breakfast.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Being ridiculously blessed beyond any merit of my own, the biggest problem facing me today is deciding what to have for breakfast.\n",
    "\n",
    "I don't know what I want for breakfast _right now_, but I've got a set of fairly consistent preferences:\n",
    "\n",
    " 1. Eggs benedict\n",
    " 2. Bacon and eggs\n",
    " 3. French toast with bacon\n",
    " 4. Pancakes\n",
    " 5. Vegemite on toast\n",
    "\n",
    "Let's encode this in Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "breakfasts = [\n",
    "    'eggs benedict',\n",
    "    'bacon and eggs',\n",
    "    'french toast with bacon',\n",
    "    'pancakes',\n",
    "    'vegemite on toast',\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Actually, I like bacon & eggs about equally as much as I like french toast, so let's represent that by making each priority level a list of options:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "breakfasts = [\n",
    "    ['eggs benedict'],\n",
    "    ['bacon and eggs', 'french toast with bacon'],\n",
    "    ['pancakes'],\n",
    "    ['vegemite on toast'],\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What I want to do is to randomly choose a breakfast so that I'm most likely to get eggs benedict, then next most likely to get bacon and eggs or french toast, then pancakes, then vegemite coming out as least likely.\n",
    "\n",
    "There's probably some well-established way of doing this, but I'm just going to muck around and see what I can find."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since this is a [partial ordering](https://en.wikipedia.org/wiki/Partially_ordered_set), and because I've spent way too much time swotting up for job interviews my natural response is to put it into a graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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CnT9/nvr163P69GkcHR21LsckyBCxeCAcHBx49dVXjSZxEEKYjxo1atC+fXvW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\nCwADwQLAQLAAMBAsAAwECwCDAOJnu0UdEGd8AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1066e0790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import networkx\n",
    "\n",
    "g = networkx.DiGraph()\n",
    "g.add_edge('eggs benedict', 'bacon and eggs')\n",
    "g.add_edge('eggs benedict', 'french toast with bacon')\n",
    "g.add_edge('bacon and eggs', 'pancakes')\n",
    "g.add_edge('french toast with bacon', 'pancakes')\n",
    "g.add_edge('pancakes', 'vegemite on toast')\n",
    "\n",
    "networkx.draw(g, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "My initial idea is that we'd randomly pick a root node (i.e. one with no parents), and then given some fixed probability `p` (e.g. `p = 0.75`), we'd decide whether to move on to the next level, at which point we'd pick a node at random from the child nodes.\n",
    "\n",
    "However, this is way too complicated. Writing up even a little bit of the required code makes it really obvious."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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uRVRUFLZv347ly5dj//79FbgKqgkYvkSVpF69emWCQkQQGxurx4qqjojg008/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\nyRcAZvIFgJl8AWAmXwCYyRcAZgHcS8Coupv7nQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105fddf90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def make_ordering_graph(priorities):\n",
    "    \"\"\"\n",
    "    Take a list of lists and turn it into a graph \n",
    "    representing the partial ordering.\n",
    "    \"\"\"\n",
    "    g = networkx.DiGraph()\n",
    "    for i in range(len(priorities) - 1):\n",
    "        level = priorities[i]\n",
    "        next_level = priorities[i + 1]\n",
    "        for item in level:\n",
    "            for next_item in next_level:\n",
    "                g.add_edge(item, next_item)\n",
    "    return g\n",
    "\n",
    "g = make_ordering_graph(breakfasts)\n",
    "networkx.draw(g, with_labels=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've got three nested loops, and the subgraph of any one priority level and the next is fully connected, which is a hint that graphs maybe aren't the right fit. Shocking, I know.\n",
    "\n",
    "But it also helps us see the core of the idea:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'eggs benedict'"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "\n",
    "def random_preference(priorities, p):\n",
    "    \"\"\"\n",
    "    Randomly choose something from a prioritised list.\n",
    "    \"\"\"\n",
    "    assert 0 < p < 1\n",
    "    for level in priorities[:-1]:\n",
    "        if random.random() <= p:\n",
    "            return random.choice(level)\n",
    "    return random.choice(priorities[-1])\n",
    "\n",
    "random_preference(breakfasts, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "My hunch is that given the priorities and `p` we could probably construct a vector of weighted probabilities algebraically. However, that would require reaching for pen and paper to do the maths, so let's try to figure it out with computers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'bacon and eggs': 12749,\n",
       "         'eggs benedict': 49808,\n",
       "         'french toast with bacon': 12306,\n",
       "         'pancakes': 12548,\n",
       "         'vegemite on toast': 12589})"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from collections import Counter\n",
    "\n",
    "def count_results(n, f, *args, **kwargs):\n",
    "    c = Counter()\n",
    "    for i in range(n):\n",
    "        c[f(*args, **kwargs)] += 1\n",
    "    return c\n",
    "\n",
    "count_results(100000, random_preference, breakfasts, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'bacon and eggs': 936219,\n",
       "         'eggs benedict': 7502546,\n",
       "         'french toast with bacon': 935817,\n",
       "         'pancakes': 468401,\n",
       "         'vegemite on toast': 157017})"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_results(10000000, random_preference, breakfasts, 0.75)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, that makes sense. The probability of level `i` getting selected is `p * (1 - p) ** i`, and the probability of any item in that level being selected is that divided by the number of things in that level. Or something like that.\n",
    "\n",
    "I'm not certain about the way we always select something from the lowest priority when the loop falls through. Wouldn't it be better to loop back around and try again? After all, with a `p = 0.5`, we are ending up with equal chance of getting the least-favoured item as the second-least."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'bacon and eggs': 937454,\n",
       "         'eggs benedict': 7500646,\n",
       "         'french toast with bacon': 936469,\n",
       "         'pancakes': 469202,\n",
       "         'vegemite on toast': 156229})"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def random_preference_loopback(priorities, p):\n",
    "    \"\"\"\n",
    "    Randomly choose something from a prioritised list.\n",
    "    \"\"\"\n",
    "    assert 0 < p < 1\n",
    "    while True:\n",
    "        for level in priorities:\n",
    "            if random.random() <= p:\n",
    "                return random.choice(level)\n",
    "\n",
    "count_results(int(1e7), random_preference, breakfasts, 0.75)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hmm. That doesn't look much different. I wonder why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the deterministic way to generate the random weightings:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.75, 'eggs benedict'),\n",
       " (0.09375, 'bacon and eggs'),\n",
       " (0.09375, 'french toast with bacon'),\n",
       " (0.046875, 'pancakes'),\n",
       " (0.015625, 'vegemite on toast')]"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def priority_weighting(priorities, p):\n",
    "    \"\"\"\n",
    "    Generate weightings for a prioritised list.\n",
    "    \"\"\"\n",
    "    assert 0 < p < 1\n",
    "    weightings = []\n",
    "    for i, level in enumerate(priorities[:-1]):\n",
    "        q = p * (1 - p) ** i / len(level)\n",
    "        weightings.extend([(q, item) for item in level])\n",
    "    # XXX: I don't know why this is different, or why \n",
    "    # this calculation works.\n",
    "    q = (1 - p) ** (i + 1) / len(level)\n",
    "    for item in priorities[-1]:\n",
    "        weightings.extend([(q, item) for item in priorities[-1]])\n",
    "    return weightings \n",
    "\n",
    "priority_weighting(breakfasts, 0.75)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double check the numbers add up:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(zip(*priority_weighting(breakfasts, 0.75))[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.5, 'eggs benedict'),\n",
       " (0.125, 'bacon and eggs'),\n",
       " (0.125, 'french toast with bacon'),\n",
       " (0.125, 'pancakes'),\n",
       " (0.125, 'vegemite on toast')]"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "priority_weighting(breakfasts, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks like it's returning the \"right\" numbers, but as the comment in the code indicates, I don't really understand what's going on with the last level. Any thoughts?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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