{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a routine to return a Gaussian PDF value given an input; include an option to return the natural log as well"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x,mu,sig,log=False) :\n",
    "    \"\"\" Funtion to evaluate a Gaussian and log(Gaussian)\n",
    "    \"\"\"\n",
    "    if log :\n",
    "        return \n",
    "    else :\n",
    "        return "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a routine to return the likelihood of a full data set (sample), where each point is drawn from a Gaussian with known mean and standard deviation. Include a keyword option (log=) to return the log(likelihood) instead"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def likelihood(sample,mu,sig,log=False) :\n",
    "    \"\"\" Calculate likelihood or ln(L) of a sample, given a mean and standard deviation\n",
    "    \"\"\"\n",
    "    p=1\n",
    "    sum=0\n",
    "    for i,s in enumerate(sample) :\n",
    "        if log :\n",
    "            sum += \n",
    "        else :\n",
    "            p *= \n",
    "        \n",
    "    if log : return sum \n",
    "    else : return p    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw a random sample from a normal distribution with some mean and standard devation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu=\n",
    "sig=\n",
    "n=\n",
    "sample=np.random.normal(mu, sig, size=n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now numerically try a range of \"guesses\" for the mean, calculate the likelihood for each, and determine the maximum likelihood value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "lik=[]\n",
    "# array of input guesses\n",
    "x=np.arange\n",
    "for mu in x :\n",
    "    lik.append(likelihood(sample,mu,sig))\n",
    "    \n",
    "plt.plot(x,lik)\n",
    "imax=np.argmax(lik)\n",
    "print('max: ',x[imax])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Demonstrate that the maximum likelihood value is equal to the mean of the sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(x,lik)\n",
    "mean=\n",
    "plt.plot([mean,mean],plt.ylim())\n",
    "print('max: ',x[imax])\n",
    "print('mean: ',mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given what we have previously discussed in class, what is your estimate of the uncertainty of the maximum likelihood estimator (i.e. the mean!)?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<B> ANSWER HERE: </B>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Experiment with samples with different standard deviations. How does the shape/breadth of the likelihood change with the sample standard deviation?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<B> ANSWER HERE: </B>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now write a routine to return a Poisson PDF instead of a Gaussian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def poiss(x,mu,log=False) :\n",
    "    \"\"\" Funtion to evaluate a Poissonn PDF and log(Poisson)\n",
    "    \"\"\"\n",
    "    if log :\n",
    "        return \n",
    "    else :\n",
    "        return "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now do the same exercise of drawing a sample from a Poisson distribution, calculate the maximum likelihood, and compare with the mean. In particular, try this for Poisson distribution with a very low mean (<5)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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