951e64175ca28a0ce3d8afc97a0a04ae5ab0bb8e27cab77d9aa65b" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "from __future__ import division\n", "from IPython.html.widgets import interact,fixed\n", "from scipy import stats\n", "import sympy\n", "from sympy import S, init_printing, Sum, summation,expand\n", "from sympy import stats as st\n", "from sympy.abc import k\n", "import re" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Maximum likelihood estimation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\hat{p} = \\frac{1}{n} \\sum_{i=1}^n X_i$$\n", "\n", "$$\\mathbb{E}(\\hat{p}) = p$$\n", "\n", "$$\\mathbb{V}(\\hat{p}) = \\frac{p(1-p)}{n}$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sympy.plot( k*(1-k),(k,0,1),ylabel='$n^2 \\mathbb{V}$',xlabel='p',fontsize=28)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 3, "text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "b=stats.bernoulli(p=.8)\n", "samples=b.rvs(100)\n", "print var(samples)\n", "print mean(samples)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.1659\n", "0.79\n" ] } ], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "def slide_figure(n=100,m=1000,p=0.8):\n", " fig,ax=subplots()\n", " ax.axis(ymax=25)\n", " b=stats.bernoulli(p=p)\n", " v=iter(b.rvs,2)\n", " samples=b.rvs(n*1000)\n", " tmp=(samples.reshape(n,-1).mean(axis=0))\n", " ax.hist(tmp,normed=1)\n", " ax1=ax.twinx()\n", " ax1.axis(ymax=25)\n", " ax1.plot(linspace(0,1,200),stats.norm(mean(tmp),std(tmp)).pdf(linspace(0.0,1,200)),lw=3,color='r')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "interact(slide_figure, n=(100,500),p=(0.01,1,.05),m=fixed(500));" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 6 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Maximum A-Posteriori (MAP) Estimation" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy.abc import p,n,k\n", "\n", "sympy.plot(st.density(st.Beta('p',3,3))(p),(p,0,1) ,xlabel='p')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 7, "text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Maximize the MAP function to get form of estimator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "obj=sympy.expand_log(sympy.log(p**k*(1-p)**(n-k) * st.density(st.Beta('p',6,6))(p)))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "sol=sympy.solve(sympy.simplify(sympy.diff(obj,p)),p)[0]\n", "print sol" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "(k + 5)/(n + 10)\n" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\hat{p}_{MAP} = \\frac{(5+\\sum_{i=1}^n X_i )}{(n + 10)}$\n", "\n", "with corresponding expectation \n", "\n", "$\\mathbb{E} = \\frac{(5+n p )}{(n + 10)}$\n", "\n", "which is a biased estimator. The variance of this estimator is the following:\n", "\n", "$$\\mathbb{V}(\\hat{p}_{MAP}) = \\frac{n (1-p) p}{(n+10)^2}$$\n", "\n", "compare this to the variance of the maximum likelihood estimator, which is reproduced here:\n", "\n", "$$\\mathbb{V}(\\hat{p}_{ML}) = \\frac{p(1-p)}{n}$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "n=5\n", "def show_bias(n=30):\n", " sympy.plot(p,(5+n*p)/(n+10),(p,0,1),aspect_ratio=1,title='more samples reduce bias')\n", " \n", "interact(show_bias,n=(10,500,10));" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compute the variance of the MAP estimator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sum(sympy.var('x1:10'))\n", "expr=((5+(sum(sympy.var('x1:10'))))/(n+10))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "def apply_exp(expr):\n", " tmp=re.sub('x[\\d]+\\*\\*2','p',str(expand(expr)))\n", " tmp=re.sub('x[\\d]+\\*x[\\d]+','p**2',tmp)\n", " tmp=re.sub('x[\\d]+','p',tmp)\n", " return sympy.sympify(tmp)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "cell_type": "code", "collapsed": false, "input": [ "ex2 = apply_exp(expr**2)\n", "print ex2" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "8*p**2/25 + 11*p/25 + 1/9\n" ] } ], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "tmp=sympy.simplify(ex2 - (apply_exp(expr))**2 )\n", "sympy.plot(tmp,p*(1-p)/10,(p,0,1))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 14, "text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "General case" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def generate_expr(num_samples=10,alpha=6):\n", " n = sympy.symbols('n')\n", " obj=sympy.expand_log(sympy.log(p**k*(1-p)**(n-k) * st.density(st.Beta('p',alpha,alpha))(p)))\n", " sol=sympy.solve(sympy.simplify(sympy.diff(obj,p)),p)[0]\n", " expr=sol.replace(k,(sum(sympy.var('x1:%d'%(num_samples)))))\n", " expr=expr.subs(n,num_samples)\n", " ex2 = apply_exp(expr**2)\n", " ex = apply_exp(expr)\n", " return (ex,sympy.simplify(ex2-ex**2))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [ "num_samples=10\n", "X_bias,X_v = generate_expr(num_samples,alpha=2)\n", "p1=sympy.plot(X_v,(p,0,1),show=False,line_color='b',ylim=(0,.03),xlabel='p')\n", "X_bias,X_v = generate_expr(num_samples,alpha=6)\n", "p2=sympy.plot(X_v,(p,0,1),show=False,line_color='r',xlabel='p')\n", "p3=sympy.plot(p*(1-p)/num_samples,(p,0,1),show=False,line_color='g',xlabel='p')\n", "p1.append(p2[0])\n", "p1.append(p3[0])\n", "p1.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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zbo6/CcPq4td4UASpqcCMGXwlseXL+Zw1SpK/iIIoLORjFA4c4OtNL10KKMMi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\n+KP0x//U4Sccun8IMekx0g+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"text": [ "" ] } ], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "p1=sympy.plot(n*(1-p)*p/(n+10)**2,(p,0,1),show=False,line_color='b',ylim=(0,.05),xlabel='p',ylabel='variance')\n", "p2=sympy.plot((1-p)*p/n,(p,0,1),show=False,line_color='r',ylim=(0,.05),xlabel='p')\n", "p1.append(p2[0])\n", "p1.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "def show_variance(n=5):\n", " p1=sympy.plot(n*(1-p)*p/(n+10)**2,(p,0,1),show=False,line_color='b',ylim=(0,.05),xlabel='p',ylabel='variance')\n", " p2=sympy.plot((1-p)*p/n,(p,0,1),show=False,line_color='r',ylim=(0,.05),xlabel='p')\n", " p1.append(p2[0])\n", " p1.show() \n", "interact(show_variance,n=(10,120,2));\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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kZPURFQV06eJy+w7VnWnTZE27desM6TTy3MCYORNYsEBmVpponLMebDZ5bLRr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\nGEREpAkDg4iINPl/idlgn/UiREQAAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The obvious question is what is the value of a biased estimator? The key fact is that the MAP estimator is biased, yes, but it is biased according to the prior probability of $\\theta$. Suppose that the true parameter $p=1/2$ which is exactly at the peak of the prior probability function. In this case, what is the bias?\n", "\n", "$\\mathbb{E} = \\frac{(5+n p )}{(n + 10)} -p \\rightarrow 0$\n", "\n", "and the variance of the MAP estimator at this point is the following:\n", "\n", "$$\\frac{n p (1-p)}{(n+10)^2} \\rightarrow$$\n", "\n", "\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "pv=.60\n", "nsamp=30\n", "fig,ax=subplots()\n", "rv = stats.bernoulli(pv)\n", "map_est=(rv.rvs((nsamp,1000)).sum(axis=0)+5)/(nsamp+10);\n", "ml_est=(rv.rvs((nsamp,1000)).sum(axis=0))/(nsamp);\n", "_,bins,_=ax.hist(map_est,bins=20,alpha=.3,normed=True,label='MAP');\n", "ax.hist(ml_est,bins=20,alpha=.3,normed=True,label='ML');\n", "ax.vlines(map_est.mean(),0,12,lw=3,linestyle=':',color='b')\n", "ax.vlines(pv,0,12,lw=3,color='r',linestyle=':')\n", "ax.vlines(ml_est.mean(),0,12,lw=3,color='g',linestyle=':')\n", "ax.legend()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 23, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 23 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 19 } ], "metadata": {} } ] }