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sfXpb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\nkR0ONZZDa3o7zKFrafq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"text/plain": [ "<IPython.core.display.Image object>" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import Image \n", "Image('../../../python_for_probability_statistics_and_machine_learning.jpg')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "[Python for Probability, Statistics, and Machine Learning](https://www.springer.com/fr/book/9783319307152)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "np.random.seed(12345)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Conditional Expectation and Mean Square Error\n", "\n", "In this section, we work through a detailed example using conditional\n", "expectation and optimization methods. Suppose we have two fair six-sided die\n", "($X$ and $Y$) and we want to measure the sum of the two variables as $Z=X+Y$.\n", "Further, let's suppose that given $Z$, we want the best estimate of $X$ in the\n", "mean-squared-sense. Thus, we want to minimize the following:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "J(\\alpha) = \\sum ( x - \\alpha z )^2 \\mathbb{P}(x,z)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " where $\\mathbb{P}$ is the probability mass function for this problem.\n", "The idea is that when we have solved this problem, we will have a function of\n", "$Z$ that is going to be the minimum MSE estimate of $X$. We can substitute in\n", "for $Z$ in $J$ and get:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "J(\\alpha) = \\sum ( x - \\alpha (x+y) )^2 \\mathbb{P}(x,y)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Let's work out the steps in Sympy in the following:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "329*a**2/6 - 329*a/6 + 91/6\n" ] } ], "source": [ "import sympy as S\n", "from sympy.stats import density, E, Die\n", "\n", "x=Die('D1',6) # 1st six sided die\n", "y=Die('D2',6) # 2nd six sides die\n", "a=S.symbols('a')\n", "z = x+y # sum of 1st and 2nd die\n", "J = E((x-a*(x+y))**2) # expectation\n", "print S.simplify(J)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " With all that setup we can now use basic calculus to minimize the\n", "objective function $J$," ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/2\n" ] } ], "source": [ "sol,=S.solve(S.diff(J,a),a) # using calculus to minimize\n", "print sol # solution is 1/2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Programming Tip.**\n", "\n", "Sympy has a `stats` module that can do some basic work with expressions\n", "involving probability densities and expectations. The above code uses its `E`\n", "function to compute the expectation.\n", "\n", "\n", "\n", " This says that $z/2$ is the MSE estimate of $X$ given $Z$ which means\n", "geometrically (interpreting the MSE as a squared distance weighted by the\n", "probability mass function) that $z/2$ is as *close* to $x$ as we are going to\n", "get for a given $z$.\n", "\n", "Let's look at the same problem using the conditional expectation operator $\n", "\\mathbb{E}(\\cdot|z) $ and apply it to our definition of $Z$. Then" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbb{E}(z|z)=\\mathbb{E}(x+y|z)=\\mathbb{E}(x|z)+\\mathbb{E}(y|z)=z\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " using the linearity of the expectation. Now, since by the\n", "symmetry of the problem (i.e., two identical die), we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbb{E}(x|z)=\\mathbb{E}(y|z)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " we can plug this in and solve" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "2 \\mathbb{E}(x|z)=z\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " which once again gives," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbb{E}(x|z) =\\frac{z}{2}\n", "$$" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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DXItIUECACCDMCbzCltgKLEkxK9A7OiEsLbSTX0a8PK3TM92AZ+6XhJWCoE0k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\n6SvBarvLzB6XtFzSOwRDDPs/WOjrwIOS0sBv8tY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"text/plain": [ "<matplotlib.figure.Figure at 0xdf1ef28>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "\n", "from numpy import arange,array\n", "from matplotlib.pylab import subplots, cm\n", "from sympy import Integer\n", "fig,ax = subplots()\n", "v = arange(1,7) + arange(1,7)[:,None]\n", "foo=lambda i: density(z)[Integer(i)].evalf() # some tweaks to get a float out\n", "Zmass=array(map(foo,v.flat),dtype=float).reshape(6,6)\n", "\n", "pc=ax.pcolor(arange(1,8),arange(1,8),Zmass,cmap=cm.gray)\n", "_=ax.set_xticks([(i+0.5) for i in range(1,7)])\n", "_=ax.set_xticklabels([str(i) for i in range(1,7)])\n", "_=ax.set_yticks([(i+0.5) for i in range(1,7)])\n", "_=ax.set_yticklabels([str(i) for i in range(1,7)])\n", "for i in range(1,7):\n", " for j in range(1,7):\n", " _=ax.text(i+.5,j+.5,str(i+j),fontsize=18,fontweight='bold',color='goldenrod')\n", "\n", "_=ax.set_title(r'Probability Mass for $Z$',fontsize=18) \n", "_=ax.set_xlabel('$X$ values',fontsize=18)\n", "_=ax.set_ylabel('$Y$ values',fontsize=18);\n", "cb=fig.colorbar(pc)\n", "_=cb.ax.set_title(r'Probability',fontsize=12)\n", "#fig.savefig('fig-probability/Conditional_expectation_MSE_001.png')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "<!-- dom:FIGURE: [fig-probability/Conditional_expectation_MSE_001.png, width=500 frac=0.85] The values of $Z$ are in yellow with the corresponding values for $X$ and $Y$ on the axes. The gray scale colors indicate the underlying joint probability density. <div id=\"fig:Conditional_expectation_MSE_001\"></div> -->\n", "<!-- begin figure -->\n", "<div id=\"fig:Conditional_expectation_MSE_001\"></div>\n", "\n", "<p>The values of $Z$ are in yellow with the corresponding values for $X$ and $Y$ on the axes. The gray scale colors indicate the underlying joint probability density.</p>\n", "<img src=\"fig-probability/Conditional_expectation_MSE_001.png\" width=500>\n", "\n", "<!-- end figure -->\n", "\n", "\n", " which is equal to the estimate we just found by minimizing the MSE.\n", "Let's explore this further with [Figure](#fig:Conditional_expectation_MSE_001). [Figure](#fig:Conditional_expectation_MSE_001) shows the values of $Z$ in yellow with\n", "the corresponding values for $X$ and $Y$ on the axes. Suppose $z=2$, then the\n", "closest $X$ to this is $X=1$, which is what $\\mathbb{E}(x|z)=z/2=1$ gives. What\n", "happens when $Z=7$? In this case, this value is spread out diagonally along the\n", "$X$ axis so if $X=1$, then $Z$ is 6 units away, if $X=2$, then $Z$ is 5 units\n", "away and so on.\n", "\n", "Now, back to the original question, if we had $Z=7$ and we wanted\n", "to get as close as we could to this using $X$, then why not choose\n", "$X=6$ which is only one unit away from $Z$? The problem with doing\n", "that is $X=6$ only occurs 1/6 of the time, so we are not likely to\n", "get it right the other 5/6 of the time. So, 1/6 of the time we are\n", "one unit away but 5/6 of the time we are much more than one unit\n", "away. This means that the MSE score is going to be worse. Since\n", "each value of $X$ from 1 to 6 is equally likely, to play it safe,\n", "we choose $7/2$ as the estimate, which is what the conditional\n", "expectation suggests.\n", "\n", "We can check this claim with samples using Sympy below:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "MSE=8.44 using 6 vs MSE=2.43 using 7/2 \n" ] } ], "source": [ "import numpy as np\n", "from sympy import stats\n", "# Eq constrains Z\n", "samples_z7 = lambda : stats.sample(x, S.Eq(z,7)) \n", "#using 6 as an estimate\n", "mn= np.mean([(6-samples_z7())**2 for i in range(100)]) \n", "#7/2 is the MSE estimate\n", "mn0= np.mean([(7/2.-samples_z7())**2 for i in range(100)]) \n", "print 'MSE=%3.2f using 6 vs MSE=%3.2f using 7/2 ' % (mn,mn0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Programming Tip.**\n", "\n", "The `stats.sample(x, S.Eq(z,7))` function call samples the `x` variable subject\n", "to a condition on the `z` variable. In other words, it generates random samples\n", "of `x` die, given that the sum of the outcomes of that die and the `y` die add\n", "up to `z==7`.\n", "\n", "\n", "\n", " Please run the above code repeatedly in the Jupyter/IPython\n", "Notebook corresponding to this section until you have convinced\n", "yourself that the $\\mathbb{E}(x|z)$ gives the lower MSE every\n", "time. To push this reasoning, let's consider the case where the\n", "die is so biased so that the outcome of `6` is ten times more\n", "probable than any of the other outcomes. That is," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbb{P}(6) = 2/3\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " whereas $\\mathbb{P}(1)=\\mathbb{P}(2)=\\ldots=\\mathbb{P}(5)=1/15$.\n", "We can explore this using Sympy as in the following:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# here 6 is ten times more probable than any other outcome\n", "x=stats.FiniteRV('D3',{1:1/15., 2:1/15., \n", " 3:1/15., 4:1/15.,\n", " 5:1/15., 6:2/3.})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " As before, we construct the sum of the two dice, and plot the\n", "corresponding probability mass function in [Figure](#fig:Conditional_expectation_MSE_002). As compared with [Figure](#fig:Conditional_expectation_MSE_001), the probability mass has been shifted\n", "away from the smaller numbers." ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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OfHKlJKdj2/1EQ10ggrd8NmX1H8q1SgN4J59MYPcfAQgdXEugYTkufy2B3Q86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\nX+hj0RC/fiHfq9B6aWpN4/H1J3H9/EOERJlXVse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"text/plain": [ "<matplotlib.figure.Figure at 0xc22f198>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "z = x + y\n", "foo=lambda i: density(z)[S.Integer(i)].evalf() # some tweaks to get a float out\n", "v = np.arange(1,7) + np.arange(1,7)[:,None]\n", "Zmass=np.array(map(foo,v.flat),dtype=float).reshape(6,6)\n", "\n", "from matplotlib.pylab import subplots, cm\n", "fig,ax=subplots()\n", "pc=ax.pcolor(np.arange(1,8),np.arange(1,8),Zmass,cmap=cm.gray)\n", "_=ax.set_xticks([(i+0.5) for i in range(1,7)])\n", "_=ax.set_xticklabels([str(i) for i in range(1,7)])\n", "_=ax.set_yticks([(i+0.5) for i in range(1,7)])\n", "_=ax.set_yticklabels([str(i) for i in range(1,7)])\n", "for i in range(1,7):\n", " for j in range(1,7):\n", " _=ax.text(i+.5,j+.5,str(i+j),fontsize=18,fontweight='bold',color='goldenrod')\n", "\n", "_=ax.set_title(r'Probability Mass for $Z$; Nonuniform case',fontsize=16) \n", "_=ax.set_xlabel('$X$ values',fontsize=18)\n", "_=ax.set_ylabel('$Y$ values',fontsize=18);\n", "cb=fig.colorbar(pc)\n", "_=cb.ax.set_title(r'Probability',fontsize=12)\n", "#fig.savefig('fig-probability/Conditional_expectation_MSE_002.png')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "<!-- dom:FIGURE: [fig-probability/Conditional_expectation_MSE_002.png, width=500 frac=0.85] The values of $Z$ are in yellow with the corresponding values for $X$ and $Y$ on the axes. <div id=\"fig:Conditional_expectation_MSE_002\"></div> -->\n", "<!-- begin figure -->\n", "<div id=\"fig:Conditional_expectation_MSE_002\"></div>\n", "\n", "<p>The values of $Z$ are in yellow with the corresponding values for $X$ and $Y$ on the axes.</p>\n", "<img src=\"fig-probability/Conditional_expectation_MSE_002.png\" width=500>\n", "\n", "<!-- end figure -->\n", "\n", "\n", "Let's see what the conditional expectation says about how we can estimate $X$\n", "from $Z$." ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "5.00000000000000" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E(x, S.Eq(z,7)) # conditional expectation E(x|z=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Now that we have $\\mathbb{E}(x|z=7) = 5$, we can generate\n", "samples as before and see if this gives the minimum MSE." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "MSE=4.08 using 6 vs MSE=2.20 using 5 \n" ] } ], "source": [ "samples_z7 = lambda : stats.sample(x, S.Eq(z,7)) \n", "#using 6 as an estimate\n", "mn= np.mean([(6-samples_z7())**2 for i in range(100)]) \n", "#5 is the MSE estimate\n", "mn0= np.mean([(5-samples_z7())**2 for i in range(100)]) \n", "print 'MSE=%3.2f using 6 vs MSE=%3.2f using 5 ' % (mn,mn0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using a simple example, we have emphasized the connection between minimum mean\n", "squared error problems and conditional expectation. Hopefully, the last two\n", "figures helped expose the role of the probability density. Next, we'll\n", "continue revealing the true power of the conditional expectation as we\n", "continue to develop corresponding geometric intuition." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.11" } }, "nbformat": 4, "nbformat_minor": 0 }