395 lines
15 KiB
Plaintext
395 lines
15 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "view-in-github",
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"colab_type": "text"
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},
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"source": [
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"<a href=\"https://colab.research.google.com/github/udlbook/udlbook/blob/main/Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "t9vk9Elugvmi"
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},
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"source": [
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"# **Notebook 17.1: Latent variable models**\n",
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"\n",
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"This notebook investigates a non-linear latent variable model similar to that in figures 17.2 and 17.3 of the book.\n",
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"\n",
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"Work through the cells below, running each cell in turn. In various places you will see the words \"TODO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n",
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"\n",
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"Contact me at udlbookmail@gmail.com if you find any mistakes or have any suggestions."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "OLComQyvCIJ7"
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import scipy\n",
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"from matplotlib.colors import ListedColormap\n",
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"from matplotlib import cm"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "IyVn-Gi-p7wf"
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},
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"source": [
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"We'll assume that our base distribution over the latent variables is a 1D standard normal so that\n",
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"\n",
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"\\begin{equation}\n",
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"Pr(z) = \\text{Norm}_{z}[0,1]\n",
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"\\end{equation}\n",
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"\n",
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"As in figure 17.2, we'll assume that the output is two dimensional, we need to define a function that maps from the 1D latent variable to two dimensions. Usually, we would use a neural network, but in this case, we'll just define an arbitrary relationship.\n",
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"\n",
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"\\begin{align}\n",
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"x_{1} &=& 0.5\\cdot\\exp\\Bigl[\\sin\\bigl[2+ 3.675 z \\bigr]\\Bigr]\\\\\n",
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"x_{2} &=& \\sin\\bigl[2+ 2.85 z \\bigr]\n",
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"\\end{align}"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "ZIfQwhd-AV6L"
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},
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"outputs": [],
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"source": [
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"# The function that maps z to x1 and x2\n",
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"def f(z):\n",
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" x_1 = np.exp(np.sin(2+z*3.675)) * 0.5\n",
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" x_2 = np.cos(2+z*2.85)\n",
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" return x_1, x_2"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "KB9FU34onW1j"
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},
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"source": [
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"Let's plot the 3D relation between the two observed variables $x_{1}$ and $x_{2}$ and the latent variables $z$ as in figure 17.2 of the book. We'll use the opacity to represent the prior probability $Pr(z)$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "lW08xqAgnP4q"
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},
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"outputs": [],
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"source": [
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"def draw_3d_projection(z,pr_z, x1,x2):\n",
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" alpha = pr_z / np.max(pr_z)\n",
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" ax = plt.axes(projection='3d')\n",
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" fig = plt.gcf()\n",
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" fig.set_size_inches(18.5, 10.5)\n",
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" for i in range(len(z)-1):\n",
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" ax.plot([z[i],z[i+1]],[x1[i],x1[i+1]],[x2[i],x2[i+1]],'r-', alpha=pr_z[i])\n",
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" ax.set_xlabel('$z$',)\n",
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" ax.set_ylabel('$x_1$')\n",
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" ax.set_zlabel('$x_2$')\n",
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" ax.set_xlim(-3,3)\n",
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" ax.set_ylim(0,2)\n",
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" ax.set_zlim(-1,1)\n",
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" ax.set_box_aspect((3,1,1))\n",
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" plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "9DUTauMi6tPk"
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},
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"outputs": [],
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"source": [
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"# Compute the prior\n",
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"def get_prior(z):\n",
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" return scipy.stats.multivariate_normal.pdf(z)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "PAzHq461VqvF"
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},
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"outputs": [],
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"source": [
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"# Define the latent variable values\n",
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"z = np.arange(-3.0,3.0,0.01)\n",
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"# Find the probability distribution over z\n",
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"pr_z = get_prior(z)\n",
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"# Compute x1 and x2 for each z\n",
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"x1,x2 = f(z)\n",
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"# Plot the function\n",
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"draw_3d_projection(z,pr_z, x1,x2)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "sQg2gKR5zMrF"
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},
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"source": [
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"The likelihood is defined as:\n",
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"\\begin{align}\n",
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" Pr(x_1,x_2|z) &=& \\text{Norm}_{[x_1,x_2]}\\Bigl[\\mathbf{f}[z],\\sigma^{2}\\mathbf{I}\\Bigr]\n",
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"\\end{align}\n",
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"\n",
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"so we will also need to define the noise level $\\sigma^2$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "In_Vg4_0nva3"
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},
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"outputs": [],
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"source": [
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"sigma_sq = 0.04"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "6P6d-AgAqxXZ"
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},
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"outputs": [],
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"source": [
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"# Draws a heatmap to represent a probability distribution, possibly with samples overlaed\n",
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"def plot_heatmap(x1_mesh,x2_mesh,y_mesh, x1_samples=None, x2_samples=None, title=None):\n",
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" # Define pretty colormap\n",
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" my_colormap_vals_hex =('2a0902', '2b0a03', '2c0b04', '2d0c05', '2e0c06', '2f0d07', '300d08', '310e09', '320f0a', '330f0b', '34100b', '35110c', '36110d', '37120e', '38120f', '39130f', '3a1410', '3b1411', '3c1511', '3d1612', '3e1613', '3f1713', '401714', '411814', '421915', '431915', '451a16', '461b16', '471b17', '481c17', '491d18', '4a1d18', '4b1e19', '4c1f19', '4d1f1a', '4e201b', '50211b', '51211c', '52221c', '53231d', '54231d', '55241e', '56251e', '57261f', '58261f', '592720', '5b2821', '5c2821', '5d2922', '5e2a22', '5f2b23', '602b23', '612c24', '622d25', '632e25', '652e26', '662f26', '673027', '683027', '693128', '6a3229', '6b3329', '6c342a', '6d342a', '6f352b', '70362c', '71372c', '72372d', '73382e', '74392e', '753a2f', '763a2f', '773b30', '783c31', '7a3d31', '7b3e32', '7c3e33', '7d3f33', '7e4034', '7f4134', '804235', '814236', '824336', '834437', '854538', '864638', '874739', '88473a', '89483a', '8a493b', '8b4a3c', '8c4b3c', '8d4c3d', '8e4c3e', '8f4d3f', '904e3f', '924f40', '935041', '945141', '955242', '965343', '975343', '985444', '995545', '9a5646', '9b5746', '9c5847', '9d5948', '9e5a49', '9f5a49', 'a05b4a', 'a15c4b', 'a35d4b', 'a45e4c', 'a55f4d', 'a6604e', 'a7614e', 'a8624f', 'a96350', 'aa6451', 'ab6552', 'ac6552', 'ad6653', 'ae6754', 'af6855', 'b06955', 'b16a56', 'b26b57', 'b36c58', 'b46d59', 'b56e59', 'b66f5a', 'b7705b', 'b8715c', 'b9725d', 'ba735d', 'bb745e', 'bc755f', 'bd7660', 'be7761', 'bf7862', 'c07962', 'c17a63', 'c27b64', 'c27c65', 'c37d66', 'c47e67', 'c57f68', 'c68068', 'c78169', 'c8826a', 'c9836b', 'ca846c', 'cb856d', 'cc866e', 'cd876f', 'ce886f', 'ce8970', 'cf8a71', 'd08b72', 'd18c73', 'd28d74', 'd38e75', 'd48f76', 'd59077', 'd59178', 'd69279', 'd7937a', 'd8957b', 'd9967b', 'da977c', 'da987d', 'db997e', 'dc9a7f', 'dd9b80', 'de9c81', 'de9d82', 'df9e83', 'e09f84', 'e1a185', 'e2a286', 'e2a387', 'e3a488', 'e4a589', 'e5a68a', 'e5a78b', 'e6a88c', 'e7aa8d', 'e7ab8e', 'e8ac8f', 'e9ad90', 'eaae91', 'eaaf92', 'ebb093', 'ecb295', 'ecb396', 'edb497', 'eeb598', 'eeb699', 'efb79a', 'efb99b', 'f0ba9c', 'f1bb9d', 'f1bc9e', 'f2bd9f', 'f2bfa1', 'f3c0a2', 'f3c1a3', 'f4c2a4', 'f5c3a5', 'f5c5a6', 'f6c6a7', 'f6c7a8', 'f7c8aa', 'f7c9ab', 'f8cbac', 'f8ccad', 'f8cdae', 'f9ceb0', 'f9d0b1', 'fad1b2', 'fad2b3', 'fbd3b4', 'fbd5b6', 'fbd6b7', 'fcd7b8', 'fcd8b9', 'fcdaba', 'fddbbc', 'fddcbd', 'fddebe', 'fddfbf', 'fee0c1', 'fee1c2', 'fee3c3', 'fee4c5', 'ffe5c6', 'ffe7c7', 'ffe8c9', 'ffe9ca', 'ffebcb', 'ffeccd', 'ffedce', 'ffefcf', 'fff0d1', 'fff2d2', 'fff3d3', 'fff4d5', 'fff6d6', 'fff7d8', 'fff8d9', 'fffada', 'fffbdc', 'fffcdd', 'fffedf', 'ffffe0')\n",
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" my_colormap_vals_dec = np.array([int(element,base=16) for element in my_colormap_vals_hex])\n",
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" r = np.floor(my_colormap_vals_dec/(256*256))\n",
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" g = np.floor((my_colormap_vals_dec - r *256 *256)/256)\n",
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" b = np.floor(my_colormap_vals_dec - r * 256 *256 - g * 256)\n",
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" my_colormap = ListedColormap(np.vstack((r,g,b)).transpose()/255.0)\n",
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"\n",
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" fig,ax = plt.subplots()\n",
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" fig.set_size_inches(8,8)\n",
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" ax.contourf(x1_mesh,x2_mesh,y_mesh,256,cmap=my_colormap)\n",
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" ax.contour(x1_mesh,x2_mesh,y_mesh,8,colors=['#80808080'])\n",
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" if title is not None:\n",
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" ax.set_title(title);\n",
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" if x1_samples is not None:\n",
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" ax.plot(x1_samples, x2_samples, 'c.')\n",
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" ax.set_xlim([-0.5,2.5])\n",
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" ax.set_ylim([-1.5,1.5])\n",
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" ax.set_xlabel('$x_1$'); ax.set_ylabel('$x_2$')\n",
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" plt.show()\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "diYKb7_ZgjlJ"
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},
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"outputs": [],
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"source": [
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"# Returns the likelihood\n",
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"def get_likelihood(x1_mesh, x2_mesh, z_val):\n",
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" # Find the corresponding x1 and x2 values\n",
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" x1,x2 = f(z_val)\n",
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"\n",
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" # Calculate the probability for a mesh of x1,x2 values.\n",
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" mn = scipy.stats.multivariate_normal([x1, x2], [[sigma_sq, 0], [0, sigma_sq]])\n",
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" pr_x1_x2_given_z_val = mn.pdf(np.dstack((x1_mesh, x2_mesh)))\n",
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" return pr_x1_x2_given_z_val"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "0X4NwixzqxtZ"
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},
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"source": [
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"Now let's plot the likelihood $Pr(x_1,x_2|z)$ as in fig 17.3b in the book."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "hWfqK-Oz5_DT"
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},
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"outputs": [],
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"source": [
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"# Choose some z value\n",
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"z_val = 1.8\n",
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"\n",
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"# Compute the conditional distribution on a grid\n",
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"x1_mesh, x2_mesh = np.meshgrid(np.arange(-0.5,2.5,0.01), np.arange(-1.5,1.5,0.01))\n",
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"pr_x1_x2_given_z_val = get_likelihood(x1_mesh,x2_mesh, z_val)\n",
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"\n",
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"# Plot the result\n",
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"plot_heatmap(x1_mesh, x2_mesh, pr_x1_x2_given_z_val, title=\"Conditional distribution $Pr(x_1,x_2|z)$\")\n",
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"\n",
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"# TODO -- Experiment with different values of z and make sure that you understand the what is happening."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "25xqXnmFo-PH"
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},
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"source": [
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"The data density is found by marginalizing over the latent variables $z$:\n",
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"\n",
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"\\begin{align}\n",
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" Pr(x_1,x_2) &=& \\int Pr(x_1,x_2, z) dz \\nonumber \\\\\n",
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" &=& \\int Pr(x_1,x_2 | z) \\cdot Pr(z)dz\\nonumber \\\\\n",
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" &=& \\int \\text{Norm}_{[x_1,x_2]}\\Bigl[\\mathbf{f}[z],\\sigma^{2}\\mathbf{I}\\Bigr]\\cdot \\text{Norm}_{z}\\left[\\mathbf{0},\\mathbf{I}\\right]dz.\n",
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"\\end{align}"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "H0Ijce9VzeCO"
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},
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"outputs": [],
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"source": [
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"# TODO Compute the data density\n",
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"# We can't integrate this function in closed form\n",
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"# So let's approximate it as a sum over the z values (z = np.arange(-3,3,0.01))\n",
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"# You will need the functions get_likelihood() and get_prior()\n",
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"# To make this a valid probability distribution, you need to multiply\n",
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"# By the z-increment (0.01)\n",
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"# Replace this line\n",
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"pr_x1_x2 = np.zeros_like(x1_mesh)\n",
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"\n",
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"\n",
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"# Plot the result\n",
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"plot_heatmap(x1_mesh, x2_mesh, pr_x1_x2, title=\"Data density $Pr(x_1,x_2)$\")\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "W264N7By_h9y"
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},
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"source": [
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"Now let's draw some samples from the model"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "Li3mK_I48k0k"
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},
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"outputs": [],
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"source": [
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"def draw_samples(n_sample):\n",
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" # TODO Write this routine to draw n_sample samples from the model\n",
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" # First draw a random value of z from the prior (a standard normal distribution)\n",
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" # Then draw a sample from Pr(x1,x2|z)\n",
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" # Replace this line\n",
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" x1_samples=0; x2_samples = 0;\n",
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"\n",
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" return x1_samples, x2_samples"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "D7N7oqLe-eJO"
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},
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"source": [
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"Let's plot those samples on top of the heat map."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "XRmWv99B-BWO"
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},
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"outputs": [],
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"source": [
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"x1_samples, x2_samples = draw_samples(500)\n",
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"# Plot the result\n",
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"plot_heatmap(x1_mesh, x2_mesh, pr_x1_x2, x1_samples, x2_samples, title=\"Data density $Pr(x_1,x_2)$\")\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "PwOjzPD5_1OF"
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},
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"outputs": [],
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"source": [
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"# Return the posterior distribution\n",
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"def get_posterior(x1,x2):\n",
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" z = np.arange(-3,3, 0.01)\n",
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" # TODO -- write this function\n",
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" # Again, we can't integrate, but we can sum\n",
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" # We don't know the constant in the denominator of equation 17.19, but we can just normalize\n",
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" # by the sum of the numerators for all values of z\n",
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" # Replace this line:\n",
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" pr_z_given_x1_x2 = np.ones_like(z)\n",
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"\n",
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"\n",
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" return z, pr_z_given_x1_x2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"id": "PKFUY42K-Tp7"
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},
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"outputs": [],
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"source": [
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"x1 = 0.9; x2 = -0.9\n",
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"z, pr_z_given_x1_x2 = get_posterior(x1,x2)\n",
|
|
"\n",
|
|
"\n",
|
|
"fig, ax = plt.subplots()\n",
|
|
"ax.plot(z, pr_z_given_x1_x2, 'r-')\n",
|
|
"ax.set_xlabel(\"Latent variable $z$\")\n",
|
|
"ax.set_ylabel(\"Posterior probability $Pr(z|x_{1},x_{2})$\")\n",
|
|
"ax.set_xlim([-3,3])\n",
|
|
"ax.set_ylim([0,1.5 * np.max(pr_z_given_x1_x2)])\n",
|
|
"plt.show()"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"colab": {
|
|
"provenance": [],
|
|
"include_colab_link": true
|
|
},
|
|
"kernelspec": {
|
|
"display_name": "Python 3",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"name": "python"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 0
|
|
} |