examples/sampling/transformation-with-and-without-jacobian.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Need for Jacobian adjustment in parameter transformation\n",
    "\n",
    "This example illustrates the importance of including a Jacobian term when sampling from a transformed parameter space (if you want to obtain samples on the untransformed space, that is).\n",
    "We will show the difference between sampling under parameter transformation with and without Jacobian adjustment, and how _not using_ Jacobian adjustment gives an _incorrect_ result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pints\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Set some random seed so this notebook can be reproduced\n",
    "np.random.seed(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Before we get started\n",
    "\n",
    "We will use a 1-dimensional beta distribution as an illustration:\n",
    "\n",
    "$$f(x; a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} x^{a-1} (1-x)^{b-1},$$\n",
    "\n",
    "with $a=2$ and $b=2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import beta\n",
    "\n",
    "class BetaLogPDF(pints.LogPDF):\n",
    "    def n_parameters(self):\n",
    "        return 1\n",
    "\n",
    "    def __call__(self, x):\n",
    "        if 0. < x[0] < 1.:\n",
    "            return np.log(beta.pdf(x, a=2, b=2))[0]\n",
    "        else:\n",
    "            return -np.inf\n",
    "\n",
    "log_pdf = BetaLogPDF()\n",
    "\n",
    "# Generate some initial positions\n",
    "x0 = np.random.uniform([0], [1], size=(4, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We try to sample it with MCMC."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running...\n",
      "Done!\n"
     ]
    }
   ],
   "source": [
    "# Create an adaptive covariance MCMC routine\n",
    "mcmc = pints.MCMCController(log_pdf, 4, x0, method=pints.HaarioBardenetACMC)\n",
    "\n",
    "# Set maximum number of iterations\n",
    "mcmc.set_max_iterations(10000)\n",
    "\n",
    "# Disable logging\n",
    "mcmc.set_log_to_screen(False)\n",
    "\n",
    "# Number of chains\n",
    "num_chains = 4\n",
    "\n",
    "# Run!\n",
    "print('Running...')\n",
    "chains = mcmc.run()\n",
    "print('Done!')\n",
    "\n",
    "# Discard warm-up\n",
    "chains = [chain[1000:] for chain in chains]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we inspect the MCMC samples and how it compares to the analytical form of the beta distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": [
    "plt.hist(chains[0], bins=15, density=True, label='samples')\n",
    "x = np.linspace(0, 1, 100)\n",
    "plt.plot(x, beta.pdf(x, a=2, b=2), label='true')\n",
    "plt.legend()\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel('PDF')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far so good, we see that we are able to sample from the original space $x \\in [0, 1]$, let's call this the \"model space\" where it's meaningful to be within $0$ and $1$ for some problem of interest."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transformation\n",
    "\n",
    "Now we consider a transformation, we will use the [logit (or log-odds) transformation](https://pints.readthedocs.io/en/latest/transformations.html#pints.LogitTransformation):\n",
    "\n",
    "$$y = \\text{logit}(x) = \\log\\left(\\frac{x}{1-x}\\right),$$\n",
    "\n",
    "which transforms the constrained model parameter $x$ to an unconstrained search space $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "transformation = pints.LogitTransformation(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now compare two transformations of the beta distribution: without and with Jacobian adjustment."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A naive transformation of PDF _without_ Jacobian adjustment and how it gets wrong\n",
    "\n",
    "The first one is the naive transformation which we simply transform the parameters as if using the [wrapping the model](../optimisation/transformed-parameters-model-wrapper.ipynb) without applying Jacobian adjustment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "class NaiveTransformedLogPDF(pints.LogPDF):\n",
    "    \"\"\"Transforming LogPDF without Jacobian adjustment\"\"\"\n",
    "    def __init__(self, log_pdf, transform):\n",
    "        self._log_pdf = log_pdf\n",
    "        self._transform = transform\n",
    "    \n",
    "    def n_parameters(self):\n",
    "        return self._log_pdf.n_parameters()\n",
    "    \n",
    "    def __call__(self, y):\n",
    "        # Transform from search space y back to model space x\n",
    "        x = self._transform.to_model(y)\n",
    "        # Then we call the log-pdf in the model space x\n",
    "        return self._log_pdf(x)\n",
    "\n",
    "naive_trans_log_pdf = NaiveTransformedLogPDF(log_pdf, transformation)\n",
    "\n",
    "# Transform the initial position to search space\n",
    "y0 = [transformation.to_search(x) for x in x0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then sample this naive transformed beta distribution using MCMC in the search space $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running...\n",
      "Done!\n"
     ]
    }
   ],
   "source": [
    "# Create an adaptive covariance MCMC routine\n",
    "mcmc = pints.MCMCController(\n",
    "    naive_trans_log_pdf,  # Naive transformation without Jacobian\n",
    "    4,\n",
    "    y0,                   # Input is in the search space y\n",
    "    method=pints.HaarioBardenetACMC)\n",
    "\n",
    "# Set maximum number of iterations\n",
    "mcmc.set_max_iterations(10000)\n",
    "\n",
    "# Disable logging\n",
    "mcmc.set_log_to_screen(False)\n",
    "\n",
    "# Number of chains\n",
    "num_chains = 4\n",
    "\n",
    "# Run!\n",
    "print('Running...')\n",
    "naive_chains_in_y = mcmc.run()\n",
    "print('Done!')\n",
    "\n",
    "# Discard warm-up\n",
    "naive_chains_in_y = [naive_chain[1000:] for naive_chain in naive_chains_in_y]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a look at what the samples look like in the search space $y$ first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "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": [
    "plt.hist(naive_chains_in_y[0], bins=15, density=True)\n",
    "plt.xlabel(r'$y$')\n",
    "plt.ylabel('PDF')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, not quite sure what that means, $y$ is the search space parameter which we are not interested in it!\n",
    "\n",
    "Let's backward transform these samples _directly_ back to the model space $x$ and hope for the best..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": [
    "# Transform the samples from y back to x\n",
    "naive_chains_in_x = np.zeros(naive_chains_in_y[0].shape)\n",
    "for i, y in enumerate(naive_chains_in_y[0]):\n",
    "    naive_chains_in_x[i, :] = transformation.to_model(y)\n",
    "\n",
    "plt.hist(naive_chains_in_x, bins=15, density=True, label='samples')\n",
    "x = np.linspace(0, 1, 100)\n",
    "plt.plot(x, beta.pdf(x, a=2, b=2), label='true')\n",
    "plt.legend()\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel('PDF (without Jacobian adjustment)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sampled distribution has gone horribly wrong using this naive transformation without Jacobian adjustment!!!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transforming PDF using Pints (with Jacobian adjustment)\n",
    "\n",
    "This time we will use [pints.TransformedLogPDF](https://pints.readthedocs.io/en/latest/transformations.html#pints.TransformedLogPDF) which will handle all the necessary correction and adjustment for the transformation!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "pints_trans_log_pdf = pints.TransformedLogPDF(log_pdf, transformation)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will repeat the same procedure as before: sampling in the search space $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running...\n",
      "Done!\n"
     ]
    }
   ],
   "source": [
    "# Create an adaptive covariance MCMC routine\n",
    "mcmc = pints.MCMCController(\n",
    "    pints_trans_log_pdf,  # Transformation using pints\n",
    "    4,\n",
    "    y0,                   # Again in search space y\n",
    "    method=pints.HaarioBardenetACMC)\n",
    "\n",
    "# Set maximum number of iterations\n",
    "mcmc.set_max_iterations(10000)\n",
    "\n",
    "# Disable logging\n",
    "mcmc.set_log_to_screen(False)\n",
    "\n",
    "# Number of chains\n",
    "num_chains = 4\n",
    "\n",
    "# Run!\n",
    "print('Running...')\n",
    "pints_chains_in_y = mcmc.run()\n",
    "print('Done!')\n",
    "\n",
    "# Discard warm-up\n",
    "pints_chains_in_y = [pints_chain[1000:] for pints_chain in pints_chains_in_y]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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": [
    "plt.hist(pints_chains_in_y[0], bins=15, density=True)\n",
    "plt.xlabel(r'$y$')\n",
    "plt.ylabel('PDF')\n",
    "plt.show()\n",
    "\n",
    "# Transform the samples from y back to x\n",
    "pints_chains_in_x = np.zeros(pints_chains_in_y[0].shape)\n",
    "for i, y in enumerate(pints_chains_in_y[0]):\n",
    "    pints_chains_in_x[i, :] = transformation.to_model(y)\n",
    "\n",
    "plt.hist(pints_chains_in_x, bins=15, density=True, label='samples')\n",
    "x = np.linspace(0, 1, 100)\n",
    "plt.plot(x, beta.pdf(x, a=2, b=2), label='true')\n",
    "plt.legend()\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel('PDF (with Jacobian adjustment)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is look very much like the correct beta distribution when we use the transform provided by Pints!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Remarks\n",
    "\n",
    "This example simply tries to show the importance of the Jacobian adjustment when applying parameter transformation for PDFs (if you are interested in the original untransformed parameters)! For practical use, all the [pints.Transformation](https://pints.readthedocs.io/en/latest/transformations.html#pints.Transformation) classes can be directly passed to the `Controller`s as shown in [another example notebook](./transformed-parameters.ipynb)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}