notebooks/combinator_exploration.ipynb

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   "source": [
    "@file:DependsOn(\"org.jetbrains.kotlinx:kotlinx-coroutines-core:1.3.9\")"
   ]
  },
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import kotlinx.coroutines.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "val POPCOUNT = 10000\n",
    "\n",
    "fun <A, B> Iterable<A>.pmap(f: suspend (A) -> B): List<B> = runBlocking {\n",
    "    map { async { f(it) } }.awaitAll()\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "       if(!window.letsPlotCallQueue) {\n",
       "           window.letsPlotCallQueue = [];\n",
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       "       window.letsPlotCall = function(f) {\n",
       "           window.letsPlotCallQueue.push(f);\n",
       "       };\n",
       "       (function() {\n",
       "           var script = document.createElement(\"script\");\n",
       "           script.type = \"text/javascript\";\n",
       "           script.src = \"https://dl.bintray.com/jetbrains/lets-plot/lets-plot-1.5.2.min.js\";\n",
       "           script.onload = function() {\n",
       "               window.letsPlotCall = function(f) {f();};\n",
       "               window.letsPlotCallQueue.forEach(function(f) {f();});\n",
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       "               \n",
       "               \n",
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       "           script.onerror = function(event) {\n",
       "               window.letsPlotCall = function(f) {};\n",
       "               window.letsPlotCallQueue = [];\n",
       "               var div = document.createElement(\"div\");\n",
       "               div.style.color = 'darkred';\n",
       "               div.textContent = 'Error loading Lets-Plot JS';\n",
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       "           };\n",
       "           var e = document.getElementById(\"R5cPjm\");\n",
       "           e.appendChild(script);\n",
       "       })();\n",
       "   </script>"
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   ],
   "source": [
    "%use lets-plot"
   ]
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   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "val rand = kotlin.random.Random(1)\n",
    "\n",
    "fun pdf(x: Double) = exp(-x * x / 2) / sqrt(2 * PI)\n",
    "\n",
    "fun pdf(x: Double, μ: Double, σ: Double) = pdf((x - μ) / σ) / σ\n",
    "\n",
    "tailrec fun cdf(z: Double, sum: Double = 0.0, term: Double = z, i: Int = 3): Double =\n",
    "    when {\n",
    "        z < -8.0 -> 0.0\n",
    "        z > 8.0 -> 1.0\n",
    "        sum + term == sum -> 0.5 + sum * pdf(z)\n",
    "        else -> cdf(z, sum + term, term * z * z / i, i + 2)\n",
    "    }\n",
    "\n",
    "fun cdf(z: Double, μ: Double, σ: Double) = cdf((z - μ) / σ)\n",
    "\n",
    "fun inverseCDF(y: Double, μ: Double = 0.0, σ: Double = 1.0) =\n",
    "    inverseCDF(y, -8.0, 8.0, μ, σ)\n",
    "\n",
    "val precision = 0.00000001\n",
    "tailrec fun inverseCDF(\n",
    "    y: Double, lo: Double, hi: Double,\n",
    "    μ: Double, σ: Double,\n",
    "    mid: Double = lo + (hi - lo) / 2\n",
    "): Double = when {\n",
    "    hi - lo < precision -> mid\n",
    "    cdf(mid, μ, σ) > y -> inverseCDF(y, lo, mid, μ, σ)\n",
    "    else -> inverseCDF(y, mid, hi, μ, σ)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "fun compare(vararg samplers: (Double) -> Double) =\n",
    "    compare(*samplers.map { f -> \n",
    "            (1..POPCOUNT).pmap { f(rand.nextDouble()) } \n",
    "        }.toTypedArray()\n",
    "    )\n",
    "\n",
    "fun compare(vararg samples: List<Double>) =\n",
    "    lets_plot(mapOf<String, Any>(\n",
    "        \"x\" to samples.fold(listOf<Double>()) { acc, function ->\n",
    "            acc + function\n",
    "        },\n",
    "        \"\" to samples.mapIndexed { i, s -> List(s.size) { \"PDF$i\" } }.flatten()\n",
    "    )).let {  \n",
    "        it + geom_density(alpha = .3) { x = \"x\"; fill = \"\" } + ggsize(500, 250)\n",
    "    }"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Suppose we have two normal distributions."
   ]
  },
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       "'mapping':{\n",
       "},\n",
       "'data':{\n",
       "},\n",
       "'ggsize':{\n",
       "'width':500,\n",
       "'height':250\n",
       "},\n",
       "'kind':\"plot\",\n",
       "'scales':[],\n",
       "'layers':[{\n",
       "'stat':\"density\",\n",
       "'mapping':{\n",
       "'x':\"x\",\n",
       "'fill':\"\"\n",
       "},\n",
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       "},\n",
       "'alpha':0.3,\n",
       "'position':\"identity\",\n",
       "'geom':\"density\"\n",
       "}]\n",
       "};\n",
       "           var plotContainer = document.getElementById(\"3qlc79\");\n",
       "           window.letsPlotCall(function() {{\n",
       "               LetsPlot.buildPlotFromProcessedSpecs(plotSpec, -1, -1, plotContainer);\n",
       "           }});\n",
       "       })();    \n",
       "   </script>"
      ]
     },
     "execution_count": 50,
     "metadata": {
      "new_classpath": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val prA = { it: Double -> inverseCDF(it) }\n",
    "val prB = { it: Double -> prA(it) * 1.4 + 3.5 }\n",
    "\n",
    "compare(prA, prB)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How can we combine these distributions? One way is to simply average them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "   <script type=\"text/javascript\" data-lets-plot-script=\"plot\">\n",
       "       (function() {\n",
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       "},\n",
       "'alpha':0.3,\n",
       "'position':\"identity\",\n",
       "'geom':\"density\"\n",
       "}]\n",
       "};\n",
       "           var plotContainer = document.getElementById(\"iuHrcA\");\n",
       "           window.letsPlotCall(function() {{\n",
       "               LetsPlot.buildPlotFromProcessedSpecs(plotSpec, -1, -1, plotContainer);\n",
       "           }});\n",
       "       })();    \n",
       "   </script>"
      ]
     },
     "execution_count": 55,
     "metadata": {
      "new_classpath": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val prC = { it: Double -> (prA(it) + prB(it)) / 2.0 }\n",
    "compare(prA, prB, prC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another way is to mix them somehow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "   <div id=\"LNuPPo\"></div>\n",
       "   <script type=\"text/javascript\" data-lets-plot-script=\"plot\">\n",
       "       (function() {\n",
       "           var plotSpec={\n",
       "'mapping':{\n",
       "},\n",
       "'data':{\n",
       "},\n",
       "'ggsize':{\n",
       "'width':500,\n",
       "'height':250\n",
       "},\n",
       "'kind':\"plot\",\n",
       "'scales':[],\n",
       "'layers':[{\n",
       "'stat':\"density\",\n",
       "'mapping':{\n",
       "'x':\"x\",\n",
       "'fill':\"\"\n",
       "},\n",
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       "},\n",
       "'alpha':0.3,\n",
       "'position':\"identity\",\n",
       "'geom':\"density\"\n",
       "}]\n",
       "};\n",
       "           var plotContainer = document.getElementById(\"LNuPPo\");\n",
       "           window.letsPlotCall(function() {{\n",
       "               LetsPlot.buildPlotFromProcessedSpecs(plotSpec, -1, -1, plotContainer);\n",
       "           }});\n",
       "       })();    \n",
       "   </script>"
      ]
     },
     "execution_count": 66,
     "metadata": {
      "new_classpath": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val prD = { it: Double -> \n",
    "    if(rand.nextBoolean()) prA(it) else prB(it) \n",
    "}\n",
    "\n",
    "compare(prA, prB, prD)"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Kotlin",
   "language": "kotlin",
   "name": "kotlin"
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   "file_extension": ".kt",
   "mimetype": "text/x-kotlin",
   "name": "kotlin",
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   "version": "1.4.30-dev-2223"
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}