diff --git a/02_Discrete_Bayes.ipynb b/02_Discrete_Bayes.ipynb index 72d35c8..e413720 100644 --- a/02_Discrete_Bayes.ipynb +++ b/02_Discrete_Bayes.ipynb @@ -16,19 +16,11 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "The autoreload extension is already loaded. To reload it, use:\n", - " %reload_ext autoreload\n" - ] - }, { "data": { "text/html": [ @@ -74,21 +66,21 @@ " }\n", " .text_cell_render h2 {\n", " font-weight: 200;\n", - " font-size: 20pt;\n", + " font-size: 16pt;\n", " font-style: italic;\n", " line-height: 100%;\n", " color:#c76c0c;\n", " margin-bottom: 0.5em;\n", " margin-top: 1.5em;\n", - " display: block;\n", - " white-space: nowrap;\n", + " display: inline;\n", + " white-space: wrap;\n", " } \n", " h3 {\n", " font-family: 'Open sans',verdana,arial,sans-serif;\n", " }\n", " .text_cell_render h3 {\n", - " font-weight: 300;\n", - " font-size: 18pt;\n", + " font-weight: 200;\n", + " font-size: 14pt;\n", " line-height: 100%;\n", " color:#d77c0c;\n", " margin-bottom: 0.5em;\n", @@ -100,8 +92,8 @@ " font-family: 'Open sans',verdana,arial,sans-serif;\n", " }\n", " .text_cell_render h4 {\n", - " font-weight: 300;\n", - " font-size: 16pt;\n", + " font-weight: 100;\n", + " font-size: 14pt;\n", " color:#d77c0c;\n", " margin-bottom: 0.5em;\n", " margin-top: 0.5em;\n", @@ -112,7 +104,7 @@ " font-family: 'Open sans',verdana,arial,sans-serif;\n", " }\n", " .text_cell_render h5 {\n", - " font-weight: 300;\n", + " font-weight: 200;\n", " font-style: normal;\n", " color: #1d3b84;\n", " font-size: 16pt;\n", @@ -123,9 +115,9 @@ " }\n", " div.text_cell_render{\n", " font-family: 'Arimo',verdana,arial,sans-serif;\n", - " line-height: 135%;\n", - " font-size: 125%;\n", - " width:750px;\n", + " line-height: 125%;\n", + " font-size: 120%;\n", + " width:740px;\n", " margin-left:auto;\n", " margin-right:auto;\n", " text-align:justify;\n", @@ -243,6 +235,9 @@ " },\n", " displayAlign: 'center', // Change this to 'center' to center equations.\n", " \"HTML-CSS\": {\n", + " availableFonts: [\"TeX\"],\n", + " preferredFont: \"TeX\",\n", + " scale:85,\n", " styles: {'.MathJax_Display': {\"margin\": 4}}\n", " }\n", " });\n", @@ -252,7 +247,7 @@ "" ] }, - "execution_count": 32, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } @@ -304,7 +299,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -319,12 +314,14 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "In Bayesian statistics this is called our *prior*, for reasons that won't yet be clear. It basically means the probability prior to incorporating measurements or other information. More completely, this is the *prior probability distribution*, but that is a mouthful and so it is normally shorted to *prior*. A *probability distribution* is just a collection of all possible probabilities for an event. Probability distributions always have to sum to 1 because *something* had to happen; the distribution just lists all the different *somethings* and the probability of each. \n", + "\n", "Now let's create a map of the hallway in another list. Suppose there are first two doors close together, and then another door quite a bit further down the hallway. We will use 1 to denote a door, and 0 to denote a wall:" ] }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -337,146 +334,21 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "So I start listening to Simon's transmissions on the network, and the first data I get from the sensor is \"door\". From this I conclude that he is in front of a door, but which one? I have no idea. I have no reason to believe he is is in front of the first, second, or third door. But what I can do is assign a probability to each door. All doors are equally likely, and there are three of them, so I assign a probability of 1/3 to each door. " + "So I start listening to Simon's transmissions on the network, and the first data I get from the sensor is \"door\". For the moment assume the sensor always returns the correct answer. From this I conclude that he is in front of a door, but which one? I have no idea. I have no reason to believe he is is in front of the first, second, or third door. But what I can do is assign a probability to each door. All doors are equally likely, and there are three of them, so I assign a probability of 1/3 to each door. 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -505,7 +377,7 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -547,7 +419,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -585,18 +457,20 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Unfortunately I have yet to come across a perfect sensor. Perhaps the sensor would not detect a door if Simon sat in front of it while scratching himself, or it might report there is a door if he is facing towards the wall, not down the hallway. So in practice when I get a report 'door' I cannot assign 1/3 as the probability for each door. I have to assign something less than 1/3 to each door, and then assign a small probability to each blank wall position. At this point it doesn't matter exactly what numbers we assign; let us say that the probably of 'door' being correct is 0.6, and the probability of being incorrect is 0.2, which is another way of saying it is about 3 times more likely to be right than wrong. How would we do this?\n", + "Unfortunately I have yet to come across a perfect sensor. Perhaps the sensor would not detect a door if Simon sat in front of it while scratching himself, or it might report there is a door if he is facing towards the wall instead of down the hallway. So in practice when I get a report 'door' I cannot assign 1/3 as the probability for each door. I have to assign something less than 1/3 to each door, and then assign a small probability to each blank wall position. \n", + "\n", + "At this point it doesn't matter exactly what numbers we assign; let us say that the probability of the sensor being right is 3 times more likely to be right than wrong. How would we do this?\n", "\n", "At first this may seem like an insurmountable problem. If the sensor is noisy it casts doubt on every piece of data. How can we conclude anything if we are always unsure?\n", "\n", - "The key, as with the problem above, is probabilities. We are already comfortable with assigning a probabilistic belief about the location of the dog; now we just have to incorporate the additional uncertainty caused by the sensor noise. Say we think there is a 50% chance that our dog is in front of a specific door and then we get a reading of 'door'. Well, we think that is only likely to be true 0.6 of the time, so we multiply: $0.5 * 0.6= 0.3$. Likewise, if we think the chances that our dog is in front of a wall is 0.1, and the reading is 'door', we would multiply the probability by the chances of a miss: $0.1 * 0.2 = 0.02$.\n", + "The key, as with the problem above, is probabilities. We are already comfortable with assigning a probabilistic belief about the location of the dog; now we just have to incorporate the additional uncertainty caused by the sensor noise. Lets say we get a reading of 'door'. We already said that the sensor is three times as likely to be correct as incorrect, so we should scale the probability distribution by 3 where ever there is a door. If we do that the result will no longer be a probability distribution, but we will learn how to correct that in a moment.\n", "\n", - "However, we more or less chose 0.6 and 0.2 at random; if we multiply the `pos_belief` array by these values the end result will no longer represent a true probability distribution. " + "Let's look at that in Python code. Here I use the variable `z` to denote the measurement as that is the customary choice in the literature. " ] }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 7, "metadata": { "collapsed": false, "scrolled": true @@ -606,141 +480,15 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0.12 0.12 0.04 0.04 0.04 0.04 0.04 0.04 0.12 0.04]\n", - "sum = 0.64\n" + "[ 0.6 0.6 0.2 0.2 0.2 0.2 0.2 0.2 0.6 0.2]\n", + "sum = 3.2\n" ] }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG3ZJREFUeJzt3X9UlvX9x/EXP1JvijgZgSBMoBymEil3LG6pPDt2b1Yz\n", - "t4p0myXaDFamMncaRadU1ModlppQqw6ymoU7neM6yTzgUUMGnUFIM1NHYzM7cNN0huZCj3B9/+gr\n", - "p7sb+aEfuW7k+TjHc24+1+fD9b7fR8/94vLDdQVYlmUJAAAAwAULtLsAAAAA4FJBuAYAAAAMIVwD\n", - "AAAAhhCuAQAAAEMI1wAAAIAhhGsAAADAEMI1AAAAYEiv4bqyslIzZsxQTEyMAgMDVVJS0us33bt3\n", - "r2677TaFhIQoJiZGK1asMFIsAAAA4M96DdcnT57UDTfcoLVr18rhcCggIKDH+cePH9ftt9+uqKgo\n", - "1dXVae3atVqzZo0KCgqMFQ0AAAD4o4D+PKExNDRUGzZs0AMPPHDOOUVFRcrNzVVra6uGDx8uSVq5\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -748,16 +496,14 @@ } ], "source": [ - "def update(map_, belief, z, p_hit, p_miss):\n", + "def update(map_, belief, z, correct_scale):\n", " for i, val in enumerate(map_):\n", " if val == z:\n", - " belief[i] *= p_hit\n", - " else:\n", - " belief[i] *= p_miss\n", + " belief[i] *= correct_scale\n", "\n", "pos_belief = np.array([0.2] * 10)\n", "reading = 1 # 1 is 'door'\n", - "update(hallway, pos_belief, 1, .6, .2)\n", + "update(hallway, pos_belief, z=1, correct_scale=3.)\n", "\n", "print(pos_belief)\n", "print('sum =', sum(pos_belief))\n", @@ -768,14 +514,17 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We can see that this is not a probability distribution because it does not sum to 1.0. But we can see that the code is doing mostly the right thing - the doors are assigned a number (0.12) that is 3 times higher than the walls (0.04). So we can write a bit of code to normalize the result so that the probabilities correctly sum to 1.0." + "We can see that this is not a probability distribution because it does not sum to 1.0. But we can see that the code is doing mostly the right thing - the doors are assigned a number (0.6) that is 3 times higher than the walls (0.2). So we can write a bit of code to normalize the result so that the probabilities correctly sum to 1.0. Normalization is done by dividing each element by the sum of all elements in the list. If this is not clear you should spend a few minutes proving it to yourself algebraically.\n", + "\n", + "Also, it is a bit odd to be talking about \"3 times as likely to be right as wrong\". We are working in probabilities, so let's specify the probability of the sensor being correct, and computing the scale factor from that." ] }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 8, "metadata": { - "collapsed": false + "collapsed": false, + "scrolled": true }, "outputs": [ { @@ -789,135 +538,9 @@ }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG2BJREFUeJzt3X9U1vX9//EHP1IvijgZgSBMoBymEilXLJDSs2NsljO3\n", - "knSbJdocrExl7jSKz6kUtXKHTU2oVQdZzcKdznGdZB7wqCGDziCkmamjsZkduK6mMzQXeoT394++\n", - "XqerC/mhL3lfyP12judcvN6vF+/n9Tx6rgdvX7zfAZZlWQIAAABwyQLtLgAAAAC4UhCuAQAAAEMI\n", - "1wAAAIAhhGsAAADAEMI1AAAAYAjhGgAAADCEcA0AAAAY0mu4rq6u1qxZsxQTE6PAwECVlZX1+k33\n", - "79+vqVOnKiQkRDExMVq1apWRYgEAAAB/1mu4Pn36tG655RatX79eDodDAQEBPc4/efKk7rrrLkVF\n", - "RamhoUHr16/XunXrVFRUZKxoAAAAwB8F9OcJjaGhodq0aZMefPDBC84pKSlRfn6+3G63hg8fLkla\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -929,17 +552,15 @@ " \"\"\" Normalize probability distribution\"\"\"\n", " prob_dist /= sum(prob_dist) \n", "\n", - "def update(map_, belief, z, p_hit, p_miss):\n", + "def update(map_, belief, z, prob_correct):\n", + " scale = prob_correct / (1. - prob_correct)\n", " for i, val in enumerate(map_):\n", " if val == z:\n", - " belief[i] *= p_hit\n", - " else:\n", - " belief[i] *= p_miss\n", - "\n", - " belief = normalize(belief)\n", + " belief[i] *= scale\n", + " normalize(belief)\n", "\n", "pos_belief = np.array([0.2] * 10)\n", - "update(hallway, pos_belief, 1, .6, .2)\n", + "update(hallway, pos_belief, 1, prob_correct=.75)\n", "\n", "print('sum =', sum(pos_belief))\n", "print('probability of door =', pos_belief[0])\n", @@ -951,7 +572,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Normalization is done by dividing each element by the sum of all elements in the list. If this is not clear you should spend a few minutes proving it to yourself algebraically. We can see from the output that the sum is now 1.0, and that the probability of a door vs wall is still three times larger. The result also fits our intuition that the probability of a door must be less than 0.333, and that the probability of a wall must be greater than 0.0. Finally, it should fit our intuition that we have not yet been given any information that would allow us to distinguish between any given door or wall position, so all door positions should have the same value, and the same should be true for wall positions. " + " We can see from the output that the sum is now 1.0, and that the probability of a door vs wall is still three times larger. The result also fits our intuition that the probability of a door must be less than 0.333, and that the probability of a wall must be greater than 0.0. Finally, it should fit our intuition that we have not yet been given any information that would allow us to distinguish between any given door or wall position, so all door positions should have the same value, and the same should be true for wall positions.\n", + " \n", + "This result is called the *posterior*, which is short for *posterior probability distribution*. All this means is a probability distribution that has incorporated the measurement information. Most of the Bayesian and Kalmaning filtering literature uses these terms, so you will have to get used to them. To review, the *prior* is the probability distribution before including the measurement's information, and the *posterier* is the distribution after the measurement has been incorporated. " ] }, { @@ -971,12 +594,12 @@ "\n", "First let's deal with the simple case - assume the movement sensor is perfect, and it reports that the dog has moved one space to the right. How would we alter our `pos_belief` array?\n", "\n", - "I hope after a moment's thought it is clear that we should just shift all the values one space to the right. If we previously thought there was a 50% chance of Simon being at position 3, then after the move to the right we should believe that there is a 50% chance he is at position 4. So let's implement that. Recall that the hallway is circular, so we will use modulo arithmetic to perform the shift correctly" + "I hope after a moment's thought it is clear that we should just shift all the values one space to the right. If we previously thought there was a 50% chance of Simon being at position 3, then after the move to the right we should believe that there is a 50% chance he is at position 4. So let's implement that. Recall that the hallway is circular, so we will use modulo arithmetic to perform the shift correctly." ] }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -991,160 +614,9 @@ }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADnCAYAAADVa3rZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X9QVXX+x/HXvagJiviDUFQCXJVswx9AFmiIrtGSmzmV\n", - "TNlKkuXKmmlUazR8E9PMdBdTVynN0MoMtzabVXLVBJW0kiXSVTMTl2zlskGKZqEG5/uHw51uXOCK\n", - "p+5Fn48ZZ+RzPp9z3udyRl/3zOd8jsUwDEMAAAAALpnV3QUAAAAAlwvCNQAAAGASwjUAAABgEsI1\n", - "AAAAYBLCNQAAAGASwjUAAABgEsI1gCvCkiVL9Otf/1o+Pj6yWq2aNWuWu0tqceLi4mS1Ov63kZ+f\n", - "/7N/niEhIQoNDf3Z9g8AZiJcA2ixFi1aJKvVKqvVqo8//rjBfm+++aamTZummpoaTZs2TRkZGYqL\n", - "i1NGRoasVqtWr179C1bdslkslotqd0VISEi90P7TfV/K/gHgl9TK3QUAQHMtX77c4e+DBw922m/D\n", - "hg2SpFdffdWhT15enqRLC4ZXuhtvvFGfffaZ/P39L2k/jf0Otm3bdkn7BoBfEneuAbRIO3fu1MGD\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1667,12 +786,12 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This is not a coincidence, or the result of a carefully chosen example - it is always true of the predict step. This is inevitable; if our sensor is noisy we will lose a bit of information on every prediction. Suppose we were to perform the prediction an infinite number of times - what would the result be? If we lose information on every step, we must eventually end up with no information at all, and our probabilities will be equally distributed across the `pos_belief` array. Let's try this with 500 iterations.\n" + "This is not a coincidence, or the result of a carefully chosen example - it is always true of the predict step. This is inevitable; if our sensor is noisy we will lose a bit of information on every prediction. Suppose we were to perform the prediction an infinite number of times - what would the result be? If we lose information on every step, we must eventually end up with no information at all, and our probabilities will be equally distributed across the `pos_belief` array. Let's try this with 500 iterations." ] }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -1686,134 +805,9 @@ }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAGy5JREFUeJzt3X9QlWX+//EXP1IPRUxmIggrUC6mEiknNo6Uzo6dXas1\n", - "dyvS3bVEWxe2TGXdaSk+Uypq5Q67akJtNcjWWrjTjNsk64Cjhiw0CyGtmbq07JoNnNPqFpobOsL9\n", - "/aNvZzod5Edech/k+Zhx5pzrvi7u93mPjC9uL+47xLIsSwAAAAAuWKjdBQAAAACXCsI1AAAAYAjh\n", - "GgAAADCEcA0AAAAYQrgGAAAADCFcAwAAAIYQrgEAAABDeg3X1dXVmj17tuLi4hQaGqqysrJev+iB\n", - "Awc0ffp0RUREKC4uTqtXrzZSLAAAABDMeg3Xp0+f1g033KANGzbI4XAoJCSkx/knT57UbbfdppiY\n", - "GDU0NGjDhg1av369ioqKjBUNAAAABKOQ/jyhMTIyUps3b9b9999/3jklJSXKz8+X1+vV8OHDJUlr\n", - "1qxRSUmJPvroowuvGAAAAAhSxvdc19XV6ZZbbvEFa0lyu91qbW3V0aNHTZ8OAAAACBrGw7XH41F0\n", - "dLTf2JfvPR6P6dMBAAAAQSPc9BfsbU/2V7W3t5s+PQAAADBgoqKi/N4bv3I9ZsyYgCvUXq/XdwwA\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1869,7 +863,7 @@ }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 14, "metadata": { "collapsed": true }, @@ -1897,7 +891,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -1908,7 +902,7 @@ "array([ 0.05, 0.05, 0.05, 0.05, 0.1 , 0.45, 0.1 , 0.05, 0.05, 0.05])" ] }, - "execution_count": 46, + "execution_count": 15, "metadata": {}, "output_type": "execute_result" } @@ -1930,7 +924,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -1941,7 +935,7 @@ "array([ 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.1 , 0.45, 0.1 , 0.05])" ] }, - "execution_count": 47, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -1967,7 +961,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1978,7 +972,7 @@ "0.36" ] }, - "execution_count": 48, + "execution_count": 17, "metadata": {}, "output_type": "execute_result" } @@ -2009,7 +1003,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 18, "metadata": { "collapsed": false, "scrolled": true @@ -2019,20 +1013,20 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0.187 0.187 0.062 0.062 0.062 0.062 0.062 0.062 0.187 0.062]\n" + "[ 0.188 0.188 0.062 0.062 0.062 0.062 0.062 0.062 0.188 0.062]\n" ] } ], "source": [ "hallway = np.array([1, 1, 0, 0, 0, 0, 0, 0, 1, 0])\n", "pos_belief = np.array([.1] * 10)\n", - "update(hallway, pos_belief, 1, .6, .2)\n", + "update(hallway, pos_belief, z=1, prob_correct=.75)\n", "print(pos_belief)" ] }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -2041,140 +1035,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0.087 0.175 0.175 0.075 0.062 0.062 0.062 0.062 0.075 0.162]\n" + "[ 0.088 0.175 0.175 0.075 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2348,7 +1089,7 @@ } ], "source": [ - "update(hallway, pos_belief, 1, .6, .2)\n", + "update(hallway, pos_belief, z=1, prob_correct=.75)\n", "print(pos_belief)\n", "bp.bar_plot(pos_belief)" ] @@ -2362,144 +1103,16 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG9RJREFUeJzt3X9U1vX9//EHPxIvijgZgSBMoBymEilXLC4pPTvGZjVz\n", - "Z0W6zRJtDlamMrdG0SkTtXKHTU2oVQdZzcKdznGdZB7wqCGDziCkmamjsZkduGg6Q2OhR3h//vAr\n", - "311dyI98wftC7rdzOIfr9X6/r9fzeh7gPK43r+v99rMsyxIAAACAS+ZvdwEAAADA5YJwDQAAABhC\n", - "uAYAAAAMIVwDAAAAhhCuAQAAAEMI1wAAAIAhhGsAAADAkD7DdWVlpebMmaPo6Gj5+/urpKSkzyc9\n", - "cOCAZsyYoeDgYEVHR2v16tVGigUAAAB8WZ/hur29XTfddJM2bNggh8MhPz+/Xvc/deqU7rjjDkVG\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2508,7 +1121,7 @@ ], "source": [ "predict(pos_belief, 1, kernel)\n", - "update(hallway, pos_belief, 0, .6, .2)\n", + "update(hallway, pos_belief, z=0, prob_correct=.75)\n", "bp.bar_plot(pos_belief)" ] }, @@ -2521,144 +1134,16 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG+NJREFUeJzt3X9U1vX9//EHPxIvijgZgSAkUA5TiZQrFkjp2TE2y5k7\n", - "Jek2S7Q5WJnK3BpFp1LUyh02NaFWHWQ1C3c6x3WSecCjhgw6g5Bmpo7GZnbgoukMiYUe4f35o698\n", - "d3UhP/Il7wu4387xHHi93y9ez+t5vM55XG9evN8+lmVZAgAAAHDJfO0uAAAAABguCNcAAACAIYRr\n", - "AAAAwBDCNQAAAGAI4RoAAAAwhHANAAAAGEK4BgAAAAzpM1xXVFRo7ty5ioyMlK+vr4qLi/v8oYcO\n", - "HdKMGTMUGBioyMhIrV271kixAAAAgDfrM1y3t7fr5ptv1qZNm+RwOOTj49Pr+WfOnNGdd96p8PBw\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2667,7 +1152,7 @@ ], "source": [ "predict(pos_belief, 1, kernel)\n", - "update(hallway, pos_belief, 0, .6, .2)\n", + "update(hallway, pos_belief, z=0, prob_correct=.75)\n", "bp.bar_plot(pos_belief)" ] }, @@ -2704,7 +1189,7 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -2718,136 +1203,9 @@ }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG5dJREFUeJzt3X9U1vX9//EHP1IvijgZgSBMoBymEilXLC4pPTvGZjVz\n", - "pyLdZok2BytTmTtFsVMmauUOmxpQqw6ymgt3do7rJPOARw0ZdAYhzUodxWZ14LqaztBc6BHenz/8\n", - "yrerC/mRL3lfwP12judwvd6vF+/n9TzwPg/evq/3O8CyLEsAAAAALlqg3QUAAAAAwwXhGgAAADCE\n", - "cA0AAAAYQrgGAAAADCFcAwAAAIYQrgEAAABDCNcAAACAIX2G6+rqas2dO1cxMTEKDAxUWVlZn9/0\n", - "wIEDmjlzpkJCQhQTE6M1a9YYKRYAAADwZ32G61OnTumGG27Qxo0b5XA4FBAQ0Ov8EydO6LbbblNU\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2861,7 +1219,7 @@ "measurements = [1, 0, 1, 0, 0]\n", "\n", "for m in measurements:\n", - " update(hallway, pos_belief, m, .6, .2)\n", + " update(hallway, pos_belief, z=m, prob_correct=.75)\n", " predict(pos_belief, 1, kernel)\n", "bp.bar_plot(pos_belief)\n", "print(pos_belief)" @@ -2876,142 +1234,16 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG0FJREFUeJzt3X9U1vX9//EHYOpFESczEIQJlMNUIuWKxSWlZ8euzWrm\n", - "VpFus0Sbg5WpzJ1GsVMqauUOm5pQqw6ymoU7neM6yTzgUUMGnUFIM1NHYzM7cF1NZ2gu9Ajvzx99\n", - "5du1C/mRL3lfwP12judcvN6vF+/n9Tx6nQdvX7zfQZZlWQIAAABwyYLtLgAAAAAYLAjXAAAAgCGE\n", - "awAAAMAQwjUAAABgCOEaAAAAMIRwDQAAABhCuAYAAAAM6TFcV1ZWavbs2YqJiVFwcLBKSkp6/KYH\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3019,7 +1251,7 @@ } ], "source": [ - "update(hallway, pos_belief, m, .6, .2)\n", + "update(hallway, pos_belief, z=m, prob_correct=.75)\n", "predict(pos_belief, 1, kernel)\n", "bp.bar_plot(pos_belief)" ] @@ -3033,472 +1265,16 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAAF9CAYAAADP4URIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XtUVPXeP/D3DHe5KelwUY6Amh7TSBxvgEiF48HMPF5Q\n", - "O/bEZJmYJppPRUdFifRk57DyBl4fNE8WHDuPVpKCoSJpBSHeMRIfNWE4QYphC/kJ398fLmY5ggzI\n", - "nr3Beb/WYgXf2d/5fGcY333Ys2dvlRBCgIiIiIiI2kyt9AKIiIiIiB4WbK6JiIiIiCTC5pqIiIiI\n", - "SCJsromIiIiIJMLmmoiIiIhIImyuiYiIiIgkwuaaiIiIiEgibK6pwwgPD4darcbly5cVW8Pt27ex\n", - "evVq6PV6PPHEE7C3t4darcbGjRsVWxMRUUfXHvK9uLgYq1atQkREBP7whz/AwcEBGo0GkZGR+OKL\n", - 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sc11RUYG6ujp4enqajGs0GhgMhibnlJSU4NKlS0hPT8dHH32EHTt2oKioCM8++yyEENKt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\n19cq320eFpKUlISnn34aAwYMgMlkwnvvvQez2YxZs2YBAJK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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3510,7 +1286,7 @@ " measurements = [0, 1, 0, 1, 0, 0]\n", "\n", " for i, m in enumerate(measurements):\n", - " update(hallway, pos_belief, m, .6, .2)\n", + " update(hallway, pos_belief, z=m, prob_correct=.75)\n", " predict(pos_belief, 1, kernel)\n", " plt.subplot(3, 2, i+1)\n", " bp.bar_plot(pos_belief, title='step{}'.format(i+1))" @@ -3549,143 +1325,16 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG5dJREFUeJzt3X9U1vX9//EHP1IvijgZgSBMoBymEilXLC4pPTvGZjVz\n", - "pyLdZok2BytTmTtFsVMmauUOmxpQqw6ymgt3do7rJPOARw0ZdAYhzUodxWZ14LqaztBc6BHenz/8\n", - "yrerC/mRL3lfwP12judwvd6vF+/n9TzwPg/evq/3O8CyLEsAAAAALlqg3QUAAAAAwwXhGgAAADCE\n", - "cA0AAAAYQrgGAAAADCFcAwAAAIYQrgEAAABDCNcAAACAIX2G6+rqas2dO1cxMTEKDAxUWVlZn9/0\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3733,7 +1382,7 @@ }, { "cell_type": "code", - "execution_count": 76, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -3741,13 +1390,11 @@ "source": [ "class Train(object):\n", "\n", - " def __init__(self, track, kernel=[1.], sense_error=.1, no_sense_error=.05):\n", + " def __init__(self, track, kernel=[1.], sensor_accuracy=.9):\n", " self.track = track\n", " self.pos = 0\n", " self.kernel = kernel\n", - " self.sense_error = sense_error\n", - " self.no_sense_error = no_sense_error\n", - "\n", + " self.sensor_accuracy = sensor_accuracy\n", "\n", " def move(self, distance=1):\n", " \"\"\" move in the specified direction with some small chance of error\"\"\"\n", @@ -3772,12 +1419,11 @@ "\n", " # insert random sensor error\n", " r = random.random()\n", - " if r < self.sense_error:\n", + " if r > self.sensor_accuracy:\n", " if random.random() > 0.5:\n", " pos += 1\n", " else:\n", " pos -= 1\n", - " print(' ***sense error***')\n", " return pos" ] }, @@ -3790,39 +1436,36 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [], "source": [ - "def simulate(iterations, kernel, sense_error, \n", - " no_sense_error, move_distance,\n", - " do_print=True):\n", + "def simulate(iterations, kernel, sensor_accuracy, \n", + " move_distance, do_print=True):\n", " track = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])\n", "\n", " pos_belief = np.array([0.01] * 10)\n", " pos_belief[0] = .9\n", " normalize(pos_belief)\n", " \n", - " robot = Train(track, kernel, sense_error, no_sense_error)\n", - "\n", + " robot = Train(track, kernel, sensor_accuracy)\n", " for i in range(iterations):\n", " robot.move(distance=move_distance)\n", " m = robot.sense()\n", " if do_print:\n", - " print('time {}: pos {}, sense {}, at magnet {}'.format(\n", + " print('time {}: pos {}, sensed {}, at position {}'.format(\n", " i, robot.pos, m, track[robot.pos]))\n", "\n", "\n", - " update(track, pos_belief, m, 1. - sense_error, no_sense_error)\n", + " update(track, pos_belief, m, sensor_accuracy)\n", " if do_print:\n", " print(' update ', pos_belief)\n", " ind = np.argmax(pos_belief)\n", " if do_print:\n", " print(' predicted position is {} with confidence {:.4}%:'.format(\n", " ind, pos_belief[ind]*100))\n", - "\n", " \n", " if i < iterations - 1:\n", " predict(pos_belief, move_distance, kernel) \n", @@ -3830,11 +1473,12 @@ " print(' predict', pos_belief)\n", "\n", " bp.bar_plot(pos_belief)\n", - " print()\n", - " print('final position i', robot.pos)\n", - " i = np.argmax(pos_belief)\n", - " print('predicted position is {} with confidence {:.4}%:'.format(\n", - " i, pos_belief[i]*100))" + " if do_print:\n", + " print()\n", + " print('final position is', robot.pos)\n", + " i = np.argmax(pos_belief)\n", + " print('predicted position is {} with confidence {:.4}%:'.format(\n", + " i, pos_belief[i]*100))" ] }, { @@ -3846,7 +1490,7 @@ }, { "cell_type": "code", - "execution_count": 80, + "execution_count": 29, "metadata": { "collapsed": false, "scrolled": false @@ -3856,156 +1500,31 @@ "name": "stdout", "output_type": "stream", "text": [ - "time 0: pos 4.0, sense 4.0, at magnet 4\n", - " update [ 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.]\n", - " predicted position is 4 with confidence 100.0%:\n", - " predict [ 0. 0. 0. 0. 0. 0. 0. 0. 1. 0.]\n", - "time 1: pos 8.0, sense 8.0, at magnet 8\n", + "time 0: pos 4.0, sensed 4.0, at position 4\n", + " update [ 0.08 0. 0. 0. 0.91 0. 0. 0. 0. 0. ]\n", + " predicted position is 4 with confidence 91.07%:\n", + " predict [ 0. 0. 0. 0. 0.08 0. 0. 0. 0.91 0. ]\n", + "time 1: pos 8.0, sensed 8.0, at position 8\n", " update [ 0. 0. 0. 0. 0. 0. 0. 0. 1. 0.]\n", - " predicted position is 8 with confidence 100.0%:\n", + " predicted position is 8 with confidence 99.99%:\n", " predict [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 0.]\n", - "time 2: pos 2.0, sense 2.0, at magnet 2\n", + "time 2: pos 2.0, sensed 2.0, at position 2\n", " update [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 0.]\n", " predicted position is 2 with confidence 100.0%:\n", " predict [ 0. 0. 0. 0. 0. 0. 1. 0. 0. 0.]\n", - "time 3: pos 6.0, sense 6.0, at magnet 6\n", + "time 3: pos 6.0, sensed 6.0, at position 6\n", " update [ 0. 0. 0. 0. 0. 0. 1. 0. 0. 0.]\n", " predicted position is 6 with confidence 100.0%:\n", "\n", - "final position is : 6.0\n", + "final position is 6.0\n", "predicted position is 6 with confidence 100.0%:\n" ] }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAGztJREFUeJzt3X9QVXX+x/EXYOKlyK+mIAoJlIupxCo3Jq6Wzo7eWfth\n", - "7lSku2uJtq5smcq601LslCtq5Q67akJt2yDVVrjTjNsk64Cjhiw0CwGtmbkWu2Qj97a6huSGjnC+\n", - "f/SV7969yA/94LkXno8ZZ+Rzz7nnxXuuMy+Oh3NCLMuyBAAAAOCyhdodAAAAABgoKNcAAACAIZRr\n", - "AAAAwBDKNQAAAGAI5RoAAAAwhHINAAAAGEK5BgAAAAzpsVxXVFRo3rx5io2NVWhoqIqLi3t804MH\n", - "D2rmzJmKiIhQbGys1q1bZyQsAAAAEMh6LNdnzpzRzTffrM2bN8vhcCgkJKTb7U+fPq05c+YoJiZG\n", 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4013,11 +1532,12 @@ } ], "source": [ + "import random\n", + "\n", "random.seed(3)\n", "np.set_printoptions(precision=2, suppress=True)\n", - "simulate(4, kernel=[1.], sense_error=0.,\n", - " no_sense_error=0., move_distance=4,\n", - " do_print=True)" + "simulate(4, kernel=[1.], sensor_accuracy=.999,\n", + " move_distance=4, do_print=True)" ] }, { @@ -4029,171 +1549,40 @@ }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 30, "metadata": { - "collapsed": false, - "scrolled": false + "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "time 0: pos 4.0, sense 4.0, at magnet 4\n", - " update [ 0.85 0.01 0.01 0.01 0.08 0.01 0.01 0.01 0.01 0.01]\n", - " predicted position is 0 with confidence 84.91%:\n", - " predict [ 0.01 0.01 0.01 0.09 0.68 0.09 0.01 0.02 0.06 0.02]\n", - "time 1: pos 8.0, sense 8.0, at magnet 8\n", - " update [ 0.01 0.01 0.01 0.07 0.47 0.07 0.01 0.01 0.35 0.01]\n", - " predicted position is 4 with confidence 47.44%:\n", - " predict [ 0.01 0.04 0.28 0.04 0.01 0.01 0.01 0.1 0.39 0.1 ]\n", - " ***sense error***\n", - "time 2: pos 2.0, sense 1.0, at magnet 2\n", - " update [ 0.01 0.27 0.21 0.03 0.01 0.01 0.01 0.08 0.3 0.08]\n", - " predicted position is 8 with confidence 29.96%:\n", - " predict [ 0.02 0.09 0.25 0.09 0.04 0.24 0.2 0.05 0.01 0.01]\n", - "time 3: pos 6.0, sense 6.0, at magnet 6\n", - " update [ 0.01 0.04 0.11 0.04 0.02 0.1 0.67 0.02 0. 0. ]\n", - " predicted position is 6 with confidence 66.84%:\n", + "time 0: pos 4.0, sensed 4.0, at position 4\n", + " update [ 0.84 0.01 0.01 0.01 0.08 0.01 0.01 0.01 0.01 0.01]\n", + " predicted position is 0 with confidence 84.11%:\n", + " predict [ 0.01 0.01 0.01 0.09 0.67 0.09 0.01 0.02 0.07 0.02]\n", + "time 1: pos 8.0, sensed 8.0, at position 8\n", + " update [ 0.01 0.01 0.01 0.06 0.43 0.06 0.01 0.01 0.4 0.01]\n", + " predicted position is 4 with confidence 43.44%:\n", + " predict [ 0.01 0.05 0.32 0.05 0.01 0.01 0.01 0.09 0.36 0.09]\n", + "time 2: pos 2.0, sensed 2.0, at position 2\n", + " update [ 0. 0.01 0.81 0.01 0. 0. 0. 0.03 0.1 0.03]\n", + " predicted position is 2 with confidence 81.09%:\n", + " predict [ 0.01 0.03 0.09 0.03 0.01 0.09 0.65 0.09 0. 0. ]\n", + "time 3: pos 5.0, sensed 5.0, at position 5\n", + " update [ 0. 0.02 0.05 0.02 0. 0.48 0.37 0.05 0. 0. ]\n", + " predicted position is 5 with confidence 47.83%:\n", "\n", - "final position is : 6.0\n", - "predicted position is 6 with confidence 66.84%:\n" + "final position is 5.0\n", + "predicted position is 5 with confidence 47.83%:\n" ] }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG/tJREFUeJzt3X9QVXUe//EXPxIvRX4zAkFJoFxMJVa5MXG1cnbsztoP\n", - "c2eLdHct0dZky1TW3ZaiKVfUyh02NaG2bZBqK9xpxm2SdcBRQxaahZDWTF2LXbKRe1tdQ3IDRzjf\n", - "P/rKd+9e5Id+4Fzg+ZhxBj7nfPi877vr9LrHw/kEWZZlCQAAAMAlC7a7AAAAAGCoIFwDAAAAhhCu\n", - 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4202,10 +1591,8 @@ ], "source": [ "random.seed(3)\n", - "np.set_printoptions(precision=2, suppress=True)\n", - "simulate(4, kernel=[.1, .8, .1], sense_error=0.2,\n", - " no_sense_error=0.1, move_distance=4,\n", - " do_print=True)" + "simulate(4, kernel=[.1, .8, .1], sensor_accuracy=.9,\n", + " move_distance=4, do_print=True)" ] }, { @@ -4214,158 +1601,22 @@ "source": [ "Here we see that there was a sense error at time 1, but we are still quite confident in our position. \n", "\n", - "Now lets run a very long simulation to see if the estimate degrades or fails over time." + "Now lets run a very long simulation and see how the filter responds to errors." ] }, { "cell_type": "code", - "execution_count": 83, + "execution_count": 31, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n", - "final position is : 6.0\n", - "predicted position is 6 with confidence 43.52%:\n" - ] - }, { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtcAAADaCAYAAABtj26qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAG6RJREFUeJzt3X9UlvUd//EXPxLBiG9GIAgJlMNUIuWOE7eUnh3jjGrm\n", - "zop0myXaHKxMZe40ip0yUSt3WGpArXWQ1Szc6RzXSeYBjxoy6AxCmpk6is3swE3TGZILPcL1/aOv\n", - "fHd3Iz/0I9cNPB/neA58rs+H632/vTm+uPxwXT6WZVkCAAAAcNl87S4AAAAAGCkI1wAAAIAhhGsA\n", - "AADAEMI1AAAAYAjhGgAAADCEcA0AAAAYQrgGAAAADOk3XFdVVWnevHmKioqSr6+vSktL+/2iBw8e\n", - "1OzZsxUUFKSoqCitXbvWSLEAAACAN+s3XJ85c0a33HKLNm3apMDAQPn4+PQ5//Tp07rrrrsUERGh\n", - "+vp6bdq0SRs3blRBQYGxogEAAABv5DOYJzQGBwersLBQDz300EXnFBcXKzc3V21tbQoICJAkrVu3\n", - "TsXFxfr8888vv2IAAADASxnfc11bW6s77rijJ1hLUlpamlpaWnTs2DHTpwMAAAC8hvFw7XK5FB4e\n", - "7jZ24XOXy2X6dAAAAIDX8Df9Bfvbk/2/2tvbTZ8eAAAAGDIhISFunxu/cj1hwgSPK9RtbW09xwAA\n", - "AICRyni4TklJ0f79+3X27NmescrKSk2cOFGTJk0yfToAAADAa/S7LeTMmTNqamqSJHV3d+vYsWNq\n", - "bGzUddddp+joaOXm5qqurk67d++WJP3oRz/SmjVrtHjxYuXl5eno0aN6/vnn9cwzz/R5nm9fUjet\n", - 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uz9r8nTdHS780gvlu/drWrm+rta1d31ZrW7u+pfLd6JnrnJwcBAUFISgoCJWVlUhISEBQ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\nsbu7i8vLS+zs7KC8vBwbGxvSZojH43A4HDAYDMj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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -4373,9 +1624,80 @@ } ], "source": [ - "simulate(4, kernel=[.1, .8, .1], sense_error=0.2,\n", - " no_sense_error=0.1, move_distance=4,\n", - " do_print=False)" + "with figsize(y=5):\n", + " for i in range (4):\n", + " random.seed(3)\n", + " plt.subplot(321+i)\n", + " simulate(148+i, kernel=[.1, .8, .1], sensor_accuracy=.8,\n", + " move_distance=4, do_print=False)\n", + " plt.title ('iteration {}'.format(148+i))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that there was a problem on iteration 149 as the confidence degrades. But within a few iterations the filter is able to correct itself and regain confidence in the estimated position." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Bayes Theorem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We developed the math in this chapter merely by reasoning about the information we have at each moment. In the process we discovered *Bayes Theorem*. We will go into the specifics of the math of Bayes theorem later in the book. For now we will take a more intuitive approach. Recall from the preface that Bayes theorem tells us how to compute the probability of an event given previous information. That is exactly what we have been doing in this chapter. With luck our code should match the Bayes Theorem equation! \n", + "\n", + "Bayes theorem is written as\n", + "\n", + "$$P(A|B) = \\frac{P(B | A)\\, P(A)}{P(B)}\\cdot$$\n", + "\n", + "If you are not familiar with this notation, let's review. $P(A)$ means the probability of event $A$. If $A$ is the event of a fair coin landing heads, then $P(A) = 0.5$.\n", + "\n", + "$P(A|B)$ is called a *conditional probability*. That is, it represents the probability of $A$ happening *if* $B$ happened. For example, it is more likely to rain today if it also rained yesterday. We'd write that as $P(rain_{yesterday}|rain_{today})$.\n", + "\n", + "In Bayesian statistics $P(A)$ is the *prior*, and $P(A|B)$ is the *posterior*. To see why, let's rewrite the equation in terms of our problem. We will use $x_i$ for the position at *i*, and $Z$ for the measurement. Hence, we want to know $P(x_i|Z)$, that is, the probability of the dog being at $x_i$ given the measurement $Z$. \n", + "\n", + "So, let's plug that into the equation and solve it.\n", + "\n", + "$$P(x_i|Z) = \\frac{P(Z|x_i) P(x_i)}{P(Z)}$$\n", + "\n", + "That looks ugly, but it is actually quite simple. Let's just figure out what each term on the right means. First is $P(Z|x_i)$. This is the probability for the measurement at every cell $x_i$. $P(Z)$ is just the probability of the measurement. We multiply those together. This is just the unnormalized multiplication in the `update()` function. \n", + "\n", + " for i, val in enumerate(map_):\n", + " if val == z:\n", + " belief[i] *= correct_scale\n", + " else:\n", + " belief[i] *= 1.\n", + "\n", + "I added the `else` here, which has no mathematical effect, to point out that every element in $x$ (called `belief` in the code) is multiplied by a probability. \n", + "\n", + "The last term to consider is the denominator $P(Z)$. This is the probability of getting the measurement $Z$ without taking the location into account. We compute that by taking the sum of $x$, or `sum(belief)` in the code. That is how we compute the normalization! So, the `update()` function is doing nothing more than computing Bayes theorem. I could have just given you Bayes theorem and then written a function, but I doubt that would have been illuminating unless you already know Bayesian statistics. Instead, we figured out what to do just by reasoning about the situation, and so of course the resulting code ended up implementing Bayes theorem. Students spend a lot of time struggling to understand this theorem; I hope you found it relatively straightforward." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Total Probability Theorem\n", + "\n", + "We know now the formal mathematics behind the `update()` function; what about the `predict()` function? `predict()` implements the *total probability theorem*. Let's recall what `predict()` computed. It computed the probability of being at any given position given the probability of all the possible movement events. Let's express that as an equation. The probability of being at any position at time t can be written as $P(X_i^t)$. We computed that as the sum of the prior at time t-1 $P(X_i^{t-1})$ multiplied by the probability of moving from cell $x_j$ to $x_i$. That is\n", + "\n", + "$$P(X_i^t) = \\sum_j P(X_i^{t-1}) P(x_i | x_j)$$\n", + "\n", + "That equation is called the *total probability theorem*. Quoting from Wikipedia [*] \"It expresses the total probability of an outcome which can be realized via several distinct events\". Again, I could have just given you that equation and implemented `predict()`, but your chances of understanding why the equation works would be slim. As a reminder, here is the code that computes this equation\n", + "\n", + " for i in range(N):\n", + " for k in range (kN):\n", + " index = (i + (width-k) - offset) % N\n", + " result[i] += prob_dist[index] * kernel[k]" ] }, { @@ -4389,18 +1711,20 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The code is very small, but the result is huge! We will go into the math more later, but we have implemented a form of a Bayesian filter. It is commonly called a Histogram filter. The Kalman filter is also a Bayesian filter, and uses this same logic to produce its results. The math is a bit more complicated, but not by much. For now, we will just explain that Bayesian statistics compute the likelihood of some estimate about the present based on imperfect knowledge of the past. If we know there are two doors in a row, and the sensor reported two doors in a row, it is likely that we are positioned near those doors. Bayesian statistics just formalize that example, and Bayesian filters formalize filtering data based on that math by implementing the predict->update->predict->update process. \n", - "\n", - "We have learned how to start with no information and derive information from noisy sensors. Even though the sensors in this chapter are very noisy (most sensors are more than 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the predict step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowledge, allowing us to converge on the most likely result.\n", + "The code is very small, but the result is huge! We have implemented a form of a Bayesian filter. We have learned how to start with no information and derive information from noisy sensors. Even though the sensors in this chapter are very noisy (most sensors are more than 80% accurate, for example) we quickly converge on the most likely position for our dog. We have learned how the predict step always degrades our knowledge, but the addition of another measurement, even when it might have noise in it, improves our knowledge, allowing us to converge on the most likely result.\n", "\n", "If you followed the math carefully you will realize that all of this math is exact. The bar charts that we are displaying are not an *estimate* or *guess* - they are mathematically exact results that exactly represent our knowledge. The knowledge is probabilistic, to be sure, but it is exact, and correct.\n", "\n", - "However, we are a long way from tracking an airplane or a car. This code only handles the 1 dimensional case, whereas cars and planes operate in 2 or 3 dimensions. Also, our position vector is *multimodal*. It expresses multiple beliefs at once. Imagine if your GPS told you \"it's 20% likely that you are here, but 10% likely that you are on this other road, and 5% likely that you are at one of 14 other locations\". That would not be very useful information. Also, the data is discrete. We split an area into 10 (or whatever) different locations, whereas in most real world applications we want to work with continuous data. We want to be able to represent moving 1 km, 1 meter, 1 mm, or any arbitrary amount, such as 2.347 cm. \n", + "Furthermore, through basic reasoning we were able to discover two extremely important theorems: Bayes theorem and the total probability theorem. I hope you spent time on those section as in almost any other source they will express filtering algorithms in terms of these two theorems. It will be your job to understand what these equations mean and how to turn them into code. \n", + "\n", + "This book is mostly about the Kalman filter. In the g-h filter chapter I told you that the Kalman filter is a type of g-h filter. It is also a type of Bayesian filter. It also uses Bayes theorem and the total probability theorem to filter data, although with a different set of assumptions and conditions than used in this chapter.\n", + "\n", + "The discrete Bayes filter allows us to filter sensors and track an object, but we are a long way from tracking an airplane or a car. This code only handles the 1 dimensional case, whereas cars and planes operate in 2 or 3 dimensions. Also, our position vector is *multimodal*. It expresses multiple beliefs at once. Imagine if your GPS told you \"it's 20% likely that you are here, but 10% likely that you are on this other road, and 5% likely that you are at one of 14 other locations\". That would not be very useful information. Also, the data is discrete. We split an area into 10 (or whatever) different locations, whereas in most real world applications we want to work with continuous data. We want to be able to represent moving 1 km, 1 meter, 1 mm, or any arbitrary amount, such as 2.347 cm. \n", "\n", "Finally, the bar charts may strike you as being a bit less certain than we would want. A 25% certainty may not give you a lot of confidence in the answer. Of course, what is important here is the ratio of this probability to the other probabilities in your vector. If the next largest bar is 23% then we are not very knowledgeable about our position, whereas if the next largest is 3% we are in fact quite certain. But this is not clear or intuitive. However, there is an extremely important insight that Kalman filters implement that will significantly improve our accuracy from the same data.\n", "\n", "\n", - "**If you can understand this chapter you will be able to understand and implement Kalman filters.** I cannot stress this enough. If anything is murky, go back and reread this chapter and play with the code. The rest of this book will build on the algorithms that we use here. If you don't intuitively understand why this histogram filter works, and can't at least work through the math, you will have little success with the rest of the material. However, if you grasp the fundamental insight - multiplying probabilities when we measure, and shifting probabilities when we update, leading to a converging solution - then you understand everything important you need to grasp the Kalman filter. " + "**If you can understand this chapter you will be able to understand and implement Kalman filters.** I cannot stress this enough. If anything is murky, go back and reread this chapter and play with the code. The rest of this book will build on the algorithms that we use here. If you don't intuitively understand why this filter works, and can't at least work through the math, you will have little success with the rest of the material. However, if you grasp the fundamental insight - multiplying probabilities when we measure, and shifting probabilities when we update leads to a converging solution - then you understand everything important you need for the Kalman filter. " ] }, { @@ -4435,6 +1759,9 @@ " \n", "* [?] Wikipedia. \"Convolution\"\n", "http://en.wikipedia.org/wiki/Convolution\n", + "\n", + "* [?] Wikipedia. \"Law of total probability\"\n", + " http://en.wikipedia.org/wiki/Law_of_total_probability\n", " " ] } diff --git a/styles/custom2.css b/styles/custom2.css index 45ce820..c358811 100644 --- a/styles/custom2.css +++ b/styles/custom2.css @@ -40,21 +40,21 @@ } .text_cell_render h2 { font-weight: 200; - font-size: 20pt; + font-size: 16pt; font-style: italic; line-height: 100%; color:#c76c0c; margin-bottom: 0.5em; margin-top: 1.5em; - display: block; - white-space: nowrap; + display: inline; + white-space: wrap; } h3 { font-family: 'Open sans',verdana,arial,sans-serif; } .text_cell_render h3 { - font-weight: 300; - font-size: 18pt; + font-weight: 200; + font-size: 14pt; line-height: 100%; color:#d77c0c; margin-bottom: 0.5em; @@ -66,8 +66,8 @@ font-family: 'Open sans',verdana,arial,sans-serif; } .text_cell_render h4 { - font-weight: 300; - font-size: 16pt; + font-weight: 100; + font-size: 14pt; color:#d77c0c; margin-bottom: 0.5em; margin-top: 0.5em; @@ -78,7 +78,7 @@ font-family: 'Open sans',verdana,arial,sans-serif; } .text_cell_render h5 { - font-weight: 300; + font-weight: 200; font-style: normal; color: #1d3b84; font-size: 16pt; @@ -209,6 +209,9 @@ }, displayAlign: 'center', // Change this to 'center' to center equations. "HTML-CSS": { + availableFonts: ["TeX"], + preferredFont: "TeX", + scale:85, styles: {'.MathJax_Display': {"margin": 4}} } });