1{ 2 "cells": [ 3 { 4 "cell_type": "markdown", 5 "metadata": {}, 6 "source": [ 7 "<a href=\"http://agrum.org\" target=\"blank\"><img src=\"http://agrum.gitlab.io/theme/img/logoAgrum.png\" align=\"left\" style=\"height:100px\"/></a><a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\"><img alt=\"Creative Commons License\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc/4.0/88x31.png\" /></a><br />This pyAgrum's notebook is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\">Creative Commons Attribution-NonCommercial 4.0 International License</a>." 8 ] 9 }, 10 { 11 "cell_type": "code", 12 "execution_count": 1, 13 "metadata": { 14 "ExecuteTime": { 15 "end_time": "2019-02-25T15:55:40.871554Z", 16 "start_time": "2019-02-25T15:55:40.618127Z" 17 } 18 }, 19 "outputs": [], 20 "source": [ 21 "from pylab import *\n", 22 "import matplotlib.pyplot as plt" 23 ] 24 }, 25 { 26 "cell_type": "markdown", 27 "metadata": {}, 28 "source": [ 29 "# Quasi-continuous BN" 30 ] 31 }, 32 { 33 "cell_type": "markdown", 34 "metadata": {}, 35 "source": [ 36 "aGrUM cannot (currently) deal with with continuous variables. However, a discrete variable with a large enough domain size is an approximation of such variables." 37 ] 38 }, 39 { 40 "cell_type": "code", 41 "execution_count": 2, 42 "metadata": { 43 "ExecuteTime": { 44 "end_time": "2019-02-25T15:55:40.883085Z", 45 "start_time": "2019-02-25T15:55:40.874972Z" 46 } 47 }, 48 "outputs": [], 49 "source": [ 50 "import pyAgrum as gum\n", 51 "import pyAgrum.lib.notebook as gnb\n", 52 "\n", 53 "#nbr of states for quasi continuous variables. You can change the value \n", 54 "#but be careful of the quadratic behavior of both memory and time complexity\n", 55 "#in this example.\n", 56 "NB=300 " 57 ] 58 }, 59 { 60 "cell_type": "code", 61 "execution_count": 3, 62 "metadata": { 63 "ExecuteTime": { 64 "end_time": "2019-02-25T15:55:43.253694Z", 65 "start_time": "2019-02-25T15:55:40.886136Z" 66 } 67 }, 68 "outputs": [ 69 { 70 "name": "stdout", 71 "output_type": "stream", 72 "text": [ 73 "BN{nodes: 2, arcs: 1, domainSize: 600, dim: 602}\n" 74 ] 75 }, 76 { 77 "data": { 78 "image/svg+xml": [ 79 "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"62pt\" height=\"116pt\" viewBox=\"0.00 0.00 62.00 116.00\">\n", 80 "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 112)\">\n", 81 "<title>G</title>\n", 82 "<!-- A -->\n", 83 "<g id=\"node1\" class=\"node\">\n", 84 "<title>A</title>\n", 85 "<g id=\"a_node1\"><a xlink:title=\"(0) A\">\n", 86 "<ellipse fill=\"#404040\" stroke=\"black\" cx=\"27\" cy=\"-90\" rx=\"27\" ry=\"18\"/>\n", 87 "<text text-anchor=\"middle\" x=\"27\" y=\"-86.3\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"white\">A</text>\n", 88 "</a>\n", 89 "</g>\n", 90 "</g>\n", 91 "<!-- B -->\n", 92 "<g id=\"node2\" class=\"node\">\n", 93 "<title>B</title>\n", 94 "<g id=\"a_node2\"><a xlink:title=\"(1) B\">\n", 95 "<ellipse fill=\"#404040\" stroke=\"black\" cx=\"27\" cy=\"-18\" rx=\"27\" ry=\"18\"/>\n", 96 "<text text-anchor=\"middle\" x=\"27\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"white\">B</text>\n", 97 "</a>\n", 98 "</g>\n", 99 "</g>\n", 100 "<!-- A->B -->\n", 101 "<g id=\"edge1\" class=\"edge\">\n", 102 "<title>A->B</title>\n", 103 "<g id=\"a_edge1\"><a xlink:title=\"(0, 1) : \">\n", 104 "<path fill=\"none\" stroke=\"#4a4a4a\" d=\"M27,-71.7C27,-63.98 27,-54.71 27,-46.11\"/>\n", 105 "<polygon fill=\"#4a4a4a\" stroke=\"#4a4a4a\" points=\"30.5,-46.1 27,-36.1 23.5,-46.1 30.5,-46.1\"/>\n", 106 "</a>\n", 107 "</g>\n", 108 "</g>\n", 109 "</g>\n", 110 "</svg>" 111 ], 112 "text/plain": [ 113 "<IPython.core.display.SVG object>" 114 ] 115 }, 116 "metadata": {}, 117 "output_type": "display_data" 118 } 119 ], 120 "source": [ 121 "bn=gum.BayesNet(\"Quasi-Continuous\")\n", 122 "a=bn.add(gum.LabelizedVariable(\"A\",\"A binary variable\",2))\n", 123 "b=bn.add(gum.RangeVariable(\"B\",\"A range variable\",0,NB-1))\n", 124 "bn.addArc(a,b)\n", 125 "print(bn)\n", 126 "gnb.showBN(bn)" 127 ] 128 }, 129 { 130 "cell_type": "code", 131 "execution_count": 4, 132 "metadata": { 133 "ExecuteTime": { 134 "end_time": "2019-02-25T15:55:43.489345Z", 135 "start_time": "2019-02-25T15:55:43.255896Z" 136 } 137 }, 138 "outputs": [ 139 { 140 "data": { 141 "image/svg+xml": [ 142 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 143 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 144 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 145 "<svg height=\"60.198125pt\" version=\"1.1\" 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"source": [ 458 "## CPT for quasi-continuous variables (with parents)" 459 ] 460 }, 461 { 462 "cell_type": "markdown", 463 "metadata": {}, 464 "source": [ 465 "Using python (and scipy), it is easy to find pdf for continuous variable" 466 ] 467 }, 468 { 469 "cell_type": "code", 470 "execution_count": 5, 471 "metadata": { 472 "ExecuteTime": { 473 "end_time": "2019-02-25T15:55:43.818717Z", 474 "start_time": "2019-02-25T15:55:43.491316Z" 475 } 476 }, 477 "outputs": [ 478 { 479 "data": { 480 "image/svg+xml": [ 481 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 482 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 483 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 484 "<svg height=\"263.68875pt\" version=\"1.1\" viewBox=\"0 0 384.828125 263.68875\" width=\"384.828125pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 485 " <metadata>\n", 486 " <rdf:RDF xmlns:cc=\"http://creativecommons.org/ns#\" 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1335 "plot(normalize(norm(),-4,1,NB)) # P(B|A=0) is a Gaussian distribution\n", 1336 "plot(normalize(maxwell(),0.5,6,NB)) # P(B|A=1) is a Maxwell-Boltzmann distribution\n", 1337 "title(\"P(B | A)\")\n", 1338 "legend([\"P(B|A=0)\",\"P(B|A=1)\"],loc='best')\n", 1339 "\n", 1340 "bn.cpt(b)[{'A':0}]=normalize(norm(),-4,1,NB)\n", 1341 "bn.cpt(b)[{'A':1}]=normalize(maxwell(),0.5,6,NB)" 1342 ] 1343 }, 1344 { 1345 "cell_type": "markdown", 1346 "metadata": {}, 1347 "source": [ 1348 "## Inference" 1349 ] 1350 }, 1351 { 1352 "cell_type": "code", 1353 "execution_count": 6, 1354 "metadata": { 1355 "ExecuteTime": { 1356 "end_time": "2019-02-25T15:55:43.964206Z", 1357 "start_time": "2019-02-25T15:55:43.820396Z" 1358 } 1359 }, 1360 "outputs": [ 1361 { 1362 "data": { 1363 "image/svg+xml": [ 1364 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 1365 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 1366 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 1367 "<svg 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"2019-02-25T15:55:46.140064Z", 2041 "start_time": "2019-02-25T15:55:43.965869Z" 2042 } 2043 }, 2044 "outputs": [ 2045 { 2046 "data": { 2047 "image/svg+xml": [ 2048 "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"62pt\" height=\"188pt\" viewBox=\"0.00 0.00 62.00 188.00\">\n", 2049 "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 184)\">\n", 2050 "<title>G</title>\n", 2051 "<!-- A -->\n", 2052 "<g id=\"node1\" class=\"node\">\n", 2053 "<title>A</title>\n", 2054 "<g id=\"a_node1\"><a xlink:title=\"(0) A\">\n", 2055 "<ellipse fill=\"#404040\" stroke=\"black\" cx=\"27\" cy=\"-162\" rx=\"27\" ry=\"18\"/>\n", 2056 "<text text-anchor=\"middle\" x=\"27\" y=\"-158.3\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"white\">A</text>\n", 2057 "</a>\n", 2058 "</g>\n", 2059 "</g>\n", 2060 "<!-- B -->\n", 2061 "<g id=\"node3\" class=\"node\">\n", 2062 "<title>B</title>\n", 2063 "<g id=\"a_node3\"><a xlink:title=\"(1) B\">\n", 2064 "<ellipse fill=\"#404040\" stroke=\"black\" cx=\"27\" cy=\"-90\" rx=\"27\" ry=\"18\"/>\n", 2065 "<text text-anchor=\"middle\" x=\"27\" y=\"-86.3\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"white\">B</text>\n", 2066 "</a>\n", 2067 "</g>\n", 2068 "</g>\n", 2069 "<!-- A->B -->\n", 2070 "<g id=\"edge1\" class=\"edge\">\n", 2071 "<title>A->B</title>\n", 2072 "<g id=\"a_edge1\"><a xlink:title=\"(0, 1) : \">\n", 2073 "<path fill=\"none\" stroke=\"#4a4a4a\" d=\"M27,-143.7C27,-135.98 27,-126.71 27,-118.11\"/>\n", 2074 "<polygon fill=\"#4a4a4a\" stroke=\"#4a4a4a\" points=\"30.5,-118.1 27,-108.1 23.5,-118.1 30.5,-118.1\"/>\n", 2075 "</a>\n", 2076 "</g>\n", 2077 "</g>\n", 2078 "<!-- C -->\n", 2079 "<g id=\"node2\" class=\"node\">\n", 2080 "<title>C</title>\n", 2081 "<g id=\"a_node2\"><a xlink:title=\"(2) C\">\n", 2082 "<ellipse fill=\"#404040\" stroke=\"black\" cx=\"27\" cy=\"-18\" rx=\"27\" ry=\"18\"/>\n", 2083 "<text text-anchor=\"middle\" x=\"27\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\" fill=\"white\">C</text>\n", 2084 "</a>\n", 2085 "</g>\n", 2086 "</g>\n", 2087 "<!-- B->C -->\n", 2088 "<g id=\"edge2\" class=\"edge\">\n", 2089 "<title>B->C</title>\n", 2090 "<g id=\"a_edge2\"><a xlink:title=\"(1, 2) : \">\n", 2091 "<path fill=\"none\" stroke=\"#4a4a4a\" d=\"M27,-71.7C27,-63.98 27,-54.71 27,-46.11\"/>\n", 2092 "<polygon fill=\"#4a4a4a\" stroke=\"#4a4a4a\" points=\"30.5,-46.1 27,-36.1 23.5,-46.1 30.5,-46.1\"/>\n", 2093 "</a>\n", 2094 "</g>\n", 2095 "</g>\n", 2096 "</g>\n", 2097 "</svg>" 2098 ], 2099 "text/plain": [ 2100 "<IPython.core.display.SVG object>" 2101 ] 2102 }, 2103 "metadata": {}, 2104 "output_type": "display_data" 2105 } 2106 ], 2107 "source": [ 2108 "c=bn.add(gum.RangeVariable(\"C\",\"Another quasi continuous variable\",0,NB-1))\n", 2109 "bn.addArc(b,c)\n", 2110 "gnb.showBN(bn) # B and C are quasi-continouous" 2111 ] 2112 }, 2113 { 2114 "cell_type": "markdown", 2115 "metadata": {}, 2116 "source": [ 2117 "Even if this BN is quite small (and linear), the size of nodes $B$ et $C$ are rather big and creates a complex model (NBxNB parameters in $P(C|B)$)." 2118 ] 2119 }, 2120 { 2121 "cell_type": "code", 2122 "execution_count": 8, 2123 "metadata": { 2124 "ExecuteTime": { 2125 "end_time": "2019-02-25T15:55:46.149488Z", 2126 "start_time": "2019-02-25T15:55:46.145347Z" 2127 } 2128 }, 2129 "outputs": [ 2130 { 2131 "name": "stdout", 2132 "output_type": "stream", 2133 "text": [ 2134 "nombre de paramètres du bn : 90299\n", 2135 "domaine du bn : 10^5.2552725051033065\n" 2136 ] 2137 } 2138 ], 2139 "source": [ 2140 "print(\"nombre de paramètres du bn : {0}\".format(bn.dim()))\n", 2141 "print(\"domaine du bn : 10^{0}\".format(bn.log10DomainSize()))" 2142 ] 2143 }, 2144 { 2145 "cell_type": "code", 2146 "execution_count": 9, 2147 "metadata": { 2148 "ExecuteTime": { 2149 "end_time": "2019-02-25T15:55:47.047755Z", 2150 "start_time": "2019-02-25T15:55:46.152718Z" 2151 } 2152 }, 2153 "outputs": [ 2154 { 2155 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"plot(ie.posterior(c)[:])\n", 7945 "t=title(\"P(C) with prior p(A)=[0.1,0.9]\")\n", 7946 " \n", 7947 "gnb.showPosterior(bn,target=\"C\",evs={})" 7948 ] 7949 }, 7950 { 7951 "cell_type": "markdown", 7952 "metadata": {}, 7953 "source": [ 7954 "# inference with evidence" 7955 ] 7956 }, 7957 { 7958 "cell_type": "markdown", 7959 "metadata": {}, 7960 "source": [ 7961 "We want to compute \n", 7962 "<ul><li>\n", 7963 "$P(A | C=\\frac{2*NB}{3})$ \n", 7964 "</li><li>\n", 7965 "$P(B | C=\\frac{2*NB}{3})$\n", 7966 "</li></ul>" 7967 ] 7968 }, 7969 { 7970 "cell_type": "code", 7971 "execution_count": 13, 7972 "metadata": { 7973 "ExecuteTime": { 7974 "end_time": "2019-02-25T15:55:47.634595Z", 7975 "start_time": "2019-02-25T15:55:47.406804Z" 7976 }, 7977 "tags": [] 7978 }, 7979 "outputs": [ 7980 { 7981 "data": { 7982 "image/svg+xml": [ 7983 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 7984 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 7985 " 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between Gaussian and Maxwell-Boltzman distributions" 10448 ] 10449 }, 10450 { 10451 "cell_type": "markdown", 10452 "metadata": {}, 10453 "source": [ 10454 "What is the behaviour of $P(A | B=i)$ when $i$ varies ? I.e. we perform a MAP decision between the two models ($A=0$ for the Gaussian distribution and $A=1$ for the Maxwell-Boltzman distribution)." 10455 ] 10456 }, 10457 { 10458 "cell_type": "code", 10459 "execution_count": 14, 10460 "metadata": { 10461 "ExecuteTime": { 10462 "end_time": "2019-02-25T15:55:47.973938Z", 10463 "start_time": "2019-02-25T15:55:47.636398Z" 10464 } 10465 }, 10466 "outputs": [ 10467 { 10468 "data": { 10469 "image/svg+xml": [ 10470 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 10471 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 10472 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 10473 "<svg height=\"263.68875pt\" version=\"1.1\" viewBox=\"0 0 372.103125 263.68875\" width=\"372.103125pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 10474 " <metadata>\n", 10475 " <rdf:RDF xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:dc=\"http://purl.org/dc/elements/1.1/\" 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size 432x288 with 1 Axes>" 11525 ] 11526 }, 11527 "metadata": {}, 11528 "output_type": "display_data" 11529 } 11530 ], 11531 "source": [ 11532 "bn.cpt(a)[:]=[0.1, 0.9]\n", 11533 "ie=gum.LazyPropagation(bn)\n", 11534 "p0=[]\n", 11535 "p1=[]\n", 11536 "for i in range(300):\n", 11537 " ie.setEvidence({'B':i})\n", 11538 " ie.makeInference() \n", 11539 " p0.append(ie.posterior(a)[0]) \n", 11540 " p1.append(ie.posterior(a)[1])\n", 11541 "plot(p0)\n", 11542 "plot(p1)\n", 11543 "title(\"P( A | B=i) with prior p(A)=[0.1,0.9]\")\n", 11544 "legend([\"A=0\",\"A=1\"],loc='best')\n", 11545 "inters=(transpose(p0)<transpose(p1)).argmin()\n", 11546 "text(inters-2,p0[inters],\n", 11547 " \"{0},{1:5.4f} \".format(inters,p0[inters]),\n", 11548 " bbox=dict(facecolor='red', alpha=0.1),ha='right');" 11549 ] 11550 }, 11551 { 11552 "cell_type": "markdown", 11553 "metadata": {}, 11554 "source": [ 11555 "i.e. if $i<164$ then $A=1$ else $A=0$" 11556 ] 11557 }, 11558 { 11559 "cell_type": "markdown", 11560 "metadata": {}, 11561 "source": [ 11562 "### Changing the prior $P(A)$" 11563 ] 11564 }, 11565 { 11566 "cell_type": "code", 11567 "execution_count": 15, 11568 "metadata": { 11569 "ExecuteTime": { 11570 "end_time": "2019-02-25T15:55:48.387971Z", 11571 "start_time": "2019-02-25T15:55:47.977284Z" 11572 } 11573 }, 11574 "outputs": [ 11575 { 11576 "data": { 11577 "image/svg+xml": [ 11578 "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n", 11579 "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", 11580 " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", 11581 "<svg height=\"263.68875pt\" version=\"1.1\" viewBox=\"0 0 372.103125 263.68875\" width=\"372.103125pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", 11582 " <metadata>\n", 11583 " <rdf:RDF xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n", 11584 " <cc:Work>\n", 11585 " 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