10-Chapter 10. Big Entropy and Generalized Linear Model.ipynb

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   "source": [
    "using DataFrames\n",
    "using StatsBase\n",
    "using StatsPlots\n",
    "using Random\n",
    "using Distributions\n",
    "\n",
    "default(label=false);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53cf8fea",
   "metadata": {},
   "source": [
    "# 10.1 Maximum entropy"
   ]
  },
  {
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   "id": "56788d55",
   "metadata": {},
   "source": [
    "Code 10.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "47699d1d",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<div class=\"data-frame\"><p>5 rows × 5 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th></tr><tr><th></th><th title=\"Int64\">Int64</th><th title=\"Int64\">Int64</th><th title=\"Int64\">Int64</th><th title=\"Int64\">Int64</th><th title=\"Int64\">Int64</th></tr></thead><tbody><tr><th>1</th><td>0</td><td>0</td><td>0</td><td>1</td><td>2</td></tr><tr><th>2</th><td>0</td><td>1</td><td>2</td><td>2</td><td>2</td></tr><tr><th>3</th><td>10</td><td>8</td><td>6</td><td>4</td><td>2</td></tr><tr><th>4</th><td>0</td><td>1</td><td>2</td><td>2</td><td>2</td></tr><tr><th>5</th><td>0</td><td>0</td><td>0</td><td>1</td><td>2</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccccc}\n",
       "\t& A & B & C & D & E\\\\\n",
       "\t\\hline\n",
       "\t& Int64 & Int64 & Int64 & Int64 & Int64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0 & 0 & 0 & 1 & 2 \\\\\n",
       "\t2 & 0 & 1 & 2 & 2 & 2 \\\\\n",
       "\t3 & 10 & 8 & 6 & 4 & 2 \\\\\n",
       "\t4 & 0 & 1 & 2 & 2 & 2 \\\\\n",
       "\t5 & 0 & 0 & 0 & 1 & 2 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m5×5 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m A     \u001b[0m\u001b[1m B     \u001b[0m\u001b[1m C     \u001b[0m\u001b[1m D     \u001b[0m\u001b[1m E     \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Int64 \u001b[0m\u001b[90m Int64 \u001b[0m\u001b[90m Int64 \u001b[0m\u001b[90m Int64 \u001b[0m\n",
       "─────┼───────────────────────────────────\n",
       "   1 │     0      0      0      1      2\n",
       "   2 │     0      1      2      2      2\n",
       "   3 │    10      8      6      4      2\n",
       "   4 │     0      1      2      2      2\n",
       "   5 │     0      0      0      1      2"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "p = DataFrame(\n",
    "    :A => [0, 0, 10, 0, 0],\n",
    "    :B => [0, 1, 8, 1, 0],\n",
    "    :C => [0, 2, 6, 2, 0],\n",
    "    :D => [1, 2, 4, 2, 1],\n",
    "    :E => [2, 2, 2, 2, 2],\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff9f01b9",
   "metadata": {},
   "source": [
    "Code 10.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "781409e2",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<div class=\"data-frame\"><p>5 rows × 5 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th></tr><tr><th></th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>0.0</td><td>0.0</td><td>0.0</td><td>0.1</td><td>0.2</td></tr><tr><th>2</th><td>0.0</td><td>0.1</td><td>0.2</td><td>0.2</td><td>0.2</td></tr><tr><th>3</th><td>1.0</td><td>0.8</td><td>0.6</td><td>0.4</td><td>0.2</td></tr><tr><th>4</th><td>0.0</td><td>0.1</td><td>0.2</td><td>0.2</td><td>0.2</td></tr><tr><th>5</th><td>0.0</td><td>0.0</td><td>0.0</td><td>0.1</td><td>0.2</td></tr></tbody></table></div>"
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       "\\begin{tabular}{r|ccccc}\n",
       "\t& A & B & C & D & E\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.0 & 0.0 & 0.0 & 0.1 & 0.2 \\\\\n",
       "\t2 & 0.0 & 0.1 & 0.2 & 0.2 & 0.2 \\\\\n",
       "\t3 & 1.0 & 0.8 & 0.6 & 0.4 & 0.2 \\\\\n",
       "\t4 & 0.0 & 0.1 & 0.2 & 0.2 & 0.2 \\\\\n",
       "\t5 & 0.0 & 0.0 & 0.0 & 0.1 & 0.2 \\\\\n",
       "\\end{tabular}\n"
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       "\u001b[1m5×5 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m A       \u001b[0m\u001b[1m B       \u001b[0m\u001b[1m C       \u001b[0m\u001b[1m D       \u001b[0m\u001b[1m E       \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
       "─────┼─────────────────────────────────────────────\n",
       "   1 │     0.0      0.0      0.0      0.1      0.2\n",
       "   2 │     0.0      0.1      0.2      0.2      0.2\n",
       "   3 │     1.0      0.8      0.6      0.4      0.2\n",
       "   4 │     0.0      0.1      0.2      0.2      0.2\n",
       "   5 │     0.0      0.0      0.0      0.1      0.2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_norm = mapcols(c -> c ./ sum(c), p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5f6bf9e",
   "metadata": {},
   "source": [
    "Code 10.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "83ff0acd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class=\"data-frame\"><p>1 rows × 5 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th></tr><tr><th></th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>-0.0</td><td>0.639032</td><td>0.950271</td><td>1.47081</td><td>1.60944</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ccccc}\n",
       "\t& A & B & C & D & E\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & -0.0 & 0.639032 & 0.950271 & 1.47081 & 1.60944 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m1×5 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m A       \u001b[0m\u001b[1m B        \u001b[0m\u001b[1m C        \u001b[0m\u001b[1m D       \u001b[0m\u001b[1m E       \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Float64 \u001b[0m\u001b[90m Float64  \u001b[0m\u001b[90m Float64  \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
       "─────┼───────────────────────────────────────────────\n",
       "   1 │    -0.0  0.639032  0.950271  1.47081  1.60944"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "ent_vals = mapcols(entropy, p_norm)"
   ]
  },
  {
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   "source": [
    "Code 10.4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d8452580",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 0.0\n",
       " 0.4499809670330265\n",
       " 0.7138866999945523\n",
       " 1.0540064381607679\n",
       " 1.163867667027579"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ways = [1, 90, 1260, 37800, 113400]\n",
    "logwayspp = log.(ways)/10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "302ab875",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "txt = text.(names(ent_vals), :bottom, :right, 11)\n",
    "scatter(logwayspp, collect(ent_vals[1,:]), txt=txt, xlab=\"log(ways) per pebble\", ylab=\"entropy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60f17fd6",
   "metadata": {},
   "source": [
    "Code 10.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a1b60614",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0\n",
       " 1.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = [\n",
    "    [1/4, 1/4, 1/4, 1/4],\n",
    "    [2/6, 1/6, 1/6, 2/6],\n",
    "    [1/6, 2/6, 2/6, 1/6],\n",
    "    [1/8, 4/8, 2/8, 1/8],\n",
    "]\n",
    "\n",
    "map(x -> sum(x .* [0, 1, 1, 2]), p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39d68bfe",
   "metadata": {},
   "source": [
    "Code 10.6\n",
    "\n",
    "Could be simplified with just `map(entropy, p)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "068ef485",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 1.3862943611198906\n",
       " 1.3296613488547582\n",
       " 1.3296613488547582\n",
       " 1.2130075659799042"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compute entropy of each distribution\n",
    "map(x -> -sum(x.*log.(x)), p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef4e1032",
   "metadata": {},
   "source": [
    "Code 10.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "9f93bd1b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 0.09000000000000002\n",
       " 0.21000000000000002\n",
       " 0.21000000000000002\n",
       " 0.48999999999999994"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.7\n",
    "A = [\n",
    "    (1-p)^2, p*(1-p), (1-p)*p, p^2,\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "303f11a3",
   "metadata": {},
   "source": [
    "Code 10.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8761745f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.221728604109787"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-sum(A.*log.(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88abff1b",
   "metadata": {},
   "source": [
    "Code 10.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "ef0273ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sim_p (generic function with 2 methods)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sim_p(G::Float64 = 1.4)\n",
    "    p = rand(Uniform(), 3)\n",
    "    x4 = (G * sum(p) - p[2] - p[3])/(2-G)\n",
    "    push!(p, x4)\n",
    "    p ./= sum(p)\n",
    "    (entropy(p), p)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f83dc7a",
   "metadata": {},
   "source": [
    "Code 10.10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "2adb138e",
   "metadata": {},
   "outputs": [
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       "  1875.09,551.112 1885.71,533.692 1896.33,516.274 1906.96,498.201 1917.58,479.16 1928.2,459.019 1938.82,437.711 1949.44,415.386 1960.07,392.525 1970.69,369.749 \n",
       "  1981.31,347.441 1991.93,325.496 2002.55,303.442 2013.17,280.839 2023.8,257.692 2034.42,234.565 2045.04,212.353 2055.66,191.851 2066.28,173.374 2076.91,156.65 \n",
       "  2087.53,141.04 2098.15,125.883 2108.77,110.868 2119.39,96.864 2130.02,87.9763 2140.64,94.546 2151.26,134.285 2161.88,227.444 2172.5,385.259 2183.12,598.235 \n",
       "  2193.75,834.729 2204.37,1053.89 2214.99,1224.59 2225.61,1336.79 2236.23,1399.16 2246.86,1428.53 2257.48,1440.26 2268.1,1444.23 2278.72,1445.37 2289.34,1445.65 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(1)\n",
    "cnt = 10^5\n",
    "H = [sim_p(1.4) for _ ∈ 1:cnt];\n",
    "density(first.(H))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b2c9e37",
   "metadata": {},
   "source": [
    "Code 10.11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "3b8792d9",
   "metadata": {},
   "outputs": [],
   "source": [
    "entropies = first.(H)\n",
    "distributions = last.(H);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bcf159a0",
   "metadata": {},
   "source": [
    "Code 10.12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0b04fbb2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2217282654319392"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maximum(entropies)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83f90a3d",
   "metadata": {},
   "source": [
    "Code 10.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "2cfba671",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 0.08983102505773202\n",
       " 0.21011303987855837\n",
       " 0.2102249100059776\n",
       " 0.48983102505773196"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distributions[findmax(entropies)[2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0ab8c7b",
   "metadata": {},
   "source": [
    "# 10.2 Generalized linear models\n",
    "\n",
    "No code here, wheee!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f115fd49",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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