book/autodiff-tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tutorial on Automatic Differentiation\n",
    "\n",
    "(by Lukas Heinrich. See: [pyhep2020-autodiff-tutorial](https://github.com/lukasheinrich/pyhep2020-autodiff-tutorial) )\n",
    "\n",
    "\n",
    "### Introduction\n",
    "\n",
    "Welcome to this tutorial on automatic differentiation. Automatic Differentiation is a method to compute exact derivatives of functions implements as **programs**. It's a widely applicable method and famously is used in\n",
    "many Machine learning optimization problems. E.g. neural networks, which are parametrized by weights $\\text{NN}(\\text{weights})$ are trained by (stocastic) **gradient** descent to find the minimum of the loss function $L$ where \n",
    "\n",
    "\n",
    "$$\\text{weights}_\\text{opt} = \\text{argmin}_\\text{weights} L(\\text{weights}) \\hspace{1cm} \\nabla L(\\text{weights}) = 0$$\n",
    "\n",
    "\n",
    "This means that efficient algorithms to compute derivatives are crucial.\n",
    "\n",
    "Aside from ML, many other use-cases require gradients: standard statistical analysis in HEP (fitting, hypothesis testing, ...) requires gradients. Uncertainty propagation (e.g. track parameters) uses gradients, etc..\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pyhf\n",
    "pyhf.set_backend('jax')\n",
    "import jax\n",
    "import jaxlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import jax.numpy as jnp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Other approaches to differentiation\n",
    "\n",
    "Before diving into automatic differentiation, let's review how my might otherwise compute derivatives\n",
    "\n",
    "\n",
    "\n",
    "#### Finite Differences\n",
    "\n",
    "\n",
    "A common appraoch to approximate gradients of a black-box function is to evaluate it\n",
    "at close-by points $x$ and $x+Δx$ and \n",
    "\n",
    "$\\frac{\\partial f}{\\partial x} \\approx \\frac{f(x) - f(x+\\Delta x}{\\Delta x}$ if $\\Delta x$ is sufficiently small\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def black_box_func(x):\n",
    "    return x**3+30\n",
    "\n",
    "def true_gradient_func(x):\n",
    "    return 3*x**2\n",
    "\n",
    "\n",
    "def plot_gradients(nsteps,title):\n",
    "    xi = np.linspace(-5,5,nsteps)\n",
    "    yi = black_box_func(xi)\n",
    "\n",
    "    approx_gradient = np.gradient(yi,xi)\n",
    "    true_gradient   = true_gradient_func(xi)\n",
    "\n",
    "    plt.plot(xi,yi, label = 'black-box func')\n",
    "    plt.scatter(xi,yi)\n",
    "\n",
    "    plt.plot(xi,approx_gradient, label = 'finite diff grad')\n",
    "    plt.scatter(xi,approx_gradient)\n",
    "\n",
    "    plt.plot(xi,true_gradient, label = 'true grad')\n",
    "    plt.scatter(xi,true_gradient)\n",
    "\n",
    "    plt.legend()\n",
    "    plt.title(title)\n",
    "    plt.show()\n",
    "    \n",
    "    \n",
    "plot_gradients(11, title = 'it is pretty bad if Δx is too large')\n",
    "plot_gradients(41, title = 'it gets better at the cost of many evaluations') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"width: 500px\">\n",
    "while only approximate, finite differences is *simple*. I don't need to know \n",
    "anything about the function beyond having the ability \n",
    "to *evaluate* it\n",
    "</p>\n",
    "<p style=\"width: 500px\">\n",
    "This way I can compute gradients of functions encoded as a computer\n",
    "program, and it works in any programming language\n",
    "</p>\n",
    "\n",
    "<p style=\"width: 500px\">\n",
    "For multivariate (possibly vector-valued) functions $\\vec{f}(\\vec{x}) = f_i(x_1,x_2,\\dots,x_n)$ one needs to compute a finite difference\n",
    "gradient for each partial derivative $\\frac{\\partial f}{\\partial x}$ in order to get the\n",
    "full jacobian / total derivative $df_i = J_{ik} dx_k\\; J_{ik} = \\frac{\\partial f_i}{\\partial x_k}$\n",
    "    \n",
    "In high dimensions, the number of required evaluations explodes!\n",
    "<p>\n",
    "\n",
    "\n",
    "**Finite Differences**:\n",
    "\n",
    "* Pro: easy to to, works in any language, no \"framework needed\"\n",
    "* Con: inaccurate unless one does a lot of evaluations\n",
    "* Con does not scale to large dimensions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolic Differentiation in a CAS\n",
    "\n",
    "Computer Algebra Systems (CAS), such as Mathematica (or sympy)\n",
    "can manipulate functional *expressions* and know about differentiation rules (and many other things)\n",
    "\n",
    "If the function / the prograrm which we want to derive is available as such an expression the \n",
    "symbolic differentiation can produce **exact gradients**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3}$"
      ],
      "text/plain": [
       "x**3"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy\n",
    "\n",
    "def function(x):\n",
    "    return x**3\n",
    "\n",
    "def true_deriv(x):\n",
    "    return 3*x**2\n",
    "\n",
    "symbolic_x = sympy.symbols('x')\n",
    "symbolic_func = function(symbolic_x)\n",
    "symbolic_func"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using `lambdify` we can turn it into a normal python function we can evaluate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f96f85724c0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xi = np.linspace(-5,5,11)\n",
    "yi = sympy.lambdify(symbolic_x,symbolic_func)(xi)\n",
    "plt.plot(xi,yi)\n",
    "plt.scatter(xi,yi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`symbolic_func` is now an experssion which we can differentiate *symbolically*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 x^{2}$"
      ],
      "text/plain": [
       "3*x**2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "symbolic_deriv = symbolic_func.diff(symbolic_x)\n",
    "symbolic_deriv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_symbolic(nsteps,title):\n",
    "    xi = np.linspace(-5,5,nsteps)\n",
    "    yi = sympy.lambdify(symbolic_x,symbolic_func)(xi)\n",
    "    plt.scatter(xi,yi)\n",
    "    plt.plot(xi,yi, label = 'function')\n",
    "\n",
    "\n",
    "\n",
    "    yi = true_deriv(xi)\n",
    "    plt.plot(xi,yi)\n",
    "    plt.scatter(xi,yi, label = 'true deriv')\n",
    "\n",
    "    yi = sympy.lambdify(symbolic_x,symbolic_deriv)(xi)\n",
    "    plt.plot(xi,yi)\n",
    "    plt.scatter(xi,yi, label = 'symbolic deriv')\n",
    "\n",
    "    plt.legend()\n",
    "    plt.title(title)\n",
    "    plt.show()\n",
    "    \n",
    "    \n",
    "plot_symbolic(11,title = 'the symbolid derivative is always exact')\n",
    "plot_symbolic(4, title = 'it does not matter where/how often you evaluate it')    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Chain Rule in CAS\n",
    "\n",
    "We can even handle function compositions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x^{2} \\right)}$"
      ],
      "text/plain": [
       "cos(x**2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f1(x):\n",
    "    #standard operations are overloaded\n",
    "    return x**2\n",
    "\n",
    "def f2(x):\n",
    "    #note here we use a special cos function from sympy\n",
    "    #instead of e.g. np.cos or math.cos\n",
    "    return sympy.cos(x) \n",
    "    \n",
    "\n",
    "composition = f2(f1(symbolic_x))\n",
    "\n",
    "composition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 2 x \\sin{\\left(x^{2} \\right)}$"
      ],
      "text/plain": [
       "-2*x*sin(x**2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "composition.diff(symbolic_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since `sympy` knows about the chain rule it can differentiate accordingly"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problems with Symbolic Differentiation\n",
    "\n",
    "This looks great! We get exact derivatives. However,\n",
    "there are drawbacks\n",
    "\n",
    "1. Need to implement it in CAS\n",
    "\n",
    "Most functions we are interested in are not implemented \n",
    "e.g. Mathematica. Rather we have loads of C, C++, Python\n",
    "code that we are interested in. \n",
    "\n",
    "But ok, `sympy` alleviates this to some degree. The functions\n",
    "`f1` and `f2` are fairly generic since they use operator\n",
    "overloading. So a symbolic program and a \"normal\" program\n",
    "could only differ by a few import statements\n",
    "\n",
    "\n",
    "\n",
    "```python\n",
    "from sympy import cos\n",
    "\n",
    "def f1(x):\n",
    "    return x**2\n",
    "\n",
    "def f2(x):\n",
    "    return cos(x) \n",
    "```\n",
    "\n",
    "versus:\n",
    "\n",
    "```python\n",
    "from math import cos\n",
    "\n",
    "def f1(x):\n",
    "    return x**2\n",
    "\n",
    "def f2(x):\n",
    "    return cos(x) \n",
    "```\n",
    "\n",
    "\n",
    "Note the code is almost exactly the same\n",
    "\n",
    "But not all our functions are so simple!\n",
    "\n",
    "\n",
    "**Expression swell**\n",
    "\n",
    "Let's look at a quadratic map which is applied a few times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 243 x^{2} + 729 x + 81 \\left(x^{2} + 3 x + 4\\right)^{2} + 27 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 9 \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 3 \\left(27 x^{2} + 81 x + 9 \\left(x^{2} + 3 x + 4\\right)^{2} + 3 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 160\\right)^{2} + \\left(81 x^{2} + 243 x + 27 \\left(x^{2} + 3 x + 4\\right)^{2} + 9 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 3 \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + \\left(27 x^{2} + 81 x + 9 \\left(x^{2} + 3 x + 4\\right)^{2} + 3 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 160\\right)^{2} + 484\\right)^{2} + 1456$"
      ],
      "text/plain": [
       "243*x**2 + 729*x + 81*(x**2 + 3*x + 4)**2 + 27*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 9*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 3*(27*x**2 + 81*x + 9*(x**2 + 3*x + 4)**2 + 3*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + (9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 160)**2 + (81*x**2 + 243*x + 27*(x**2 + 3*x + 4)**2 + 9*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 3*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + (27*x**2 + 81*x + 9*(x**2 + 3*x + 4)**2 + 3*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + (9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 160)**2 + 484)**2 + 1456"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def quadmap(x):\n",
    "    return x**2 + 3*x + 4\n",
    "\n",
    "def func(x):\n",
    "    for i in range(6):\n",
    "        x = quadmap(x)\n",
    "    return x\n",
    "\n",
    "quad_6_times = func(symbolic_x)\n",
    "quad_6_times"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks pretty intimidating. What happened? \n",
    "Symbolic programs run through the prgram and \n",
    "accumulate the full program into a single expression\n",
    "\n",
    "If we would just blindly differentiate this it would look like this"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 486 x + 81 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 27 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 9 \\left(36 x + 6 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 2 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 54\\right) \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right) + 3 \\left(108 x + 18 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 6 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 2 \\left(36 x + 6 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 2 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 54\\right) \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right) + 162\\right) \\left(27 x^{2} + 81 x + 9 \\left(x^{2} + 3 x + 4\\right)^{2} + 3 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 160\\right) + \\left(324 x + 54 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 6 \\left(36 x + 6 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 2 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 54\\right) \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right) + 2 \\left(108 x + 18 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 6 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 2 \\left(36 x + 6 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 2 \\left(12 x + 2 \\left(4 x + 6\\right) \\left(x^{2} + 3 x + 4\\right) + 18\\right) \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right) + 54\\right) \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right) + 162\\right) \\left(27 x^{2} + 81 x + 9 \\left(x^{2} + 3 x + 4\\right)^{2} + 3 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 160\\right) + 486\\right) \\left(81 x^{2} + 243 x + 27 \\left(x^{2} + 3 x + 4\\right)^{2} + 9 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 3 \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + \\left(27 x^{2} + 81 x + 9 \\left(x^{2} + 3 x + 4\\right)^{2} + 3 \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + \\left(9 x^{2} + 27 x + 3 \\left(x^{2} + 3 x + 4\\right)^{2} + \\left(3 x^{2} + 9 x + \\left(x^{2} + 3 x + 4\\right)^{2} + 16\\right)^{2} + 52\\right)^{2} + 160\\right)^{2} + 484\\right) + 729$"
      ],
      "text/plain": [
       "486*x + 81*(4*x + 6)*(x**2 + 3*x + 4) + 27*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 9*(36*x + 6*(4*x + 6)*(x**2 + 3*x + 4) + 2*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 54)*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52) + 3*(108*x + 18*(4*x + 6)*(x**2 + 3*x + 4) + 6*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 2*(36*x + 6*(4*x + 6)*(x**2 + 3*x + 4) + 2*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 54)*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52) + 162)*(27*x**2 + 81*x + 9*(x**2 + 3*x + 4)**2 + 3*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + (9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 160) + (324*x + 54*(4*x + 6)*(x**2 + 3*x + 4) + 18*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 6*(36*x + 6*(4*x + 6)*(x**2 + 3*x + 4) + 2*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 54)*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52) + 2*(108*x + 18*(4*x + 6)*(x**2 + 3*x + 4) + 6*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 2*(36*x + 6*(4*x + 6)*(x**2 + 3*x + 4) + 2*(12*x + 2*(4*x + 6)*(x**2 + 3*x + 4) + 18)*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16) + 54)*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52) + 162)*(27*x**2 + 81*x + 9*(x**2 + 3*x + 4)**2 + 3*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + (9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 160) + 486)*(81*x**2 + 243*x + 27*(x**2 + 3*x + 4)**2 + 9*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 3*(9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + (27*x**2 + 81*x + 9*(x**2 + 3*x + 4)**2 + 3*(3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + (9*x**2 + 27*x + 3*(x**2 + 3*x + 4)**2 + (3*x**2 + 9*x + (x**2 + 3*x + 4)**2 + 16)**2 + 52)**2 + 160)**2 + 484) + 729"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quad_6_times.diff(symbolic_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks even worse!\n",
    "\n",
    "Also note that that if we just blindly substitute x for some value\n",
    "e.g. x=2, we would be computing a lot of the same terms\n",
    "manyt times. E.g. in the above expression $x^2+3x+4$ appears in a \n",
    "lot of places due to the \"structure' of the original progrm\n",
    "\n",
    "If you knew the structure of the program you likely could precompute\n",
    "some of these repeating terms. However once it got all expanded all\n",
    "this knowledge about the structure is gone!\n",
    "\n",
    "Modern CAS can recover some of this by finding \"common subexpressions\" (CSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([(x0, x**2),\n",
       "  (x1, (3*x + x0 + 4)**2),\n",
       "  (x2, (9*x + 3*x0 + x1 + 16)**2),\n",
       "  (x3, (27*x + 9*x0 + 3*x1 + x2 + 52)**2),\n",
       "  (x4, (81*x + 27*x0 + 9*x1 + 3*x2 + x3 + 160)**2)],\n",
       " [729*x + 243*x0 + 81*x1 + 27*x2 + 9*x3 + 3*x4 + (243*x + 81*x0 + 27*x1 + 9*x2 + 3*x3 + x4 + 484)**2 + 1456])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cse(quad_6_times)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But it's not as automatic and may note find all relevant subexpressions. In any case it's trying hard to recover some\n",
    "of the structure that is already implicitly present in the prograam we want to differentiate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Control Flow**\n",
    "\n",
    "In addition to looping constucts like above, a lot of the functions we are interested in have \n",
    "control flow structures like if/else statements, while loops, etc..\n",
    "\n",
    "\n",
    "If we try to create a symbolic expression with conditionals we fail badly\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cannot determine truth value of Relational\n"
     ]
    }
   ],
   "source": [
    "def func(x):\n",
    "    if x > 2:\n",
    "        return x**2\n",
    "    else:\n",
    "        return x**3\n",
    "    \n",
    "try:\n",
    "    symbolic_result = func(symbolic_x)\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's too bad because this is a perfectly respectable function *almost everywhere*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xi = np.linspace(-2,5,1001)\n",
    "yi = np.asarray([func(xx) for xx in xi])\n",
    "\n",
    "plt.plot(xi,yi)\n",
    "plt.scatter(xi,yi)\n",
    "plt.title(\"pretty smooth except at x=2\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we could afford finite diffences it would compute gradients *just fine*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "g = np.gradient(yi,xi)\n",
    "plt.plot(xi,g)\n",
    "plt.scatter(xi,g)\n",
    "plt.ylim(-2,10)\n",
    "plt.title('''\\\n",
    "parabolesque gradient in x^3 region,\n",
    "linear in x^2 region as expected''');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In short: symbolic differentiation is not our saving grace.\n",
    "\n",
    "* Pro: Gradients are exact, if you can compute them\n",
    "* Con: Need to implement in CAS. Full-featured Cas not easily available in all languages\n",
    "* Con: lead to expression swell by losing any structure of the program (needs to be recovered separately0\n",
    "* Con: Cannot handle common control-flow structures like loops and conditionals easily"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What we need\n",
    "\n",
    "To recap: \n",
    "\n",
    "Finite differences is\n",
    "* easy to implement in any language\n",
    "* handles arbitrary (halting) programs but\n",
    "* is inaccurate unless we're ready to pay a large computational overhead\n",
    "\n",
    "Symbolic differentiation is:\n",
    "* exact to machine precision\n",
    "* can lead to exccessive / inefficient computation if not careful\n",
    "* cannot handle complex programs with control flow structures\n",
    "\n",
    "\n",
    "<h4> So what we need is a third approach! </h4>\n",
    "\n",
    "One, that is \n",
    "* exact\n",
    "* efficient\n",
    "* can handle arbitrayr programs\n",
    "* that is easy to implement in many languages\n",
    "\n",
    "\n",
    "This third approach is 'Automatic' differentiation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Short Interlude on Linear Transformations\n",
    "\n",
    "Before we start, let's first look at *linear transformations** from ℝᵐ → ℝⁿ:\n",
    "\n",
    "$$\n",
    "y(x) = Ax\n",
    "$$\n",
    "\n",
    "With a given basis, this is representable as a (rectangular0 matrix: \n",
    "\n",
    "$$\n",
    "y_i(x) = A_{ij}x_j\n",
    "$$\n",
    "\n",
    "\n",
    "For a given linear problem, there are few ways we can run this computation\n",
    "\n",
    "\n",
    "1. **full matrix computation**\n",
    "\n",
    "   i.e. we store the full (dense) $nm$ elements of the rectangular matrix and \n",
    "   compute an explicit matrix multiplication.\n",
    "   \n",
    "   The computation can be fully generic for any matrix\n",
    "   \n",
    "```python\n",
    "    def result(matrix, vector):\n",
    "        return np.matmul(matrix,vector)\n",
    "```\n",
    "<br/>\n",
    "\n",
    "2. **sparse matrix computation**\n",
    "\n",
    "   If many $A_{ij}=0$, it might be wasteful to expend memory on them. We can just \n",
    "   create a sparse matrix, by\n",
    "   \n",
    "   * storing only the non-zerro elements \n",
    "   * storing a look-up table, where those elements are in the matrix\n",
    "   \n",
    "   The computation can be kept general\n",
    "\n",
    "```python\n",
    "    def result(sparse_matrix, vector):\n",
    "        return sparse_matmul(sparse_matrix,vector)\n",
    "```\n",
    "\n",
    "<br/>\n",
    "   \n",
    "3. **matrix-free computation**\n",
    "\n",
    "    In many cases a linear program is not explicitly given by a Matrix, but it's\n",
    "    given as *code* / a \"black-box\" function. As long as the computation in the body of \n",
    "    keeps to (hard-coded) linear transformation the program is linear. The matrix elements\n",
    "    are no longer explicitly enumerated and stored in a data structure\n",
    "    but implicitly defined in the source code.\n",
    "    \n",
    "    This is not anymore a generic computation, but each linear transformation is its own\n",
    "    program. At the same time this is also the most memory efficient  representation. No\n",
    "    lookup table is needed since all constants are hard-coded.\n",
    "    \n",
    "    \n",
    "```python\n",
    "    def linear_program(vector):\n",
    "        z1,z2 = 0,0\n",
    "        z1 += A_11*x1\n",
    "        z2 += A_12*x2\n",
    "        z2 += A_22*x2\n",
    "        return [z1,z2]\n",
    "```\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Recovering Matrix Elements from matrix-free computations\n",
    "\n",
    "\n",
    "#### Matrix-vector products\n",
    "\n",
    "In the matrix-free setting, the program does not give access to the matrix elements,\n",
    "but only computes \"matrix-vector products\" (MVP)\n",
    "\n",
    "We can use basis vectors to recover the matrix **one column at a time**\n",
    "\n",
    "<img src=\"./assets/mvp.png\" alt=\"A Matrix Vector Product\" width=\"600\"/>\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M derived from matrix-vector products:\n",
      "[[2 0 0]\n",
      " [0 1 3]]\n"
     ]
    }
   ],
   "source": [
    "def matrix_vector_product(x):\n",
    "    x1,x2,x3 = x\n",
    "    z1,z2 = 0,0\n",
    "    z1 += 2*x1  #MVP statement 1\n",
    "    z2 += 1*x2  #MVP statement 2\n",
    "    z2 += 3*x3  #MVP statement 3\n",
    "    return np.asarray([z1,z2])\n",
    "\n",
    "M = np.concatenate([\n",
    "    matrix_vector_product(np.asarray([1,0,0])).reshape(-1,1),\n",
    "    matrix_vector_product(np.asarray([0,1,0])).reshape(-1,1),\n",
    "    matrix_vector_product(np.asarray([0,0,1])).reshape(-1,1),\n",
    "],axis=1)\n",
    "print(f'M derived from matrix-vector products:\\n{M}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Vector Matrix product (VMP)\n",
    "\n",
    "The same matrix induces a \"dual\" linear map: ℝⁿ → ℝᵐ \n",
    "$$ x_k = y_i A_{ik}$$\n",
    "\n",
    "i.e. instead of a Matrix-vector product it's now a *vector-Matrix* product (VMP)\n",
    "\n",
    "If one has access to a \"vector-Matrix\" program corresponding to a matrix $A$ one\n",
    "can again -- as in the MVP-case -- recover the matrix elements, by feeding in basis vectors.\n",
    "\n",
    "This time the matrix is built **one row at a time**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"./assets/vmp.png\" alt=\"A Matrix Vector Product\" width=\"600\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M derived from vector-matix products:\n",
      "[[2 0 0]\n",
      " [0 1 3]]\n"
     ]
    }
   ],
   "source": [
    "def vector_matrix_product(z):\n",
    "    x1,x2,x3 = 0,0,0\n",
    "    z1,z2 = z\n",
    "\n",
    "    x3 += z2*3 #VMP version of statement 3\n",
    "    x2 += z2*1 #VMP version of statement 2\n",
    "    x1 += z1*2 #VMP version of statement 1\n",
    "\n",
    "    return np.asarray([x1,x2,x3])\n",
    "\n",
    "\n",
    "M = np.concatenate([\n",
    "    vector_matrix_product(np.asarray([1,0])).reshape(1,-1),\n",
    "    vector_matrix_product(np.asarray([0,1])).reshape(1,-1),\n",
    "],axis=0)\n",
    "print(f'M derived from vector-matix products:\\n{M}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Short Recap:\n",
    "\n",
    "For a given linear transformation, characterized by a matrix $A_{ij}$ we have a forward (matrix-vector) and backward (vector-matrix) map $$y_i = A_{ij}x_k$$ $$x_j = y_i A_{ij}$$\n",
    "\n",
    "and we can use either to recover the full matrix $A_{ij}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wide versus Tall Transformation\n",
    "\n",
    "If you look at the code above, you'll notice that the number of calls necessary to the MVP or VMP program\n",
    "is related to the dimensions of matrix itself.\n",
    "\n",
    "For a $n\\times m$ matrix (for a map: ℝᵐ → ℝⁿ), you need as $m$ calls to the \"Matrix-vector\" program to \n",
    "built the full matrix one-column-at-a-time. Likewise you need $n$ calls to the \"vector-Matrix\" program\n",
    "to build the matrix one-row-at-a-time.\n",
    "\n",
    "This becomes relevant for very asymmetric maps: e.g. scalar maps from very high-dimensional spaces\n",
    "$\\mathbb{R}^{10000} \\to \\mathbb{R}$ the \"vector-Matrix\" appraoch is *vastly* more efficient than the\n",
    "\"Matrix-vector one. There's only one row, so only one call too the VMP program is needed to construct the full matrix!\n",
    "\n",
    "Similarly, functions mapping few variables into very high dimensional spaces $\\mathbb{R} \\to \\mathbb{R}^{10000}$\n",
    "it's the opposite: the \"Matrix-vector\" approach is much better suited than the \"vector-Matrix\" one (this time it's a single column!).\n",
    "\n",
    "\n",
    "## Function Compositions\n",
    "\n",
    "Of course copositions $(f\\circ g)(x) = f(g(x))$ of linear maps are also linear, so the above applies.\n",
    "\n",
    "<img src=\"./assets/composition.png\" alt=\"A Matrix Vector Product\" width=\"400\"/>\n",
    "\n",
    "Depending on whether the \"Matrix-vector\" or \"vector-Matrix\" appraoch is used, the data is propagated **forwards** or **backwards**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "|         Forward     |  Backward |\n",
    ":-------------------------:|:-------------------------:\n",
    "![](./assets/forward.png) | ![](./assets/backward.png)\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "    \n",
    "### From Matrices to Graphs\n",
    "\n",
    "The \"vector-Matrix\" or \"Matrix-vector\" picture can be generalized to arrbitrary directed acyclic graphs.\n",
    "\n",
    "* In the \"Matrix-vector\" picture the node value is the edge-weighted sum of the \"upstream nodes\".\n",
    "* In the \"vector-Matrix\" picture the node value is the edge-weighted sum of its \"downstream nodes\".\n",
    "\n",
    "(one could in principle always recove a rectangular/matrix-like version of a DAG  by inserting trivial nodes)\n",
    "\n",
    "|              |    |\n",
    ":---------- : | : ------ : \n",
    "![](./assets/graphs.png) | ![](./assets/dag.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def graph_like(x):\n",
    "    x1,x2,x3 = x\n",
    "    y1 = 2*x1+x2\n",
    "    z1,z2 = y1+2*x3,x3-y1 #note that we reach \"over\" the \"ys\" and diectly touch x_n\n",
    "    return np.asarray([z1,z2])\n",
    "\n",
    "def matrix_like(x):\n",
    "    x1,x2,x3 = x\n",
    "    y1 = 2*x1+x2\n",
    "    y2 = x3 #can just introduce a dummy variable to make it matrix-like\n",
    "    z1,z2 = y1+2*x3,y2-y1\n",
    "    return np.asarray([z1,z2])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M derived from matrix like computation:\n",
      "[[ 2  1  2]\n",
      " [-2 -1  1]]\n",
      "M derived from graph-like products:\n",
      "[[ 2  1  2]\n",
      " [-2 -1  1]]\n"
     ]
    }
   ],
   "source": [
    "M = np.concatenate([\n",
    "    matrix_like(np.asarray([1,0,0])).reshape(-1,1),\n",
    "    matrix_like(np.asarray([0,1,0])).reshape(-1,1),\n",
    "    matrix_like(np.asarray([0,0,1])).reshape(-1,1),\n",
    "],axis=1)\n",
    "print(f'M derived from matrix like computation:\\n{M}')\n",
    "\n",
    "\n",
    "M = np.concatenate([\n",
    "    graph_like(np.asarray([1,0,0])).reshape(-1,1),\n",
    "    graph_like(np.asarray([0,1,0])).reshape(-1,1),\n",
    "    graph_like(np.asarray([0,0,1])).reshape(-1,1),\n",
    "],axis=1)\n",
    "print(f'M derived from graph-like products:\\n{M}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivatives\n",
    "\n",
    "\n",
    "Why are we talking about linear transformations? After all lot of the code we write is non-linear! However, derivatives are always linear.\n",
    "\n",
    "And derivatives (the jacobian) of a composition $f\\circ g$ is the composition of linear derivatives (the jacobians\n",
    "of each map) i.e. the full jacobian Matrix is the  result of multipying all Jacobians of the composition.\n",
    "$$J = J_0 J_1 J_2 J_3 \\dots J_n $$\n",
    "\n",
    "(This is just the chain rule)\n",
    "$$z = f(y) = f(g(x))\\hspace{1cm} \\frac{\\partial f_i}{\\partial x_j} = \\frac{\\partial f_i}{\\partial z_j}\\frac{\\partial z_j}{\\partial x_k}$$ \n",
    "\n",
    "\n",
    "I.e. finding derivatives, means characterizing the jacobian matrix. From the above discussion, we can use the \"Jacobian-vector product\" (JVP, builds Jacobians column-wise) or \"vector-Jacobian product\" (builds Jacobians row-wise) approach.\n",
    "\n",
    "In the language of automatic differentiation \n",
    "\n",
    "* Jacobian-vector products (JVP) = forward mode (forward propagation)\n",
    "\n",
    "$$ Jv_n =  J_0 J_1 J_3 \\dots J_n v_n = J_0 J_1 J_2 J_3 v_3 = J_0 J_1 J_2 v_2 = J_0 J_1 v_1 = J_0 v_0 = \\text{col}$$\n",
    "\n",
    "* vector-Jacobian products (VJP) = reverse mode (reverse propagation)\n",
    "\n",
    "$$ v_0 J = v_0 J_0 J_1 J_3 \\dots J_n = v_1 J_1 J_2 J_3 \\dots J_n = v_2 J_2 J_3 \\dots J_n = v_3 J_3 \\dots J_n = \\dots = v_n J_n = \\text{row}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Let's work this out on a very simple problem\n",
    "\n",
    "\n",
    "<img src=\"./assets/intro_autodiff.png\" alt=\"A Matrix Vector Product\" width=\"700\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the forward pass we use \"Matrix-vector\" products and need to do two evaluation\n",
    "\n",
    "<img src=\"./assets/intro_fwd.png\" alt=\"A Matrix Vector Product\" width=\"700\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the backward pass we use \"vector-Matrix\" products and need to do only a single evaluation\n",
    "\n",
    "<img src=\"./assets/intro_bwd.png\" alt=\"A Matrix Vector Product\" width=\"700\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both approaches give the same result. Since this is a map from $\\mathbb{R}^2 \\to \\mathbb{R}^1$ the backward pass is more efficient than the forward pass\n",
    "\n",
    "\n",
    "Let's look at a real-life example\n",
    "\n",
    "$$z(x_1,x_2) = y + x_2 = x_1x_2 + x_2$$\n",
    "\n",
    "This is easy python code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\n"
     ]
    }
   ],
   "source": [
    "def mul_func(x1,x2):\n",
    "    return x1*x2\n",
    "\n",
    "def sum_func(x1,x2):\n",
    "    return x1+x2\n",
    "\n",
    "def function(x):\n",
    "    x1,x2 = x\n",
    "    y = mul_func(x1,x2)\n",
    "    z = sum_func(y,x2)\n",
    "    return z\n",
    "\n",
    "print(function([2,4]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the forward pass, an autodiff system needs to create a JVP implementation for each elementary operation "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mul_jvp(x1,dx1,x2,dx2):\n",
    "    y  = mul_func(x1,x2)\n",
    "    dy = x1*dx2 + x2*dx1\n",
    "    return y, dy\n",
    "\n",
    "def sum_jvp(x1,dx1,x2,dx2):\n",
    "    return sum_func(x1,x2), dx1 + dx2\n",
    "\n",
    "def function_jvp(x,dx):\n",
    "    x1,x2 = x\n",
    "    dx1,dx2 = dx\n",
    "    y, dy = mul_jvp(x1,dx1,x2,dx2)\n",
    "    z, dz = sum_jvp(y,dy, x2, dx2)\n",
    "    return z,dz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since in the forward pass we build \"column-at a time\" and our final jacobian is has shape (1x2), i.e. two columns we need two forward passes to get the full Jacobian. Not that for eacch forward pass we also get the fully computed functino value delivered on top!\n",
    "\n",
    "\n",
    "Also note that the \"JVP\" version of the functino has the same *structure* as the original function. For each call in the original program there is an equivalent call in the JVP program. However the JVP call does always two things at once\n",
    "\n",
    "1. compute the nominal result\n",
    "2. compute the differentials\n",
    "\n",
    "So it has roughly 2x the run-time as the original program (depending on the complexity of the derivatives). Said another way: computing the one-pass in the derivative has the same computational complexity as the function itself."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(12, 4)\n",
      "(12, 3)\n"
     ]
    }
   ],
   "source": [
    "print(function_jvp([2,4],[1,0]))\n",
    "print(function_jvp([2,4],[0,1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the backward pass we build \"row-at-a-time'. For each elementary operation we need to build a VJP implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mul_vjp(x1,x2,dx1,dx2,dout):\n",
    "    dx2 += dout * x1\n",
    "    dx1 += dout * x2\n",
    "    return dx1,dx2\n",
    "\n",
    "def sum_vjp(x1,x2,dx1,dx2,dout):\n",
    "    dx1 += dout * 1\n",
    "    dx2 += dout * 1\n",
    "    return dx1,dx2\n",
    "\n",
    "def function_vjp(x,dz):\n",
    "    \n",
    "    #run forward\n",
    "    x1,x2 = x\n",
    "    y = mul_func(x1,x2)\n",
    "    z = sum_func(y,x2)\n",
    "\n",
    "    #zero gradients\n",
    "    dy  = 0    \n",
    "    dx1 = 0\n",
    "    dx2 = 0\n",
    "    \n",
    "    #run backward\n",
    "    dy,dx2  = sum_vjp(y,x1, dy, dx2, dz)\n",
    "    dx1,dx2 = mul_vjp(x1,x2, dx1, dx2, dy)\n",
    "    return z,[dx1,dx2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we see the power of backward propagation (or the reverse mode)  we get all gradients of the single row ine oone go. Since this Jacobian only has one row, we're done! And we get the function value delivered on top of the gradients as well!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(12, [4.0, 3.0])\n"
     ]
    }
   ],
   "source": [
    "print(function_vjp([2,4],1.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, let's look at the \"VJP\" code. The forward pass is *exactly* the same as the original function. This just records the final result and all intermediate values, which we will need for the backward pass.\n",
    "\n",
    "Moving on to the backward pass, we see again, as in JVP, it has the same *structure* as the forward pass. For each call to a subroutine there is an equivalent call in the backward pass to compute the VJP. \n",
    "\n",
    "\n",
    "As in the JVP case, the computational complexity of one backward pass is roughly the same as the forward pass. Now unlike the JVP-case we only needed a single pass for **all the gradients** of this scalar function. So obtaining the **full gradient** of a function is only as expensive as the function itself."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Recap:\n",
    "\n",
    "Above we have built a *manual* autodiff system. Let's recap what we needed to do\n",
    "\n",
    "* define a set of operations we want to be differentiable\n",
    "* define sub-routines for nominal operations, JVP and VJP\n",
    "\n",
    "\n",
    "\n",
    "Once given a program, we had to do the following\n",
    "\n",
    "**In the forward mode**:\n",
    "\n",
    "* just replace the nominal function with the JVP one\n",
    "* for each variable in the program allocate a \"differential\" variable and pass it\n",
    "  into the JVP whereever we also pass the nominal variable\n",
    "  \n",
    "  \n",
    "**In the backward mode**:\n",
    "\n",
    "* Run the program forward, keep track of all values\n",
    "* keep track of the order of operations on a \"record\" of sorts\n",
    "* allocate \"differential\" variables for all values and initialize to zero\n",
    "* use the record to replay the order of operations backwards, passing along the \n",
    "  appropriate differential values, and updating the relevant ones with the result\n",
    "  of the VJP\n",
    "\n",
    "\n",
    "All of this is pretty mechanistic and hence \"automatable\". And given that it's a very narrow\n",
    "domain of only implementing JVP/JVP operations this is easy to do in any language.\n",
    "\n",
    "That's why it's **automatic differentiation**\n",
    "  \n",
    "  \n",
    "What we gain from this is that we get\n",
    "\n",
    "* exact derivatives (to machine precision) for arbitrary composed of the operations we define\n",
    "* complexity of a derivative-pass through the program is of same order of complexity as the original program\n",
    "* often only a single pass is necessary (e.g. scalar multi-variate functions)\n",
    "* unlike symbolic differrentiation, the structure of the program is preserved and allows naturally to avoid\n",
    "  repetitive calculations of the same values\n",
    "* (we will see that) arbitrary control flows are handles naturally\n",
    "* it's something that is easy for a comoputer do and for a progarmmer to imlpement\n",
    "\n",
    "\n",
    "\n",
    "Some notes on pros and cons:\n",
    "\n",
    "**In the forward mode**:\n",
    "\n",
    "the signature of each opeartion basically extends \n",
    "  ```c++\n",
    "  float f(float x,float y,float z)\n",
    "  ```\n",
    "  to\n",
    "  ```c++\n",
    "  pair<float> f(float x,float dx,float y,float float dy, float z,float dz)\n",
    "  ```\n",
    "  * if you use composite types (\"dual numbers\") that hold both x,dx you can basically \n",
    "    keep the signature unchanged\n",
    "    ```c++\n",
    "    f(dual x, dual x, dual z)\n",
    "    ```\n",
    "  * together with operator overloading on these dual types e.g. `dual * dual` you can \n",
    "    essentially keep the source code unchanged\n",
    "    ```c++\n",
    "    float f(float x, float y): return x*y\n",
    "    ``` \n",
    "    ->\n",
    "    ```c++\n",
    "    dual f(dual x,dual y): return x*y\n",
    "    ```\n",
    "    \n",
    "* That means it's very easy implement. And memory efficient, no superfluous values are kept when they run out of scope.\n",
    "* But forward more better for vector-value functions of few parameters\n",
    "\n",
    "\n",
    "**In the reverse mode**:\n",
    "\n",
    "* very efficient, but we need to keep track of order (need a \"tape\" of sorts)\n",
    "* since we need to access all intermediate varriables, we can run into memory bounds\n",
    "* the procedurer is a bit more complex than fwd: 1) run fwd, 2) zero grads 3) run bwd\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## I don't want to implement an autodiff system.. Aren't there libraries for this??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yes there are! And a lot of them in many languages. On the othe rhand, try finding CAS systems in each of those \n",
    "\n",
    "<img src=\"./assets/autodiff_systems.png\" alt=\"A Matrix Vector Product\" width=\"700\"/>\n",
    "\n",
    "This is PyHEP, so let's focus on Python. Here, basically what you think of as \"Machine Learning frameworks\" are at the core autodiff libraries\n",
    "\n",
    "* Tensorflow\n",
    "* PyTorch\n",
    "* JAX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's focus on jax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "import jax\n",
    "import jax.numpy as jnp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`jax.numpy` is almost a drop-in rerplacement for `numpy`. I do `import jax.numpy as jnp` but if you're daring you could do `import jax.numpy as np`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/cranmer/anaconda3/envs/stats-book-2/lib/python3.8/site-packages/jax/lib/xla_bridge.py:130: UserWarning: No GPU/TPU found, falling back to CPU.\n",
      "  warnings.warn('No GPU/TPU found, falling back to CPU.')\n"
     ]
    }
   ],
   "source": [
    "x = jnp.array([1,2,3])\n",
    "y = jnp.array([2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3 5 7]\n",
      "[ 2  6 12]\n",
      "[0.         0.69314718 1.09861229]\n",
      "[ 7.3890561  20.08553692 54.59815003]\n"
     ]
    }
   ],
   "source": [
    "print(x+y)\n",
    "print(x*y)\n",
    "print(jnp.log(x))\n",
    "print(jnp.exp(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "64.0\n",
      "48.0\n",
      "24.0\n",
      "6.0\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(f(4.0))\n",
    "print(jax.grad(f)(4.0)) #boom!\n",
    "print(jax.grad(jax.grad(f))(4.0)) #boom!\n",
    "print(jax.grad(jax.grad(jax.grad(f)))(4.0)) #boom!\n",
    "print(jax.grad(jax.grad(jax.grad(jax.grad(f))))(4.0)) #boom!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f972aeb56a0>]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xi = jnp.linspace(-5,5)\n",
    "yi = f(xi)\n",
    "\n",
    "plt.plot(xi,yi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gradient only defined for scalar-output functions. Output had shape: (50,).\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    jax.grad(f)(xi)\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Whoops, jax.grad defaults to reverse mode with a single backward pass, but through broadcasting we get a `vector -> vector` map. We can use some jax magic to \"unbroadcast\" the function, take the gradient and re-broadcast it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DeviceArray([7.50000000e+01, 6.90024990e+01, 6.32548938e+01,\n",
       "             5.77571845e+01, 5.25093711e+01, 4.75114536e+01,\n",
       "             4.27634319e+01, 3.82653061e+01, 3.40170762e+01,\n",
       "             3.00187422e+01, 2.62703040e+01, 2.27717618e+01,\n",
       "             1.95231154e+01, 1.65243648e+01, 1.37755102e+01,\n",
       "             1.12765514e+01, 9.02748855e+00, 7.02832153e+00,\n",
       "             5.27905040e+00, 3.77967514e+00, 2.53019575e+00,\n",
       "             1.53061224e+00, 7.80924615e-01, 2.81132861e-01,\n",
       "             3.12369846e-02, 3.12369846e-02, 2.81132861e-01,\n",
       "             7.80924615e-01, 1.53061224e+00, 2.53019575e+00,\n",
       "             3.77967514e+00, 5.27905040e+00, 7.02832153e+00,\n",
       "             9.02748855e+00, 1.12765514e+01, 1.37755102e+01,\n",
       "             1.65243648e+01, 1.95231154e+01, 2.27717618e+01,\n",
       "             2.62703040e+01, 3.00187422e+01, 3.40170762e+01,\n",
       "             3.82653061e+01, 4.27634319e+01, 4.75114536e+01,\n",
       "             5.25093711e+01, 5.77571845e+01, 6.32548938e+01,\n",
       "             6.90024990e+01, 7.50000000e+01], dtype=float64)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jax.vmap(jax.grad(f))(xi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "that looks better!\n",
    "\n",
    "`jax.grad(f)` just returns another function. Of course we can just \n",
    "take the gradient of that as well. And so on..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f97007d0c40>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "g1i = jax.vmap(jax.grad(f))(xi)\n",
    "g2i = jax.vmap(jax.grad(jax.grad(f)))(xi)\n",
    "g3i = jax.vmap(jax.grad(jax.grad(jax.grad(f))))(xi)\n",
    "plt.plot(xi,yi,  label = \"f\")\n",
    "plt.plot(xi,g1i, label = \"f'\")\n",
    "plt.plot(xi,g2i, label = \"f''\")\n",
    "plt.plot(xi,g3i, label = \"f'''\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Control Flow\n",
    "\n",
    "Back when discussing symbolic differentiation we hit a snag when adding \n",
    "control flow through to our prorgam. In Jax this just passes through\n",
    "transparently. \n",
    "\n",
    "\n",
    "Let's compare this to finite differences. So far the only system\n",
    "we had to compute derivatives of control-flow-ful programs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f9718b2dee0>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def control_flow_func(x):\n",
    "    if x > 2:\n",
    "        return x**2\n",
    "    else:\n",
    "        return x**3\n",
    "    \n",
    "\n",
    "first_gradient_of_cflow = jax.grad(control_flow_func)\n",
    "    \n",
    "xi = jnp.linspace(-2,5,101)\n",
    "yi = np.asarray([first_gradient_of_cflow(xx) for xx in xi])\n",
    "plt.plot(xi,yi,c = 'k')\n",
    "\n",
    "xi = jnp.linspace(-2,5,11)\n",
    "yi = np.asarray([first_gradient_of_cflow(xx) for xx in xi])\n",
    "plt.scatter(xi,yi, label = 'jax autodiff')\n",
    "\n",
    "\n",
    "\n",
    "xi = jnp.linspace(-2,5,11)\n",
    "yi = np.asarray([control_flow_func(xx) for xx in xi])\n",
    "plt.scatter(xi,np.gradient(yi,xi), label = 'finite differences')\n",
    "\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can start to see the benefits autodiff. Among other things, finite differnces becomes\n",
    "quite sensitive to exactly where the evaluation points are (e.g. wrt  to the discontinuity)\n",
    "\n",
    "\n",
    "As we compute higher derivatives, this error compounds badly for finite differences. But for\n",
    "autodiff, it's smooth sailing!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f96f8748f70>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "second_gradient_of_cflow = jax.grad(first_gradient_of_cflow)\n",
    "xi = jnp.linspace(-2,5,101)\n",
    "yi = np.asarray([second_gradient_of_cflow(xx) for xx in xi])\n",
    "plt.plot(xi,yi,c = 'k')\n",
    "\n",
    "xi = jnp.linspace(-2,5,11)\n",
    "yi = np.asarray([second_gradient_of_cflow(xx) for xx in xi])\n",
    "plt.scatter(xi,yi, label = '2nd deriv jax autodiff')\n",
    "\n",
    "xi = jnp.linspace(-2,5,11)\n",
    "yi = np.asarray([control_flow_func(xx) for xx in xi])\n",
    "plt.scatter(xi,np.gradient(np.gradient(yi),xi), label = '2nd deriv finite differences',)\n",
    "\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Custom Operations\n",
    "\n",
    "\n",
    "Not all our programs are so simple. Consider this \n",
    "\n",
    "\n",
    "```\n",
    "def func(x)\n",
    "    y_root = solve(x^2 + y^2 == 1,x = x, y_start = 2.0)\n",
    "    return y_root\n",
    "    \n",
    "```\n",
    "\n",
    "solving this often goes through some iterative algorithm like Brent bracketing\n",
    "But, differentiating through the iteration is not the right solution.\n",
    "\n",
    "We can add our own custom gradients\n",
    "\n",
    "\n",
    "Recall the implicit function theorem\n",
    "$$\n",
    "f(x,y) = x^2 + y^2 -1 = 0\n",
    "$$\n",
    "\n",
    "$$\n",
    "df = 0 \\leftrightarrow 2x dx + 2y dy = 0\n",
    "$$\n",
    "\n",
    "$$\n",
    "dy/dx = -x/y\n",
    "$$\n",
    "\n",
    "\n",
    "How do we teach this an autodiff system:\n",
    "\n",
    "\n",
    "Recall: \n",
    "\n",
    "* we can choose which operations we consider \"fundamental\"\n",
    "* we don't need to constrain ourselves to the lowest possible representationo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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eE5HL1i0iS0UkU0QyS0pKXBS6Z6utrWXdunUUbvmQtwZn8JvQj9hmG8lNRT9iR9RcCmqaAIiOjiY0NNTN2WqebHtdP25r+g3L66fww8itfBT+B6LL9pFXUU9tbS1btmxxd4pu15b78Noy0PbD2IdpRCn1jYgEYR/Wsdh5IV8aiLuhoYHt27ezY8cOjA2lbE7bxQBDGb8138dfrXNRUfb/B30jdW+3mmvY792DZyxL2VaUwEu91/FVwqv8uGQ+586lsGnTJpKSkkhMbHlYTG/Xlj283cBgEUkWkQBgEfbBt52dA9IBRGQ4EAT4xi7cJcxmM1u3buXll18mIyODqIZT/FDeI9TQxIPWX/C6dR7K8WsP9jeybPZQN2eseYtls4cS7G/EEBDElxELmJO/lDIVztu9/86eD5/HZrOxYsUK6uvr3Z2q27Ra8JRSFuBxYD1wBPhYKXVIRJ4Tkdsdi/0U+J6I7Ac+AL6jfOyEgVKKvXv38vLLL7NhwwZM9fWMsWXxqOETTqhETA+uZ+Hd9+p76rQu43zvnkEEdc0MNo7/M1sZz4OGtUxXGdRUlfPZZ5/57Pk8/aSFC5lMJg4dOsT6tWuYbt3I9YYDfGGdStD0HzNz5s3uTk/zMfn5+WzcuJETJ04wyHaUB43/4KAtic9kDum33MmUKVPcnWKX0J0HdJOAgACOHchkgfVTBkkeb1hvpyJ8DD+adqO7U9N8UGxsLGPGjEEpxalTBv5oiWKpYSUP8xHv/cNKYmKizz2Hqwuei1gsFv7+15e5ufAVwqWelQOe4fz5eual30RAQIC709N8UEBAAGPGjGHMmDFUV1dz4MAB3tyewL21b/GIWs7ydw0s/vGvCQwMdHeq3UYXvE640BtxWQ1Tq9bxWsJnNBDAvrS/sGDWnTQsX64HZdF6hIiICNLS0pg6dSoHs2YTsmYJD5ve4PUXG3jf7w4Kqhp84skM3XlAB63OyuOpT/ZyvqyGG8pX8VbCCopsvVgx5H+4/ua7EBHuvfde3V+d1qOICKMnTCH2R5s5ZBzO49a/kV71KQp84skM/WnsoOc/yqDiZBa31XzK2/0/46i1L3eWPcbfc6MuLONLhwqaZ+kVFcuTfv+Hr6zX8uugv/NDwwpsTfUXPZnhjXTBayebzUZGRgY529by4IAC/l/8GnaaB7Gw4gfURqSQX2lyd4qa1ibnqm08Yn6SFZYb+GnASn5mfRtbY51X/w3rc3jtUFNTw6pVqzh16hSPjm7it+Hvs9k6lqWWJ2gKtz8xoZ+c0DxFc6/KP617gHpjE09EbcOvCt4J/L67U+syuuC10cmTJ1m1ahV1dXUkWnNY3Hs9G63jedT8I5oM9quw+skJzZMsmz3U3qsy4TzbsBhrNfyg1zZizEYqK6YRGRVFbW0t/v7+XnN6Rh/StsJqtbJhwwaWL19OXV0d/W1nWGxcQ2bARCpve5PYyF76yQnNIzk/meEXFM5fwh/jE/M0FvpnsOlPS6moqODQoUNs377d3am6jH7S4ioqKyv59NNPyc21946VYDvHElnJHkYw8j/WExoa7uYMNc21Kioq+eblB7iVLaww3EpJ3DTKy8t54oknCA/3jL93PaZFB5lMJkaNGkVMTAzR1kIWy2r2qcFU3/ArXew0rxQVFcmUJ97lK6awwPYlwQXbMJvNbNq0yd2puYQueFeRkJBAv379aCw9xRLDp5xU/fjccAuTJk12d2qa1iXq6+vJycnhRO872GIbw7flC+Ks+WRlZVFcXNz6Cno4XfCuwmQyser9N3iYjykjkhPjf05KyiAiIiJaf7OmeaCAgACsVit1dXX8S27kgEphieFTwq0lfPXVV+5Or9P0VdoWKKVY9fF7zK9bjg3BvPAjZiYN4+zZs+5OTdO6jJ+fH1OmTGH8+PHs3LmTdRl+hDZ9yMPyKa8dC+TUqetISUlxd5odpvfwWrB969eMzHmNOCo5MvWPpIyYQEhICMOHD3d3aprW5QIDA5k2bRpP/GQZ2SOfxYwf98tnbFjn2X3p6YLnYDKZqKqqAuB8bi6NXz3HWEMO6/r/mOtnL3BzdprmHiEhIdy18AFyZrxKbyqZUvQe+/bsdndaHdamQ9rWBuJ2LHMv9tHLFLBfKXW/C/PsUmazmWUv/JlMw3AKqhv5ftO7PBOzh48D7+aeJc+4Oz1Nc7vJN84jo/x5pmX/J+u//DVT//kEBdVNHtfDSqt7eI6BuP8M3AKMAO5zDLbtvMxg4GkgTSk1EnjS9al2DavVyn+8+BofbzlMfp2NSRX/4D+jN/NFw1is057WwyZqmsO0ux5hTfgiZtsymFa7ziN7WHHVQNzfA/6slKoAUEp5xPVrpRSfffYZKzZlYg0Mp0/FXl4f8AUnrX34Sc23eWVLbusr0TQf8mLj3XxtHcNz/u8wnqMAHtXDiqsG4h4CDBGRbSKyw3EIfJmeNC6tUor169eTnZ1NTYOZ4NBQ3uizCgOK7xQtoik8wat7jdC0jsirNvOE+XEKVAyvBvw/wsoPA3jMZ8VVFy38gMHAjdgH5f6riEReupBS6nWlVKpSKjUuLs5FoTtmy5Yt7NixA4DwIH9+mXKQMcbT/KxpKfnR9qdSdM8nmnaxvpHBVBHGksK7iJYa/hC1Apup0mM+K20peG0ZiPs8sEYpZVZKnQaOYy+APVJmZiYbN2688PPS0TaWhHzN25bZfKUmISK65xNNu4LmsW+PR13PL8tvZlbwUZZYVvCT9GvcnVqbuGog7tXY9+4QkVjsh7inXJem6xw+fJi1a9de+DlI1fFExAaOkMzbIYt1zyeadhXNPaz0jwrlfb87Wdc4hqdjv8bv8Gp3p9Ymrd6WopSyiEjzQNxG4K3mgbiBTKXUGse8m0XkMGAFlimlyroy8Y5ovmHyrrvu4osvvqCxoYGZ6mv8xIrc8w4ZI8e5N0FN8wDzx/e7sDOwc9dAStYuYmzOn9m3Zyrjrp3o5uyurk3n8JRSXyqlhiilrlFK/cYx7ReOYoey+4lSaoRSarRS6sOuTLqjRIQRI0ZgNBppbGwkRU6TajjGisC7GaaLnaa12+RJk9ky4HGSDEXkrvkVZWU9bj/nIj73pIVSioyMDAJsddyrviQrcCIDptzj7rQ0zWPds/jHfGGcyTzZwqo3X8JisVBeXs6pUz3vrJbPdR5w4sQJCgvyuZsNmCSIAd95k3F9Et2dlqZ5LIPBwKSlr3Dq1RnMrfuIL9dci83gj9Vq7XEdDfjUHl7z3l2yymG04RRHxjxDbMJA/TSFpnVS7/gETl37C+KpQLLeYf/+/Rw9epTGxkZ3p3YRnyp4p0+fpjT3GAtkA3v9JzDljkfcnZKmeYVz584RHDOQz5nOXMM2ellLMZvNHD582N2pXcSnCl7G118zXW1DUITd/ScMRp9qvqZ1meDgYLKzs9nDGIqJZB4bMCgr+/fvd3dqF/GZT/zZs2dpPL2N6wyH+DphMUOGjXZ3SprmNeLi4liyZAmTJ03hPdttXGPIZ4g6ypkzZ6isrHR3ehf4TMHL2LyRubKJk6o/Nzz0S3eno2lex9/fn7lz5zJn4aNssY1hnmQQoExkZ2e7O7ULfKLg5eXlYTy5jv5SSt7EZwgLDnJ3SprmtUaPGknobb/DHzOpag/79+/vMb0k+0TB27j+c+4wbGabIZVpcz2mX1JN81gTJl3PrvhF3GzYTUNJDnl5PaO/PK8veEVFRfQ5swI/bETe9mt9C4qmdZOx33qeUiKZ57+NE8ePuzsdwAcKXsbaD0mXTP4ZOIuRE6a6Ox1N8xnhvaI5PvJJhlmPE2fTe3hdrrS0lH5nPqaWYK6Z/6y709E0nzPh9h9whn702vEiympxdzreWfBWZ+WR9sJGHvrZz5lqOMgnaiYjRoxyd1qa5nOCAgM5O/ZJBljO8V/P/5Lkp9aS9sJGt42B4XUFb3VWHk+vPMC5wmKe6fMNJSqCP1ZN95hBRjTN2xT3n80BWxJLrB/ih8WtA/94XcF7af0xTGYrU+UgaUGneblyOjXBCR4zyIimeZs//iuH/2tZSKKhhLvqPwXcN/CP1xW85sFEfhi2kSIVyYd+8y6armla98qvNLGpcSh7LSn8IGIrqrrwwvTu1qaCJyJzROSYiJwUkaeustzdIqJEJNV1KbZP38hgRssprjce4k3LLVj8wy5M1zSt+/WNDMbgH8QrFdcx0K+MOeaNF6Z3N5cMxO1YLhz4EbDT1Um2x7LZQ3nU73OqVQjvW9MB9IA8muZGzQP/bPBL44S1D0/03otfQ6VbPpOuGogb4NfA74EGF+bXbpMiKphj2MWH3EwdIXpAHk1zs+aBfwYkJPBqxXWM9M/n+9H73PKZbEuPx1caiHuy8wIiMgEYoJRaKyLLWlqRiCwFlgIkJnZNL8Pn175ADH7c+civWZqgezLWtJ6geeCf3NxBFL6xmXTLJoqLi+ndu3e35tHpixYiYgD+B/hpa8t29UDcpqoyxpT9g71Rc4jTxU7TepwBAxLZGjKLCYYTZKz9oNvju2Ig7nBgFLBZRM4AU4A17rhwcXLDXwkSM8FpS7s7tKZpbTTklsdoVP6EnF7X7aOcdXogbqVUlVIqVimVpJRKAnYAtyulMrsk45YoRcyR5RySIYy59oZuDa1pWtuNGTOODMNEpkkW2zdv6NbYrRY8pZQFaB6I+wjwcfNA3CJye1cn2FblRzbT15JL7jWLMBh0jyia1pOZRn2LMGmgKvvzbu0R2SUDcV+y7I3dvncHlG1+jWoVwrD0b3d3aE3T2il97kKO2fozSe1j27Zt3RbXK560ULUlDCzewJaQWSQluP5iiKZprhUa5M/emHkMMeRyPnMdNTU13RLXKwpe/tb3CMAC137H3alomtZGKbO+j0kFMMh6jO3bt3dLTK8oeNYDKzmskrg+TV+s0DRPMXFYElsN1zJejrBn9y7q6uq6PKbHF7ymsnMk1h3gZGw6vYL93Z2OpmltZDAIpsG3Ey01RJtz2bFjB0CX3qri8QUv5+v3Aeh93SI3Z6JpWnuNmXEP9SqQZHWWXbt2UVJSwnvvvYfNZuuSeB5f8AKOreYoyaSOd1sHLZqmdVBSQhx7gyYzUQ7R1GDijTfeoLKykvLy8i6J59EFryLvJNc0HqGg/y34GT26KZrmk/bt28dBhhItNUSpUhobGwH7eDRdwaOrxInNfwdg4A16rFlN80Rjx44letBk6lQgg9SpC9NLSkq6JJ5HF7zI019wwjiIlKGj3Z2KpmkdICIsuPtuMo3jmWI4iCj7uTu9h3cJVVfGEMtxcnvPcHcqmqZ1gsFgwDZivv1qLfYrtHoP7xLVRzcD0NA/zb2JaJrWaUFD7Dsug4MrAfsenlLK5XE8tuCZjm+mXgUSnjLJ3alomtZJ/fsncszWn9EhZURHR9PU1ER1dbXL43hswQvK206mbQhJ8VHuTkXTtE7qGxnMbkYQV7mPB+9fRFhYWJcc1npmwastIbL2JLsZqUcj0zQvYDQIp8ImEGAzEVV/mgcffJDa2lqXx/HMgndmCwBnIyZg1H3faZpXqOztOD11JoP4+HhGj3b93RcuGZdWRH4iIodFJFtE/iUiA12eqbMzW6gnGHP82C4No2la9+kd34+jKhF12r5DYzQaXR7DVePSZgGpSqkxwArgRVcn6kyd2cou21AGxkV2ZRhN07pRSmwo260jULk7wdLYJTFcMi6tUmqTUqre8eMO7AP9dI2aQqT0ONutw0mJDe2yMJqmda/kuFC+sY3AYGmAvD1dEqMtBe9K49JebQTdh4F1nUnqqs5sBeAb20iSdMHTNK+RFBPKTtswFAKOw1pXa8tA3G0mIg8AqcD0FuZ3fiDukqPYMHBYDSRZFzxN8xqxYQGowEjKA/sRU3K0S2K4YlxaAERkJvAs9iEar3gA7pKBuGsKqPWLIiQwkNiwgI6tQ9O0HkdESI4LpZhoqCnskhidHpcWQETGA/+LvdgVuz5NJ9UFlEoMyXGhiOhbUjTNmyTHhpJriYSa/C5Zv6vGpX0JCAM+EZF9InLZ8I0uU1NInrWXPpzVNC+UHBvKmcZwVE0hdMGztG06h6eU+hL48pJpv3B6PdPFebWcS00+Z8zX6oKnaV4oOTaU/SoasTSAqQJCol26fs960sLcgJgqKLJF6YKnaV4oJTaMIuV4Pr4LzuN5VsGrtf8CiogiJTbMzclomuZqSbEhFKlI+w81BS5fv2cVvGr7L6BIRZEUG+LmZDRNc7XwIH/MoX3sP/h8wXP8AppC4gkP0mPQapo3Cot1PKilC579FxAcM6CVBTVN81QD4qKoJPzCEZ0reVzBa8Sf+Lh4d2eiaVoXSY4NpcAWibnK9ffieVTBa6rMp9AWRXKcvmChad4qOTaUYhWFucLXC155HkVE6U4DNM2LJceGUqiikVp9SEuRitLdQmmaF0uMCaGYSAIbSsFmdem6PafgKUWgqYgiFUVijL4lRdO8VaCfkabgeAzYoNa1j+Z7TsFrrMbf1kBjcG8C/Vzf9bOmaT2HoVdf+wsX35riOQXP8ZiJRPR1cyKapnW10Bj7vXjKVwueqrZfsQmO7rre4zVN6xki+9jHAastyW1lyfbxmIJX42h4r/gO9pSsaZrH6JMwAKsSqot9tOBVFZ0FoHffJPcmomlal0vuHUEJkTRWnHfpej2m4JnK86hSIQzsE+vuVDRN62J9I4O7pKt3Vw3EHSgiHznm7xSRJJdmif0cXjHR9I0MdvWqNU3rYYwGodY/loD6Ipeu11UDcT8MVCilBgF/AH7vyiRXZ+VhKs+j0BbJtBc3sTrrsjGENE3zIquz8jjd1IvQphLSXtjoss+8Swbidvz8N8frFUC6uGiEndVZeTy98gBxVFBENHmVJp5eeUAXPU3zUs2f+XxrJFFSS2lllcs+864aiPvCMo5Bf6qAmE5nB7y0/hgNZjO9qbzQE6rJbOWl9cdcsXpN03qYl9Yfw2S2UkwkAL2lwmWfeZcOxN2ajgzEnV9pwoDip+ZHyVF9L5quaZr3af5s77AN58dNj1Klwi6a3hmuGoj7wjIi4gf0AsouXVFHBuLuGxmMFSNrbFM5pJIumq5pmvdp/mznqnhW2W6gmtCLpneGSwbidvz8kOP1AmCjUq4ZVHLZ7KEE+1/87Gywv5Fls4e6YvWapvUwXfmZb/WQVillEZHmgbiNwFvNA3EDmUqpNcCbwHsichIox14UXWL+ePvpwpfWHyO/0kTfyGCWzR56Ybqmad6lKz/z4qIdsXZLTU1VmZmZbomtaZr3EpE9SqnUK83zmCctNE3TOstte3giUgKcbefbYoHSLkinp8fW8fW299X4HYk9UCl1xauibit4HSEimS3tqnpzbB1fb3tfje/q2PqQVtM0n6ELnqZpPsPTCt7rPhpbx9fb3lfjuzS2R53D0zRN6wxP28PTNE3rsB5X8ETkHhE5JCI2EWnx6kxLnZI6HoHb6Zj+keNxuLbGjhaRr0TkhON71BWWmSEi+5y+GkRkvmPeOyJy2mneuHa2vdX4juWsTjHWOE3vcNvb0f5xIvKNYxtli8hCp3ntbn9nOpcVkacd04+JyOz2tLUd8X8iIocdbf2XiAx0mnfF7eDi+N8RkRKnOEuc5j3k2FYnROShS9/rgth/cIp7XEQqneZ1qu0i8paIFIvIwRbmi4j8yZFbtohMcJrX8XYrpXrUFzAcGApsBlJbWMYI5AApQACwHxjhmPcxsMjx+jXg0XbEfhF4yvH6KeD3rSwfjf1RuhDHz+8ACzrR9jbFB2pbmN7htrc1PjAEGOx43RcoACI70v6rbUenZX4AvOZ4vQj4yPF6hGP5QCDZsR5jO9vblvgznLbvo83xr7YdXBz/O8ArLfztnXJ8j3K8jnJl7EuW/yH2x0pd1fZpwATgYAvzbwXWAQJMAXa6ot09bg9PKXVEKdVax1dX7JRURAS4CXsnpGDvlHR+O8I7d2TalvcuANYpperbEcOV8S9wQdvbFF8pdVwpdcLxOh8oBtrW9c3lOtO57B3Ah0qpRqXUaeCkY30uja+U2uS0fXdg7y3IVdrS/pbMBr5SSpUrpSqAr4A5XRj7PuCDdqz/qpRSGdh3FlpyB/CustsBRIpIAp1sd48reG3UUqekMUClsndC6jy9reKVUs0j/xYC8a0sv4jL/wh+49gF/4OIBLYjdnviB4lIpojsaD6cpvNtb098AERkEva9gxynye1pf2c6l23Le1vT3nU8jH2vo9mVtkNXxL/b8TtdISLNXbV1tv1tfr/jMD4Z2Og0ubNt72h+nWp3t3YA2kxENgB9rjDrWaXUZ+6K7fyDUkqJSIuXsB3/bUZj70Wm2dPYC0UA9svp/wk81wXxByql8kQkBdgoIgewF4JWubj97wEPKaVsjsmttt9TicgDQCow3WnyZdtBKZVz5TV02OfAB0qpRhH5Pva93ZtcHKM1i4AVSimr07TuaLvLuaXgKaVmdnIVLXVKWoZ919fPsTdwWWelV4stIkUikqCUKnB8oIuvksO9wCqllNlp3c17R40i8jbws0vf5Ir4Sqk8x/dTIrIZGA98Sittd1V8EYkA1mL/B7XDad2ttv8S7elc9rxc3LlsW97bmjatQ0RmYv+HMF0p1dg8vYXt0J4PfavxlVLOHem+gf08a/N7b7zkvZtdGdvJIuCxS/LqbNs7ml/n2t2ZE49d+cXVL1r4YT9Zmcy/T7iOdMz7hItP3P+gHTFf4uKT9i9eZdkdwIxLpiU4vgvwR+CFdra51fjYT9QGOl7HAif49wWbDre9HfEDgH8BT15hXrvaf7Xt6LTMY1x80eJjx+uRXHzR4hTtv2jRlvjNH+TBbd0OLo6f4PT6TmCH43U0cNqRR5TjdbQrYzuWGwacwXHPrqva7nhvEi1ftJjLxRctdrmk3e1Nsqu/HBv1PNAIFAHrHdP7Al86LXcrcNzxx/is0/QUYBf2k9ifNG+YNsaOcXyYTwAbmn+R2A9l3rhkQ+UBhkvevxE4ABwElgNh7Wx7q/GBqY4Y+x3fH3ZF29sR/wHADOxz+hrX0fZfaTtiPwy+3fE6yNGWk462pTi991nH+44Bt3Tw7621+Bscf4fNbV3T2nZwcfzfAYcccTYBw5zeu9jxezkJfNfVsR0//5JL/nG5ou3Yz30XOP6WzmM/P/oI8IhjvmAfHjbHESPV6b0dbrd+0kLTNJ/hqVdpNU3T2k0XPE3TfIYueJqm+Qxd8DRN8xm64Gma5jN0wdM0zWfogqdpms/QBU/TNJ/x/wE7/QJVOAhobQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 360x180 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import jax\n",
    "from jax import core\n",
    "import numpy as np\n",
    "from jax.interpreters import ad\n",
    "\n",
    "import scipy\n",
    "import functools\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def findroot(f):\n",
    "    return scipy.optimize.brentq(f,a = 0,b = 10) \n",
    "\n",
    "def func(x,y):\n",
    "    return x**2 + y**2 - 1\n",
    "\n",
    "def y_for_x(x):\n",
    "    return findroot(functools.partial(func,x))\n",
    "\n",
    "xi = np.linspace(-1,1)\n",
    "yi = np.asarray([y_for_x(xx) for xx in xi])\n",
    "\n",
    "plt.plot(xi,yi)\n",
    "\n",
    "findrootjax_p = core.Primitive('findrootjax')\n",
    "findrootjax_p.def_impl(lambda x: y_for_x(x))\n",
    "ad.defvjp(findrootjax_p, lambda g, x: - x / y_for_x(x))\n",
    "\n",
    "def findrootjax(x):\n",
    "    return findrootjax_p.bind(x)\n",
    "jax.value_and_grad(findrootjax)(0.5)\n",
    "\n",
    "xi = np.linspace(-1,1,101)\n",
    "yi = [findrootjax(v) for v in xi]\n",
    "plt.plot(xi,yi)\n",
    "\n",
    "\n",
    "\n",
    "xi = np.linspace(-1,1,21)\n",
    "vg = np.asarray([np.asarray(jax.value_and_grad(findrootjax)(v)) for v in xi])\n",
    "plt.scatter(xi,vg[:,0])\n",
    "plt.quiver(xi,vg[:,0],np.ones_like(vg[:,0]),vg[:,1],\n",
    "    angles = 'uv',\n",
    "    alpha = 0.5,\n",
    ")\n",
    "plt.gcf().set_size_inches(5,2.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## In HEP\n",
    "\n",
    "\n",
    "Of course  we can use automatic differentiation\n",
    "for neural networks. But other things in HEP also \n",
    "can make use of gradients. A prime example where this is the \n",
    "case is statistical analysis\n",
    "\n",
    "For a maximum likelihood fit we want to minimize the log likelihood\n",
    "\n",
    "$\\theta^* = \\mathrm{argmin}_\\theta(\\log L)$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "import jax\n",
    "import jax.numpy as jnp\n",
    "import numpy as np\n",
    "import pyhf\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "pyhf.set_backend('jax')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DeviceArray([-4.25748227], dtype=float64)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = pyhf.simplemodels.hepdata_like([5.],[10.],[3.5])\n",
    "pars = jnp.array(m.config.suggested_init())\n",
    "data = jnp.array([15.] + m.config.auxdata)\n",
    "m.logpdf(pars,data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DeviceArray([1., 1.], dtype=float64)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bestfit = pyhf.infer.mle.fit(data,m)\n",
    "bestfit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = x,y = np.mgrid[.5:1.5:101j,.5:1.5:101j]\n",
    "\n",
    "points = np.swapaxes(grid,0,-1).reshape(-1,2)\n",
    "v = jax.vmap(m.logpdf, in_axes = (0,None))(points,data)\n",
    "v = np.swapaxes(v.reshape(101,101),0,-1)\n",
    "plt.contourf(x,y,v, levels = 100)\n",
    "plt.contour(x,y,v, levels = 20, colors = 'w')\n",
    "\n",
    "\n",
    "\n",
    "grid = x,y = np.mgrid[.5:1.5:11j,.5:1.5:11j]\n",
    "points = np.swapaxes(grid,0,-1).reshape(-1,2)\n",
    "values, gradients = jax.vmap(\n",
    "    jax.value_and_grad(\n",
    "        lambda p,d: m.logpdf(p,d)[0]\n",
    "    ), in_axes = (0,None)\n",
    ")(points,data)\n",
    "\n",
    "plt.quiver(\n",
    "    points[:,0],\n",
    "    points[:,1],\n",
    "    gradients[:,0],\n",
    "    gradients[:,1],\n",
    "    angles = 'xy',\n",
    "    scale = 75\n",
    ")\n",
    "plt.scatter(bestfit[0],bestfit[1], c = 'r')\n",
    "\n",
    "plt.xlim(0.5,1.5)\n",
    "plt.ylim(0.5,1.5)\n",
    "plt.gcf().set_size_inches(5,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Thanks for joining the Tutorial!\n",
    "\n",
    "\n",
    "\n",
    "<img src=\"./assets/schmidhuber.png\" alt=\"A Matrix Vector Product\" width=\"700\"/>\n"
   ]
  }
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