04-ParameterLearning.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parameter Learning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using RDatasets\n",
    "using PGFPlots\n",
    "using Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Maximum likelihood parameter learning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>SepalLength</th><th>SepalWidth</th><th>PetalLength</th><th>PetalWidth</th><th>Species</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Categorical…</th></tr></thead><tbody><p>150 rows × 5 columns</p><tr><th>1</th><td>5.1</td><td>3.5</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>2</th><td>4.9</td><td>3.0</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>3</th><td>4.7</td><td>3.2</td><td>1.3</td><td>0.2</td><td>setosa</td></tr><tr><th>4</th><td>4.6</td><td>3.1</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>5</th><td>5.0</td><td>3.6</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>6</th><td>5.4</td><td>3.9</td><td>1.7</td><td>0.4</td><td>setosa</td></tr><tr><th>7</th><td>4.6</td><td>3.4</td><td>1.4</td><td>0.3</td><td>setosa</td></tr><tr><th>8</th><td>5.0</td><td>3.4</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>9</th><td>4.4</td><td>2.9</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>10</th><td>4.9</td><td>3.1</td><td>1.5</td><td>0.1</td><td>setosa</td></tr><tr><th>11</th><td>5.4</td><td>3.7</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>12</th><td>4.8</td><td>3.4</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>13</th><td>4.8</td><td>3.0</td><td>1.4</td><td>0.1</td><td>setosa</td></tr><tr><th>14</th><td>4.3</td><td>3.0</td><td>1.1</td><td>0.1</td><td>setosa</td></tr><tr><th>15</th><td>5.8</td><td>4.0</td><td>1.2</td><td>0.2</td><td>setosa</td></tr><tr><th>16</th><td>5.7</td><td>4.4</td><td>1.5</td><td>0.4</td><td>setosa</td></tr><tr><th>17</th><td>5.4</td><td>3.9</td><td>1.3</td><td>0.4</td><td>setosa</td></tr><tr><th>18</th><td>5.1</td><td>3.5</td><td>1.4</td><td>0.3</td><td>setosa</td></tr><tr><th>19</th><td>5.7</td><td>3.8</td><td>1.7</td><td>0.3</td><td>setosa</td></tr><tr><th>20</th><td>5.1</td><td>3.8</td><td>1.5</td><td>0.3</td><td>setosa</td></tr><tr><th>21</th><td>5.4</td><td>3.4</td><td>1.7</td><td>0.2</td><td>setosa</td></tr><tr><th>22</th><td>5.1</td><td>3.7</td><td>1.5</td><td>0.4</td><td>setosa</td></tr><tr><th>23</th><td>4.6</td><td>3.6</td><td>1.0</td><td>0.2</td><td>setosa</td></tr><tr><th>24</th><td>5.1</td><td>3.3</td><td>1.7</td><td>0.5</td><td>setosa</td></tr><tr><th>25</th><td>4.8</td><td>3.4</td><td>1.9</td><td>0.2</td><td>setosa</td></tr><tr><th>26</th><td>5.0</td><td>3.0</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>27</th><td>5.0</td><td>3.4</td><td>1.6</td><td>0.4</td><td>setosa</td></tr><tr><th>28</th><td>5.2</td><td>3.5</td><td>1.5</td><td>0.2</td><td>setosa</td></tr><tr><th>29</th><td>5.2</td><td>3.4</td><td>1.4</td><td>0.2</td><td>setosa</td></tr><tr><th>30</th><td>4.7</td><td>3.2</td><td>1.6</td><td>0.2</td><td>setosa</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccccc}\n",
       "\t& SepalLength & SepalWidth & PetalLength & PetalWidth & Species\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Categorical…\\\\\n",
       "\t\\hline\n",
       "\t1 & 5.1 & 3.5 & 1.4 & 0.2 & setosa \\\\\n",
       "\t2 & 4.9 & 3.0 & 1.4 & 0.2 & setosa \\\\\n",
       "\t3 & 4.7 & 3.2 & 1.3 & 0.2 & setosa \\\\\n",
       "\t4 & 4.6 & 3.1 & 1.5 & 0.2 & setosa \\\\\n",
       "\t5 & 5.0 & 3.6 & 1.4 & 0.2 & setosa \\\\\n",
       "\t6 & 5.4 & 3.9 & 1.7 & 0.4 & setosa \\\\\n",
       "\t7 & 4.6 & 3.4 & 1.4 & 0.3 & setosa \\\\\n",
       "\t8 & 5.0 & 3.4 & 1.5 & 0.2 & setosa \\\\\n",
       "\t9 & 4.4 & 2.9 & 1.4 & 0.2 & setosa \\\\\n",
       "\t10 & 4.9 & 3.1 & 1.5 & 0.1 & setosa \\\\\n",
       "\t11 & 5.4 & 3.7 & 1.5 & 0.2 & setosa \\\\\n",
       "\t12 & 4.8 & 3.4 & 1.6 & 0.2 & setosa \\\\\n",
       "\t13 & 4.8 & 3.0 & 1.4 & 0.1 & setosa \\\\\n",
       "\t14 & 4.3 & 3.0 & 1.1 & 0.1 & setosa \\\\\n",
       "\t15 & 5.8 & 4.0 & 1.2 & 0.2 & setosa \\\\\n",
       "\t16 & 5.7 & 4.4 & 1.5 & 0.4 & setosa \\\\\n",
       "\t17 & 5.4 & 3.9 & 1.3 & 0.4 & setosa \\\\\n",
       "\t18 & 5.1 & 3.5 & 1.4 & 0.3 & setosa \\\\\n",
       "\t19 & 5.7 & 3.8 & 1.7 & 0.3 & setosa \\\\\n",
       "\t20 & 5.1 & 3.8 & 1.5 & 0.3 & setosa \\\\\n",
       "\t21 & 5.4 & 3.4 & 1.7 & 0.2 & setosa \\\\\n",
       "\t22 & 5.1 & 3.7 & 1.5 & 0.4 & setosa \\\\\n",
       "\t23 & 4.6 & 3.6 & 1.0 & 0.2 & setosa \\\\\n",
       "\t24 & 5.1 & 3.3 & 1.7 & 0.5 & setosa \\\\\n",
       "\t25 & 4.8 & 3.4 & 1.9 & 0.2 & setosa \\\\\n",
       "\t26 & 5.0 & 3.0 & 1.6 & 0.2 & setosa \\\\\n",
       "\t27 & 5.0 & 3.4 & 1.6 & 0.4 & setosa \\\\\n",
       "\t28 & 5.2 & 3.5 & 1.5 & 0.2 & setosa \\\\\n",
       "\t29 & 5.2 & 3.4 & 1.4 & 0.2 & setosa \\\\\n",
       "\t30 & 4.7 & 3.2 & 1.6 & 0.2 & setosa \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "150×5 DataFrame\n",
       "│ Row │ SepalLength │ SepalWidth │ PetalLength │ PetalWidth │ Species      │\n",
       "│     │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mCategorical…\u001b[39m │\n",
       "├─────┼─────────────┼────────────┼─────────────┼────────────┼──────────────┤\n",
       "│ 1   │ 5.1         │ 3.5        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 2   │ 4.9         │ 3.0        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 3   │ 4.7         │ 3.2        │ 1.3         │ 0.2        │ setosa       │\n",
       "│ 4   │ 4.6         │ 3.1        │ 1.5         │ 0.2        │ setosa       │\n",
       "│ 5   │ 5.0         │ 3.6        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 6   │ 5.4         │ 3.9        │ 1.7         │ 0.4        │ setosa       │\n",
       "│ 7   │ 4.6         │ 3.4        │ 1.4         │ 0.3        │ setosa       │\n",
       "│ 8   │ 5.0         │ 3.4        │ 1.5         │ 0.2        │ setosa       │\n",
       "│ 9   │ 4.4         │ 2.9        │ 1.4         │ 0.2        │ setosa       │\n",
       "│ 10  │ 4.9         │ 3.1        │ 1.5         │ 0.1        │ setosa       │\n",
       "⋮\n",
       "│ 140 │ 6.9         │ 3.1        │ 5.4         │ 2.1        │ virginica    │\n",
       "│ 141 │ 6.7         │ 3.1        │ 5.6         │ 2.4        │ virginica    │\n",
       "│ 142 │ 6.9         │ 3.1        │ 5.1         │ 2.3        │ virginica    │\n",
       "│ 143 │ 5.8         │ 2.7        │ 5.1         │ 1.9        │ virginica    │\n",
       "│ 144 │ 6.8         │ 3.2        │ 5.9         │ 2.3        │ virginica    │\n",
       "│ 145 │ 6.7         │ 3.3        │ 5.7         │ 2.5        │ virginica    │\n",
       "│ 146 │ 6.7         │ 3.0        │ 5.2         │ 2.3        │ virginica    │\n",
       "│ 147 │ 6.3         │ 2.5        │ 5.0         │ 1.9        │ virginica    │\n",
       "│ 148 │ 6.5         │ 3.0        │ 5.2         │ 2.0        │ virginica    │\n",
       "│ 149 │ 6.2         │ 3.4        │ 5.4         │ 2.3        │ virginica    │\n",
       "│ 150 │ 5.9         │ 3.0        │ 5.1         │ 1.8        │ virginica    │"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = dataset(\"datasets\", \"iris\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      ],
      "text/plain": [
       "PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 10, false, false, \"fill=blue!10\", :default)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = D[!,:SepalLength]\n",
    "Plots.Histogram(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 10, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 0.039904951290522954 0.04446259219010051 … 0.01805070677356521 0.015902083920377806], \"none\", nothing, nothing, nothing, nothing, nothing)], nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Normal{Float64}(μ=5.843333333333335, σ=0.8253012917851409)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dist = fit_mle(Normal, d)\n",
    "g = Axis([\n",
    "    Plots.Histogram(d, density=true),\n",
    "    Plots.Linear(x->pdf(dist,x), (4,8))\n",
    "]\n",
    ")\n",
    "display(g)\n",
    "display(dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayesian parameter learning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# small helper function for plotting Beta distributions\n",
    "PGFPlots.plot(d::Beta) = plot(x->pdf(d, x), (0,1));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prior = Beta(6,2)\n",
    "plot(prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Beta{Float64}(α=6.0, β=2.0)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "posterior (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "posterior(d::Beta, x) = Beta(d.α + sum(x .== 1), d.β + sum(x .== 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Beta{Float64}(α=7.0, β=4.0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The posterior function is provided by distributions.jl\n",
    "post = posterior(prior, [0, 0, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(post)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nonparametric parameter learning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "\n"
      ],
      "text/plain": [
       "Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 30, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 0.010975748980430503 0.015834910845550776 … 0.03862508876301471 0.030995020645305776], \"none\", nothing, nothing, nothing, nothing, nothing)], nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\")"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bandwith = 0.2\n",
    "# kernel\n",
    "K(x) = pdf(Normal(0,bandwith), x)\n",
    "# nonparametric density function\n",
    "p(x) = sum([K(x - o) for o in d])/length(d)\n",
    "# plot result, comparing with histogram\n",
    "g = Axis([\n",
    "    Plots.Histogram(d, density=true, bins=30),\n",
    "    Plots.Linear(p, (4,8))\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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       "\n"
      ],
      "text/plain": [
       "GroupPlot(Axis[Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 30, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 9.824307565862008e-197 1.2278858091106003e-147 … 5.1575940161736055e-9 5.129732417806102e-23], \"none\", nothing, nothing, nothing, nothing, nothing)], 0.01, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\"), Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 30, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 0.0003223233420382349 0.0010399646823790979 … 0.025970453038656697 0.017322130298512928], \"none\", nothing, nothing, nothing, nothing, nothing)], 0.1, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\"), Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 30, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 0.010975748980430503 0.015834910845550776 … 0.03862508876301471 0.030995020645305776], \"none\", nothing, nothing, nothing, nothing, nothing)], 0.2, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\"), Axis(PGFPlots.Plots.Plot[PGFPlots.Plots.Histogram(Real[5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9  …  6.7, 6.9, 5.8, 6.8, 6.7, 6.7, 6.3, 6.5, 6.2, 5.9], 30, true, false, \"fill=blue!10\", :default), PGFPlots.Plots.Linear(Real[4.0 4.040404040404041 … 7.959595959595959 8.0; 0.061604521449853206 0.06897924373352007 … 0.046481671197790723 0.04282982879962451], \"none\", nothing, nothing, nothing, nothing, nothing)], 0.5, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, \"axis\")], (2, 2), nothing, nothing)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bandwidths = [0.01, 0.1, 0.2, 0.5]\n",
    "gp = GroupPlot(2,2)\n",
    "for bandwidth in bandwidths\n",
    "    # kernel\n",
    "    local K(x) = pdf(Normal(0,bandwidth), x)\n",
    "    # nonparametric density function\n",
    "    local p(x) = sum([K(x - o) for o in d])/length(d)\n",
    "    # plot result, comparing with histogram\n",
    "    local g = Axis([\n",
    "        Plots.Histogram(d, density=true, bins=30),\n",
    "        Plots.Linear(p, (4,8))\n",
    "        ], title = bandwidth)\n",
    "    push!(gp, g)\n",
    "end\n",
    "gp"
   ]
  }
 ],
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