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   "source": [
    "# Making Custom Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By using the ``InterpolatedUnivariateDistribution`` class, you can easily create a single-variable distribution by specifying its PDF as a callable function. Here, we'll demonstrate this functionality by implementing the asymmetric Lorentz distribution of [Stancik and Brauns](http://www.sciencedirect.com/science/article/pii/S0924203108000453)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preamble"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As always, we start by setting up the Python environment for inline plotting and true division."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/cgranade/anaconda/envs/qinfer-binder/lib/python3.5/site-packages/matplotlib/__init__.py:872: UserWarning: axes.color_cycle is deprecated and replaced with axes.prop_cycle; please use the latter.\n",
      "  warnings.warn(self.msg_depr % (key, alt_key))\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "try: plt.style.use('ggplot')\n",
    "except: pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we import the ``InterpolatedUnivariateDistribution`` class."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/cgranade/anaconda/envs/qinfer-binder/lib/python3.5/site-packages/qinfer/metrics.py:51: UserWarning: Could not import scikit-learn. Some features may not work.\n",
      "  warnings.warn(\"Could not import scikit-learn. Some features may not work.\")\n",
      "/home/cgranade/anaconda/envs/qinfer-binder/lib/python3.5/site-packages/IPython/parallel.py:13: ShimWarning: The `IPython.parallel` package has been deprecated. You should import from ipyparallel instead.\n",
      "  \"You should import from ipyparallel instead.\", ShimWarning)\n",
      "/home/cgranade/anaconda/envs/qinfer-binder/lib/python3.5/site-packages/qinfer/parallel.py:53: UserWarning: Could not import IPython parallel. Parallelization support will be disabled.\n",
      "  \"Could not import IPython parallel. \"\n"
     ]
    }
   ],
   "source": [
    "from qinfer.distributions import InterpolatedUnivariateDistribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defining Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The asymmetric Lorentz distribution is defined by letting the scale parameter $\\gamma$ of a Lorentz distribution be a function of the random variable $x$,\n",
    "$$\n",
    "    \\gamma(x) = \\frac{2\\gamma_0}{1 + \\exp(a [x - x_0])}.\n",
    "$$\n",
    "It is straightforward to implement this in a vectorized way by defining this function and then substituting it into the PDF of a Lorentz distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def asym_lorentz_scale(x, x_0, gamma_0, a):\n",
    "    return 2 * gamma_0 / (1 + np.exp(a * (x - x_0)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def asym_lorentz_pdf(x, x_0, gamma_0, a):\n",
    "    gamma = asym_lorentz_scale(x, x_0, gamma_0, a)\n",
    "    return 2 * gamma / (np.pi * gamma_0 * (1 + 4 * ((x - x_0) / (gamma_0))**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have this, we can pass the PDF as a lambda function to ``InterpolatedUnivariateDistribution`` in order to specify\n",
    "the values of the location $x_0$, the nominal scale $\\gamma_0$ and the asymmetry parameter $a$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "dist = InterpolatedUnivariateDistribution(lambda x: asym_lorentz_pdf(x, 0, 1, 2), 2, 1200)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The resulting distribution can be sampled like any other, such that we can quickly check that it produces something of the desired shape."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "hist(dist.sample(n=10000), bins=100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that making this distribution object is fast enough that it can conceivably be embedded within a likelihood function itself, so as to enable using the method of hyperparameters to estimate the parameters of the asymmetric Lorentz distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 764 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit dist = InterpolatedUnivariateDistribution(lambda x: asym_lorentz_pdf(x, 0, 1, 2), 2, 120)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 957 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit dist = InterpolatedUnivariateDistribution(lambda x: asym_lorentz_pdf(x, 0, 1, 2), 2, 1200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 2.65 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit dist = InterpolatedUnivariateDistribution(lambda x: asym_lorentz_pdf(x, 0, 1, 2), 2, 12000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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