From 167a904de548fd7a5394113484d28f8f62aae913 Mon Sep 17 00:00:00 2001 From: Roger Labbe Date: Mon, 11 May 2015 21:21:56 -0700 Subject: [PATCH] All examples use 10 cells now. Due to uneven editing some of the examples used 10 hallway locations, and others used less. I normalized on 10. --- .gitignore | 11 ++--- 02_Discrete_Bayes.ipynb | 94 ++++++++++++++++------------------------- 2 files changed, 42 insertions(+), 63 deletions(-) diff --git a/.gitignore b/.gitignore index 08cb68b..457d77a 100644 --- a/.gitignore +++ b/.gitignore @@ -1,5 +1,6 @@ -*.bak -.ipynb_checkpoints -__pycache__ -*.pyc -short.pdf +*.bak +.ipynb_checkpoints +__pycache__ +*.pyc +short.pdf +13* diff --git a/02_Discrete_Bayes.ipynb b/02_Discrete_Bayes.ipynb index 4b2fc2b..355c38d 100644 --- a/02_Discrete_Bayes.ipynb +++ b/02_Discrete_Bayes.ipynb @@ -16,7 +16,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -250,7 +250,7 @@ "" ] }, - "execution_count": 1, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -302,7 +302,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -324,7 +324,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -342,7 +342,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -351,7 +351,7 @@ "data": { "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -378,7 +378,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -420,7 +420,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -471,7 +471,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 8, "metadata": { "collapsed": false, "scrolled": true @@ -489,7 +489,7 @@ "data": { "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -522,7 +522,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 9, "metadata": { "collapsed": false, "scrolled": true @@ -541,7 +541,7 @@ "data": { "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -600,7 +600,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -609,15 +609,15 @@ "name": "stdout", "output_type": "stream", "text": [ - "belief before predict = [ 0.4 0.1 0.2 0.3]\n", - "belief after predict = [ 0.3 0.4 0.1 0.2]\n" + "belief before predict = [ 0.35 0.1 0.2 0.3 0. 0. 0. 0. 0. 0.05]\n", + "belief after predict = [ 0.05 0.35 0.1 0.2 0.3 0. 0. 0. 0. 0. ]\n" ] }, { "data": { - "image/png": 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UKH17cnIyRo8ebXIfPz8/zJ07F6Wlpfp518nJyWjfvj1cXFwM+j54G52IiIiI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\n", "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -635,7 +635,7 @@ " result[i] = pos_belief[(i-move) % n]\n", " pos_belief[:] = result # copy back to original array\n", " \n", - "pos_belief = np.array([.4, .1, .2, .3])\n", + "pos_belief = np.array([.35, .1, .2, .3, 0, 0, 0, 0, 0, .05])\n", "print('belief before predict =', pos_belief)\n", "bp.bar_plot(pos_belief, title='Before prediction')\n", "\n", @@ -671,7 +671,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -680,14 +680,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0. 0. 0. 0. 0.1 0.8 0.1 0. ]\n" + "[ 0. 0. 0. 0. 0.1 0.8 0.1 0. 0. 0. ]\n" ] }, { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -705,7 +705,7 @@ " pos_belief[(i-move+1) % n] * p_under) \n", " pos_belief[:] = result\n", "\n", - "pos_belief = np.array([0., 0., 0., 1., 0., 0., 0., 0.])\n", + "pos_belief = np.array([0., 0., 0., 1., 0., 0., 0., 0., 0., 0.])\n", "predict(pos_belief, 2, .8, .1, .1)\n", "print(pos_belief)\n", "bp.bar_plot(pos_belief)" @@ -720,7 +720,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 13, "metadata": { "collapsed": false, "scrolled": true @@ -730,14 +730,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0. 0. 0. 0.04 0.38 0.52 0.06 0. ]\n" + "[ 0. 0. 0. 0.04 0.38 0.52 0.06 0. 0. 0. ]\n" ] }, { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -745,7 +745,7 @@ } ], "source": [ - "pos_belief = np.array([0, 0, .4, .6, 0, 0, 0, 0])\n", + "pos_belief = np.array([0, 0, .4, .6, 0, 0, 0, 0, 0, 0])\n", "predict(pos_belief, 2, .8, .1, .1)\n", "print(pos_belief)\n", "bp.bar_plot(pos_belief)" @@ -762,16 +762,16 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { - "image/png": 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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -858,7 +858,7 @@ "\n", "If you look at that equation and compare it to the `predict()` function you can see that they are doing the same thing. \n", "\n", - "I would love to go on and on about convolution, but we have a filter to implement. Khan Academy [?] has a good mathematical introduction to convolution, and Wikipedia has some excellent animations of convolutions [?]. But the general idea is already clear to you. You slide across an array, multiplying the neighbors of the current cell with the values of a second array. This second array is called the *kernel*. In our example above we used .8 for the probability of moving to the correct location, .1 for undershooting, and .1 for overshooting. We make a kernel of this with the array `[0.1, 0.8, 0.1]`. So all we need to do is write a loop that goes over each element of our array, multiplying by the kernel, and summing the results." + "I would love to go on and on about convolution, but we have a filter to implement. Khan Academy [4] has a good mathematical introduction to convolution, and Wikipedia has some excellent animations of convolutions [5]. But the general idea is already clear to you. You slide across an array, multiplying the neighbors of the current cell with the values of a second array. This second array is called the *kernel*. In our example above we used .8 for the probability of moving to the correct location, .1 for undershooting, and .1 for overshooting. We make a kernel of this with the array `[0.1, 0.8, 0.1]`. So all we need to do is write a loop that goes over each element of our array, multiplying by the kernel, and summing the results." ] }, { @@ -959,28 +959,6 @@ "You probably will never be able to compute exact probabilities for the kernel of a real problem. It is usually enough to be approximately right. What you *must* do is ensure that the terms of the kernel sum to 1. The kernel expresses the probability of any given move, and the sum of any probability distribution must be 1." ] }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "0.36" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "1-sum([.02, .1, .2, .2, .1, .02])" - ] - }, { "cell_type": "markdown", "metadata": {}, @@ -1698,7 +1676,7 @@ "\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", + "That equation is called the *total probability theorem*. Quoting from Wikipedia [6] \"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", @@ -1759,14 +1737,14 @@ " https://www.udacity.com/course/cs373\n", " \n", " \n", - " * [?] Khan Acadamy. \"Introduction to the Convolution\"\n", + " * [4] Khan Acadamy. \"Introduction to the Convolution\"\n", " https://www.khanacademy.org/math/differential-equations/laplace-transform/convolution-integral/v/introduction-to-the-convolution\n", " \n", " \n", - "* [?] Wikipedia. \"Convolution\"\n", + "* [5] Wikipedia. \"Convolution\"\n", "http://en.wikipedia.org/wiki/Convolution\n", "\n", - "* [?] Wikipedia. \"Law of total probability\"\n", + "* [6] Wikipedia. \"Law of total probability\"\n", " http://en.wikipedia.org/wiki/Law_of_total_probability\n", " " ]