init change of variables
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"import torch\n",
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"import torch.distributions as dist\n",
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"import numpy as np\n",
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"from scipy import stats\n",
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"import matplotlib.pyplot as plt"
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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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"source": [
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"# Transforming probability distributions\n",
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"Normalizing flows uses transformation to obtain complex posteriors. \n",
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"This notebook explores transformations of distributions.\n",
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"\n",
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"We start with a basis Gaussian $\\mathcal{N}(\\mu=1, \\sigma=0.1)$. These parameters are chose so we can look at transformations in the positive domain and don't have trouble with square rooting negative numbers.\n",
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"\n",
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"Below we plot the **pdf** and integrate to validate $\\int P(x)dx=1$."
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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": 221,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1.0"
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]
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},
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"execution_count": 221,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": 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truncated
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"text/plain": [
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"<Figure size 432x288 with 1 Axes>"
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]
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},
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"metadata": {
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"needs_background": "light"
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},
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"output_type": "display_data"
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}
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],
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"source": [
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"x = np.linspace(0.01, 2, num=100)\n",
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"\n",
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"base = stats.norm(1, 0.1)\n",
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"\n",
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"plt.plot(x, base.pdf(x))\n",
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"\n",
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"np.trapz(base.pdf(x), x)"
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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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"source": [
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"## Transformation\n",
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"\n",
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"As we apply $f(x) = y = x^2$ we see that the result of the transformation doesn't suffice the requirements of a probability distribution, as $\\int P(y)dy \\neq 1$"
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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": 222,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5.641895835477563"
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]
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},
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"execution_count": 222,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"def f(x):\n",
|
||||
" return x**2\n",
|
||||
"\n",
|
||||
"transformed = f(base.pdf(x))\n",
|
||||
"plt.plot(f(x), transformed)\n",
|
||||
"\n",
|
||||
"np.trapz(transformed, f(x))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Change of variables\n",
|
||||
"\n",
|
||||
"We want to transform $P(x)$ to $P(y)$ with $f: \\mathbb{R}^n \\mapsto \\mathbb{R}^n $ and $f(x) = y$\n",
|
||||
"The transformation blows up or shrinks the area/ volume (in N-D) of the distribution, so we need to determine a transformation so that:\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$$ \n",
|
||||
"\\begin{eqnarray}\n",
|
||||
"\\int P(x)dx &=&\\int P(y)dy = 1 \\\\\n",
|
||||
"P(x)dx &=&P(y)dy \\\\\n",
|
||||
"P(y) &=&P(x)\\frac{dx}{dy}\n",
|
||||
"\\end{eqnarray} $$\n",
|
||||
"\n",
|
||||
"So to keep an integral of 1 for $P(y)$ (the definition of a probability distribution), we need to multiply $P(x)$ with its derivative w.r.t. $y$.\n",
|
||||
"Therefore we need to express $x$ in $y$. This can be done if we only have invertible transformations.\n",
|
||||
"\n",
|
||||
"$$\\begin{eqnarray}\n",
|
||||
"f(x) &=& y \\\\\n",
|
||||
"f^{-1}(y) &=& x\n",
|
||||
"\\end{eqnarray} $$\n",
|
||||
"\n",
|
||||
"$$\\begin{eqnarray}\n",
|
||||
" P(y) &=& P(x)\\frac{\\text{d}f^{-1}(y)}{\\text{d}y} \\\\\n",
|
||||
" &=& P(x)f'^{-1}(y) \\\\\n",
|
||||
" &=& P(f^{-1}(y))f'^{-1}(y)\n",
|
||||
"\\end{eqnarray} $$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 223,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"0.9987379589284238"
|
||||
]
|
||||
},
|
||||
"execution_count": 223,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"def f_i(y):\n",
|
||||
" return y**0.5\n",
|
||||
"\n",
|
||||
"def f_i_prime(y):\n",
|
||||
" return 0.5*y**-0.5\n",
|
||||
"\n",
|
||||
"assert np.allclose(f_i(f(x)), x)\n",
|
||||
"\n",
|
||||
"y = f(x)\n",
|
||||
"px = base.pdf(x)\n",
|
||||
"\n",
|
||||
"transformed = px * f_i_prime(f_i(y))\n",
|
||||
"\n",
|
||||
"plt.plot(x, px)\n",
|
||||
"plt.plot(y, transformed)\n",
|
||||
"plt.xlim(0, 2)\n",
|
||||
"\n",
|
||||
"np.trapz(transformed, y)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 224,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"0.9987894616589735"
|
||||
]
|
||||
},
|
||||
"execution_count": 224,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD4CAYAAAATpHZ6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAgAElEQVR4nO3deXhc9X3v8fdX+2ptHtvyvuANYxuM6rAFDCRAWNM8SQlt0zRN6rZZmrZpe9t7+7RJ2/vc7k3am/ZemiYthJBLEmgp2Qhg4xC2COMFr9gGgyzZ2mzJWqxtfvePM2PJsqQ5I82Z5fB5PY8z0sw5Z74Rxx/9/J3f+R1zziEiIuGQl+kCREQkdRTqIiIholAXEQkRhbqISIgo1EVEQqQgiIPOnj3bLV26NIhDi4iE0iuvvNLunIvM9DiBhPrSpUtpbGwM4tAiIqFkZsdTcRy1X0REQkShLiISIgp1EZEQUaiLiISIQl1EJER8hbqZ/baZ7TOz18zsYTMrCbowERFJXsJQN7MFwG8CDc65y4B84MNBFyYiIsnz234pAErNrAAoA5qDK0kkQJ1vwOEfZroKkcAkDHXn3Angb4C3gBagyzn35PjtzGyrmTWaWWNbW1vqKxWZqe4W+Lc74Bs/Bz06RyWc/LRfaoB7gGXAfKDczH5x/HbOufudcw3OuYZIZMZXuoqk1vAAPHwv9Jzyvj+2LbP1iATET/vlPcAbzrk259wQ8ChwTbBliaTY8eehZTfc82UorYUjT2e6IpFA+An1t4CrzKzMzAy4GTgQbFkiKda633tccTOsuBGOPgPRaGZrEgmAn576S8C3gZ3A3tg+9wdcl0hqndoP5RGoiHjB3tsKrfsyXZVIyvma/eKc+xPn3Brn3GXOuY845waCLkwkpVr3w5xLva9X3OQ9qgUjIaQrSiX8olFoOwhz13nfz6qHukug6aeZrUskAAp1Cb/Tb8BQH8xZO/pczTLoejtzNYkERKEu4Rf/kHTOutHnqhfDmbcyU49IgBTqEn6n4qG+ZvS56kXQfxrOdWemJpGAKNQl/Fr3Q81SKCoffa56sfeoFoyEjEJdwq/1wIWtF4DqJd6jWjASMgp1CTfn4MxxqF124fPxkfoZjdQlXBTqEm79p2H4HMyaf+Hz5REoKPECXyREFOoSbmdbvMfK+gufN9MMGAklhbqEW3cs1MeP1EGhLqGkUJdwOxu7n8v4kTpA1SKFuoSOQl3CLT5Sr5x38WvVi6G/EwZ60luTSIAU6hJuZ5uhrA4Kii9+TXPVJYQU6hJu3S1QOUE/HTStUUJJoS7hdrbFW5VxIhVzvMde3a9UwkOhLuF2tmXiD0nBm6sOCnUJFT83nl5tZrvG/Ok2s99KR3EiMzI86AX2RNMZwVsLprBMoS6hUpBoA+fcIeByADPLB04AjwVcl8jM9Zz0HicbqQOUz4be9vTUI5IGybZfbgaOOud0bbVkv6kuPIorj2ikLqGSbKh/GHh4ohfMbKuZNZpZY1ub/pJIFpjqwqM4hbqEjO9QN7Mi4G7gWxO97py73znX4JxriEQiqapPZPp8jdTVfpFwSWak/j5gp3PuVFDFiKTU2WbIL4bSmsm3KZvtjdSdS19dIgFKJtTvY5LWi0hW6mmFyrneioyTKY9AdAjOdaWvLpEA+Qp1MysD3gs8Gmw5IinU2z46F30y5+eqqwUj4eAr1J1zfc65OuechjOSO3rbvPbKVMpnj24rEgK6olTCq69jNLQno6tKJWQU6hJOznktlbK6qbdTqEvIKNQlnAZ7YGQg8Ug9HvrqqUtIKNQlnOIhnainXlAEJdUaqUtoKNQlnPo6vMdEI3XQVaUSKgp1CSe/I3WIhbraLxIOCnUJp75YSJcn+KAUYksFaKQu4aBQl3BKaqSuUJfwUKhLOPW1Q0GJdyOMREpr4dwZiEaDr0skYAp1CafeDm+UPtW6L3FlteCiMNAdfF0iAVOoSzj1tfvrp8PoKo79p4OrRyRNFOoSTr3t/vrpoFCXUFGoSzj1tvubow5jQr0zuHpE0kShLuHUl8xIvdZ77D8TXD0iaaJQl/AZ7IOhPvXU5R1JoS7h05fEHHWA0urYfmq/SO7ze+ejajP7tpkdNLMDZnZ10IWJTFv8wiO/PfX8QiiepZG6hEKBz+2+BPzAOfdBMysCygKsSWRm4ot5+R2pgzdaV6hLCCQMdTObBVwP/DKAc24QGAy2LJEZSGaFxrjSGoW6hIKf9styoA34mpm9amZfMbOLrr02s61m1mhmjW1tWkdDMijeG49/AOpHaY2mNEoo+An1AmAT8M/OuSuAXuAPxm/knLvfOdfgnGuIRBLcwV0kSP2dYHnezS/8Kq3VSF1CwU+oNwFNzrmXYt9/Gy/kRbJTX6cX6HlJTO5S+0VCIuFZ75w7CbxtZqtjT90M7A+0KpGZ6O/0FulKRjzUtVKj5Di/s18+AzwUm/lyDPhYcCWJzFBf5+hVon6NXamxNIm2jUiW8RXqzrldQEPAtYikRn8nzFqQ3D5jrypVqEsO0xWlEj59p5MfqWupAAkJhbqEz3R76vF9RXKYQl3CZeict5hXMnPUQSs1Smgo1CVc+qdx4dHY7dV+kRynUJdwiYdy0u0XrdQo4aBQl3A5v0RAkqGeXwhFlRqpS85TqEu4xNsvyY7UwRutn1NPXXKbQl3CZbojdfCWFtAHpZLjFOoSLhqpyzucQl3Cpa8TCkqhsDT5fUuqNFKXnKdQl3DpPz29UTpopC6hoFCXcJnOYl5x6qlLCCjUJVz6O6EsyQuP4kqrYbgfhgdSW5NIGinUJVxmOlIHjdYlpynUJVyms5hXXHypAPXVJYf5Wk/dzN4EzgIjwLBzTmurS/aJRmProU93pF7lPWqkLjnM752PAG50zrUHVonITA10eXcvmu5IPd5+0UhdcpjaLxIe8XVbpjtSL1VPXXKf31B3wJNm9oqZbZ1oAzPbamaNZtbY1taWugpF/OqLh/o0Z79opC4h4DfUr3XObQLeB3zKzK4fv4Fz7n7nXINzriESiaS0SBFfZrJEAGikLqHgK9Sdc82xx1bgMWBzkEWJTMtMFvMCb/ndwnKN1CWnJQx1Mys3s8r418AtwGtBFyaStJmO1MEbrWukLjnMz+yXucBjZhbf/hvOuR8EWpXIdPSfBmx0auJ0lGj9F8ltCUPdOXcM2JiGWkRmpq/TC/S8/OkfQyN1yXGa0ijhMZOrSeM0Upccp1CX8JjJui9xGqlLjlOoS3hopC6iUJcQ6T89/QuP4kqrYagPhgdTU5NIminUJTz6ZrCYV5yuKpUcp1CXcBgehMGzM2+/6KpSyXEKdQmH/hmu+xKnkbrkOIW6hEMqriaF0V8KGqlLjlKoSzikaqReqpG65DaFuoTDTBfzitN9SiXHKdQlHFLWfomH+umZHUckQxTqEg6pGqlr+V3JcQp1CYf+TsgvgqLymR+rtEbtF8lZCnUJh/jVpN4S0TNTqqUCJHcp1CUcUrGYV1yJFvWS3KVQl3DoPz3zD0njNFKXHOY71M0s38xeNbMngixIZFr6Omc+Rz2utFqzXyRnJTNS/yxwIKhCRGYkFcvuxqn9IjnMV6ib2ULgDuArwZYjMg3OQV8HlM2edJOhkSjtPQP+jldaDcP9MOxze5Es4ufG0wBfBH4fqJxsAzPbCmwFWLx48cwrE/FroBuiw1BWN+HLJ87087GvvczhUz2smVfJltVzuGXdXDYtnqRdM/aq0sq5ARUtEoyEI3UzuxNodc69MtV2zrn7nXMNzrmGSCSSsgJFEurr8B4nCPXXTnTxs1/+CS1d5/jMTZdQXVbIV358jA/80/M80vj2xMeL9+b1YankID8j9WuBu83sdqAEmGVmX3fO/WKwpYn41DtxqG8/1MonH9pJdWkh3/mNa1g11/uHZve5IbY+0MgXHt/HVcvqWFxXduHxtFSA5LCEI3Xn3B865xY655YCHwaeUaBLVplgpN58pp+tD77C0rpyHvvUtecDHWBWSSF/+3OXk5dn/PYjuxgeiV54vBItvyu5S/PUJfedD/XR2S//+MwRnHPc/0tXMndWyUW7LKgu5c/ffxmvHD/NP28/euGLWn5XclhSoe6c2+6cuzOoYkSmJR7q5d7sl+MdvXyr8W3u27yYhTVlk+52z+ULuOfy+Xzx6dfZ/faYANfyu5LDNFKX3NfXEVvMqwKALz39Ovl5xqduvCThrn96z2XMrijiL75/cPTJkirvUSN1yUEKdcl9fR1eP92MI61n+Y9XT/DRa5ZO2HYZr6q0kI9es5QXjnXw+qmz3pP5BVA8Sx+USk5SqEvui4c68PdPvU5pYT6/dv1y37vf27CIovw8Hnzx+OiTuqpUcpRCXXJfXweU1XKgpZvv7mnhV65bRl1Fse/d6yqKuXNDPY/uPEHPwLD3ZGmV2i+SkxTqkvtiI/X/99O3KSrI4xPX+R+lx33k6iX0DAzz2M4m74nSWrVfJCcp1CX39XUwUlrLf+1u5r1r51JVVpj0IS5fVM36BVU88MJxnHPeVaXxW+SJ5BCFuuS2kWHoP8Nb/WV09A7y/isWTOswZsZHrl7C6609vHDMa+ecv5m1SA5RqEtuO3cGcDS2GTVlhdywavrrDt29cT7VZYU8+MLx0fZLNJp4R5EsolCX3Ba78OiFk3DnhvkUFUz/lC4pzOdDVy7kyf2n6CuoAhf1VoAUySEKdcltve0AnBoun3brZay7Ns5nJOp4rTP2V0MtGMkxCnXJbbGRevGsCJsWV8/4cOsXVLGgupQXWlzs+JoBI7lFoS45rbvzJABXXbYKM5vx8cyMW9fN4yfNsV66RuqSYxTqktMOHfOuAn3vlWtTdsz3rZ9H20i5942mNUqOUahLTmtubqLPSllWP/n9SZN15eIa8stja7PrAiTJMQp1yVmt3ecY6W1nuGjmvfSx8vKMq9ctI+qMoZ72lB5bJGgKdclLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
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||||
},
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||||
"metadata": {
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"needs_background": "light"
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||||
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|
||||
"source": [
|
||||
"# Same principle, other transformation\n",
|
||||
"\n",
|
||||
"def f(x):\n",
|
||||
" return x**0.5\n",
|
||||
"\n",
|
||||
"def f_i(y):\n",
|
||||
" return y**2\n",
|
||||
"\n",
|
||||
"def f_i_prime(y):\n",
|
||||
" return 2*x\n",
|
||||
"\n",
|
||||
"assert np.allclose(f_i(f(x)), x)\n",
|
||||
"\n",
|
||||
"y = f(x)\n",
|
||||
"px = base.pdf(x)\n",
|
||||
"\n",
|
||||
"transformed = px * f_i_prime(f_i(y))\n",
|
||||
"\n",
|
||||
"plt.plot(x, px)\n",
|
||||
"plt.plot(y, transformed)\n",
|
||||
"plt.xlim(0, 2)\n",
|
||||
"\n",
|
||||
"np.trapz(transformed, y)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Conditions\n",
|
||||
"The transformations we've used were strictly increasing. \n",
|
||||
"\n",
|
||||
"In the example we've used above $\\frac{dx}{dy}$ was positive, as the transformation was strictly increasing, in the strictly decreasing case, $\\frac{dx}{dy}$ is negative and $P(y)=-P(x)\\frac{dx}{dy}$. If we take the absolute value of the derivative we have one equation for both cases.\n",
|
||||
"\n",
|
||||
"$$ \n",
|
||||
"\\begin{eqnarray}\n",
|
||||
"P(y)&=&P(x) \\cdot \\left| \\frac{dx}{dy} \\right| \\\\\n",
|
||||
" &=& P(f^{-1}(y)) \\cdot \\left| f'^{-1}(y)\\right|\n",
|
||||
"\\end{eqnarray}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"The intuition of taking the modulus is that if the (local) rate of change of $x + dx = y + dy > 0$, than we can ensure ourselves that the rate of amount of probability is increasing, and the total amount of probability is preserved. This is ensured by conditioning $\\frac{dx}{dy} \\geq 0$.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Multiple dimensions\n",
|
||||
"In N-D $\\frac{dx}{dy}$ is expressed in the Jacobian matrix, i.e. the matrix with first order partial derivatives $\\frac{\\partial(x)}{\\partial{y}}=\\mathbf{J}$. This absolute value, is now a determinant, such that;\n",
|
||||
"\n",
|
||||
"$$ \n",
|
||||
"\\begin{eqnarray}\n",
|
||||
"P(y)&=&P(f^{-1}(y)) \\cdot \\left| \\det \\frac{\\partial{ f(y)^{-1} }}{\\partial{y}} \\right| \\\\\n",
|
||||
"P(y)&=&P(f^{-1}(y)) \\cdot \\left| \\det \\mathbf{J}_{f^{-1}}(y) \\right| \\\\\n",
|
||||
"\\end{eqnarray}\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"The geometric meaning of the determinant is a scaling factor. $\\det A$ tells us how much an n-dimensional volume is scaled by applying the transformation of $A$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 22,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
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||||
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||||
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|
||||
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||||
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||||
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|
||||
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|
||||
},
|
||||
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|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"m = dist.Normal(torch.zeros(2), scale=torch.ones(2))\n",
|
||||
"s = m.sample((1000,))\n",
|
||||
"\n",
|
||||
"plt.scatter(s[:, 0], s[:, 1], alpha=0.2)\n",
|
||||
"plt.axis('scaled');"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 23,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def f(x):\n",
|
||||
" torch.exp(x) + 2\n",
|
||||
" \n"
|
||||
]
|
||||
},
|
||||
{
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
],
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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Reference in new issue
Block a user