Poisson Counting Experiment - BAT.jl Tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# BAT.jl Tutorial - Poisson Counting Experiment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using BAT\n",
    "using Distributions \n",
    "using IntervalSets\n",
    "using ShapesOfVariables\n",
    "using Plots\n",
    "using ArraysOfArrays\n",
    "using StatsBase "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Situation\n",
    "We want to measure a source in the presence of background. \n",
    "For example, this could be a radioactive element that is to be measured in a laboratory and is therefore background from natural radioactivity.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Background only measurement\n",
    "A measurement without the signal source yields a number of $N_B=10$ counts.\n",
    "### Task: \n",
    "Perform a Bayesian analysis of this situation to estimate the parameter describing the backound count rate $\\lambda_b$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "struct Background<:AbstractDensity\n",
    "    k::Float64 # observed counts\n",
    "end\n",
    "\n",
    "function BAT.density_logval(target::Background, params::Union{NamedTuple,AbstractVector{<:Real}})\n",
    "    return logpdf(Poisson(params[1][1]), target.k) # poisson log-likelihood\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PosteriorDensity{Background,DistributionDensity{NamedPrior{(:λb,),Tuple{Uniform{Float64}},Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}},Tuple{HyperRectBounds{Float64}}}},HyperRectBounds{Float64},VarShapes{(:λb,),Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}}}}(Background(10.0), DistributionDensity{NamedPrior{(:λb,),Tuple{Uniform{Float64}},Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}},Tuple{HyperRectBounds{Float64}}}}(NamedPrior{(:λb,),Tuple{Uniform{Float64}},Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}},Tuple{HyperRectBounds{Float64}}}(\n",
       "_distributions: (λb = Uniform{Float64}(a=0.0, b=30.0),)\n",
       "_shapes: VarShapes{(:λb,),Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}}}((λb = ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}(ScalarShape{Real}(), 0, 1),), 1)\n",
       "_bounds: (λb = HyperRectBounds{Float64}(HyperRectVolume{Float64}([0.0], [30.0]), BoundsType[reflective_bounds]),)\n",
       ")\n",
       "), HyperRectBounds{Float64}(HyperRectVolume{Float64}([0.0], [30.0]), BoundsType[reflective_bounds]), VarShapes{(:λb,),Tuple{ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}}}((λb = ShapesOfVariables.VariableDataAccessor{ScalarShape{Real}}(ScalarShape{Real}(), 0, 1),), 1))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Number of observed background events\n",
    "kb = 10\n",
    "likelihood_B = Background(kb)\n",
    "\n",
    "prior_B = NamedPrior(\n",
    "    λb = 0..30\n",
    ")\n",
    "\n",
    "posterior_B = PosteriorDensity(likelihood_B, prior_B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100000"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "algorithm = MetropolisHastings()\n",
    "nchains = 8\n",
    "nsamples = 10^5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Trying to generate 8 viable MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:160\n",
      "┌ Info: Selected 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:269\n",
      "┌ Info: Begin tuning of 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:52\n",
      "┌ Info: MCMC Tuning cycle 1 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 2 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 3 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 4 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 5 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 6 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 7 finished, 8 chains, 8 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC tuning of 8 chains successful after 7 cycle(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:88\n"
     ]
    }
   ],
   "source": [
    "samples_B, sampleids_B, stats_B, chains_B = rand(MCMCSpec(algorithm, posterior_B), nsamples, nchains);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "200\n",
      "MCMCBasicStats{Float64,Float64}(BasicMvStatistics{Float64,FrequencyWeights}(1, [11.006466414848784], [10.98541022727633], [29.958334900436746], [1.3333963488624452]), BasicUvStatistics{Float64,FrequencyWeights}(OnlineUvMean{Float64}(-1.3778232120898303e7, 2.301051e6), OnlineUvVar{Float64,FrequencyWeights}(800000, 2.301051e6, 1.1534257e7, -5.987799540687421, 1.1826539093023478e6), -5.479759024799576, -16.961712973781932), [10.000006873222164])"
     ]
    }
   ],
   "source": [
    "par_names=[\"\\$\\\\lambda_b\\$\"]\n",
    "plot(posterior_B, samples_B, :λb, xlabel = par_names[1], ylabel = \"P($(par_names[1]))\")\n",
    "plot!(prior_B, :λb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MCMCBasicStats{Float64,Float64}(BasicMvStatistics{Float64,FrequencyWeights}(1, [11.006466414848784], [10.98541022727633], [29.958334900436746], [1.3333963488624452]), BasicUvStatistics{Float64,FrequencyWeights}(OnlineUvMean{Float64}(-1.3778232120898303e7, 2.301051e6), OnlineUvVar{Float64,FrequencyWeights}(800000, 2.301051e6, 1.1534257e7, -5.987799540687421, 1.1826539093023478e6), -5.479759024799576, -16.961712973781932), [10.000006873222164])"
     ]
    }
   ],
   "source": [
    "print(stats_B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Questions:\n",
    "* What is the distribution of the posterior? What is the best estimator for the parameter?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Further Background only measurement\n",
    "A second measurement without the signal source yields a number of $N_B=8$ counts.\n",
    "\n",
    "Use posterior of first backoground measurement as prior (analytic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Histogram{Int64,1,Tuple{StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}}}\n",
       "edges:\n",
       "  1.3:0.1:30.0\n",
       "weights: [1, 1, 1, 2, 4, 5, 8, 11, 22, 15  …  12, 3, 12, 4, 3, 7, 4, 5, 4, 2]\n",
       "closed: left\n",
       "isdensity: false"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# replace by analytic poisson prior\n",
    "hist_10 = fit(Histogram, flatview(samples_B.params)[1, :], FrequencyWeights(samples_B.weight), nbins = 400, closed = :left)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Number of observed background events\n",
    "kb2 = 8\n",
    "likelihood_B2 = Background(kb2)\n",
    "\n",
    "prior_B2 = NamedPrior(\n",
    "    λb = BAT.HistogramAsUvDistribution(hist_10) # replace by analytic poisson prior\n",
    ")\n",
    "\n",
    "prior_B2flat = NamedPrior(\n",
    "    λb = 0..30\n",
    ")\n",
    "\n",
    "posterior_B2 = PosteriorDensity(likelihood_B2, prior_B2)\n",
    "likelipost_B2 = PosteriorDensity(likelihood_B2, prior_B2flat)\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Trying to generate 8 viable MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:160\n",
      "┌ Info: Selected 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:269\n",
      "┌ Info: Begin tuning of 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:52\n",
      "┌ Info: MCMC Tuning cycle 1 finished, 8 chains, 8 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC tuning of 8 chains successful after 1 cycle(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:88\n",
      "┌ Info: Trying to generate 8 viable MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:160\n",
      "┌ Info: Selected 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:269\n",
      "┌ Info: Begin tuning of 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:52\n",
      "┌ Info: MCMC Tuning cycle 1 finished, 8 chains, 8 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC tuning of 8 chains successful after 1 cycle(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:88\n"
     ]
    }
   ],
   "source": [
    "samples_B2, sampleids_B2, stats_B2, chains_B2 = rand(MCMCSpec(algorithm, posterior_B2), nsamples, nchains);\n",
    "likelipost_samples_B2, likelids_B2, likeli_stats_B2, like_chains_B2 = rand(MCMCSpec(algorithm, likelipost_B2), nsamples, nchains);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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       "  725.721,620.632 732.595,620.632 732.595,597.018 739.469,597.018 739.469,547.791 746.343,547.791 746.343,500.812 753.217,500.812 753.217,476.032 760.09,476.032 \n",
       "  760.09,440.18 766.964,440.18 766.964,400.083 773.838,400.083 773.838,348.47 780.712,348.47 780.712,294.859 787.586,294.859 787.586,273.325 794.459,273.325 \n",
       "  794.459,286.922 801.333,286.922 801.333,232.951 808.207,232.951 808.207,186.388 815.081,186.388 815.081,203.287 821.955,203.287 821.955,181.365 828.829,181.365 \n",
       "  828.829,169.683 835.702,169.683 835.702,182.309 842.576,182.309 842.576,152.895 849.45,152.895 849.45,141.434 856.324,141.434 856.324,141.545 863.198,141.545 \n",
       "  863.198,156.696 870.072,156.696 870.072,124.619 876.945,124.619 876.945,191.632 883.819,191.632 883.819,165.909 890.693,165.909 890.693,142.628 897.567,142.628 \n",
       "  897.567,175.843 904.441,175.843 904.441,180.561 911.315,180.561 911.315,184.279 918.188,184.279 918.188,194.102 925.062,194.102 925.062,248.102 931.936,248.102 \n",
       "  931.936,278.709 938.81,278.709 938.81,285.563 945.684,285.563 945.684,331.016 952.557,331.016 952.557,340.339 959.431,340.339 959.431,364.647 966.305,364.647 \n",
       "  966.305,399.5 973.179,399.5 973.179,440.652 980.053,440.652 980.053,479.972 986.927,479.972 986.927,485.023 993.8,485.023 993.8,551.287 1000.67,551.287 \n",
       "  1000.67,538.717 1007.55,538.717 1007.55,551.37 1014.42,551.37 1014.42,625.71 1021.3,625.71 1021.3,628.957 1028.17,628.957 1028.17,673.189 1035.04,673.189 \n",
       "  1035.04,685.676 1041.92,685.676 1041.92,721.555 1048.79,721.555 1048.79,755.742 1055.66,755.742 1055.66,785.322 1062.54,785.322 1062.54,828.333 1069.41,828.333 \n",
       "  1069.41,851.42 1076.29,851.42 1076.29,866.488 1083.16,866.488 1083.16,890.796 1090.03,890.796 1090.03,915.049 1096.91,915.049 1096.91,971.324 1103.78,971.324 \n",
       "  1103.78,979.565 1110.66,979.565 1110.66,1020.05 1117.53,1020.05 1117.53,1024.91 1124.4,1024.91 1124.4,1037.12 1131.28,1037.12 1131.28,1061.73 1138.15,1061.73 \n",
       "  1138.15,1094.03 1145.02,1094.03 1145.02,1101.72 1151.9,1101.72 1151.9,1123.25 1158.77,1123.25 1158.77,1148.11 1165.65,1148.11 1165.65,1165.12 1172.52,1165.12 \n",
       "  1172.52,1177.83 1179.39,1177.83 1179.39,1177.55 1186.27,1177.55 1186.27,1198.78 1193.14,1198.78 1193.14,1216.82 1200.02,1216.82 1200.02,1227.25 1206.89,1227.25 \n",
       "  1206.89,1232.19 1213.76,1232.19 1213.76,1395.13 1220.64,1395.13 1220.64,1395.13 1227.51,1395.13 1227.51,1395.13 1234.38,1395.13 1234.38,1395.13 1241.26,1395.13 \n",
       "  1241.26,1395.13 1248.13,1395.13 1248.13,1395.13 1255.01,1395.13 1255.01,1395.13 1261.88,1395.13 1261.88,1395.13 1268.75,1395.13 1268.75,1395.13 1275.63,1395.13 \n",
       "  1275.63,1395.13 1282.5,1395.13 1282.5,1395.13 1289.37,1395.13 1289.37,1395.13 1296.25,1395.13 1296.25,1395.13 1303.12,1395.13 1303.12,1395.13 1310,1395.13 \n",
       "  1310,1395.13 1316.87,1395.13 1316.87,1395.13 1323.74,1395.13 1323.74,1395.13 1330.62,1395.13 1330.62,1395.13 1337.49,1395.13 1337.49,1395.13 1344.37,1395.13 \n",
       "  1344.37,1395.13 1351.24,1395.13 1351.24,1395.13 1358.11,1395.13 1358.11,1395.13 1364.99,1395.13 1364.99,1395.13 1371.86,1395.13 1371.86,1395.13 1378.73,1395.13 \n",
       "  1378.73,1395.13 1385.61,1395.13 1385.61,1395.13 1392.48,1395.13 1392.48,1395.13 1399.36,1395.13 1399.36,1395.13 1406.23,1395.13 1406.23,1395.13 1413.1,1395.13 \n",
       "  1413.1,1395.13 1419.98,1395.13 1419.98,1395.13 1426.85,1395.13 1426.85,1395.13 1433.73,1395.13 1433.73,1395.13 1440.6,1395.13 1440.6,1395.13 1447.47,1395.13 \n",
       "  1447.47,1395.13 1454.35,1395.13 1454.35,1395.13 1461.22,1395.13 1461.22,1395.13 1468.09,1395.13 1468.09,1395.13 1474.97,1395.13 1474.97,1395.13 1481.84,1395.13 \n",
       "  1481.84,1395.13 1488.72,1395.13 1488.72,1395.13 1495.59,1395.13 1495.59,1395.13 1502.46,1395.13 1502.46,1395.13 1509.34,1395.13 1509.34,1395.13 1516.21,1395.13 \n",
       "  1516.21,1395.13 1523.09,1395.13 1523.09,1395.13 1529.96,1395.13 1529.96,1395.13 1536.83,1395.13 1536.83,1395.13 1543.71,1395.13 1543.71,1395.13 1550.58,1395.13 \n",
       "  1550.58,1395.13 1557.45,1395.13 1557.45,1395.13 1564.33,1395.13 1564.33,1395.13 1571.2,1395.13 1571.2,1395.13 1578.08,1395.13 1578.08,1395.13 1584.95,1395.13 \n",
       "  1584.95,1395.13 1591.82,1395.13 1591.82,1395.13 1598.7,1395.13 1598.7,1395.13 1605.57,1395.13 1605.57,1395.13 1612.44,1395.13 1612.44,1395.13 1619.32,1395.13 \n",
       "  1619.32,1395.13 1626.19,1395.13 1626.19,1395.13 1633.07,1395.13 1633.07,1395.13 1639.94,1395.13 1639.94,1395.13 1646.81,1395.13 1646.81,1395.13 1653.69,1395.13 \n",
       "  1653.69,1395.13 1660.56,1395.13 1660.56,1395.13 1667.44,1395.13 1667.44,1395.13 1674.31,1395.13 1674.31,1395.13 1681.18,1395.13 1681.18,1395.13 1688.06,1395.13 \n",
       "  1688.06,1395.13 1694.93,1395.13 1694.93,1395.13 1701.8,1395.13 1701.8,1395.13 1708.68,1395.13 1708.68,1395.13 1715.55,1395.13 1715.55,1395.13 1722.43,1395.13 \n",
       "  1722.43,1395.13 1729.3,1395.13 1729.3,1395.13 1736.17,1395.13 1736.17,1395.13 1743.05,1395.13 1743.05,1395.13 1749.92,1395.13 1749.92,1395.13 1756.8,1395.13 \n",
       "  1756.8,1395.13 1763.67,1395.13 1763.67,1395.13 1770.54,1395.13 1770.54,1395.13 1777.42,1395.13 1777.42,1395.13 1784.29,1395.13 1784.29,1395.13 1791.16,1395.13 \n",
       "  1791.16,1395.13 1798.04,1395.13 1798.04,1395.13 1804.91,1395.13 1804.91,1395.13 1811.79,1395.13 1811.79,1395.13 1818.66,1395.13 1818.66,1395.13 1825.53,1395.13 \n",
       "  1825.53,1395.13 1832.41,1395.13 1832.41,1395.13 1839.28,1395.13 1839.28,1395.13 1846.15,1395.13 1846.15,1395.13 1853.03,1395.13 1853.03,1395.13 1859.9,1395.13 \n",
       "  1859.9,1395.13 1866.78,1395.13 1866.78,1395.13 1873.65,1395.13 1873.65,1395.13 1880.52,1395.13 1880.52,1395.13 1887.4,1395.13 1887.4,1395.13 1894.27,1395.13 \n",
       "  1894.27,1395.13 1901.15,1395.13 1901.15,1395.13 1908.02,1395.13 1908.02,1395.13 1914.89,1395.13 1914.89,1395.13 1914.89,1395.13 1914.89,1395.13 1908.02,1395.13 \n",
       "  1908.02,1395.13 1901.15,1395.13 1901.15,1395.13 1894.27,1395.13 1894.27,1395.13 1887.4,1395.13 1887.4,1395.13 1880.52,1395.13 1880.52,1395.13 1873.65,1395.13 \n",
       "  1873.65,1395.13 1866.78,1395.13 1866.78,1395.13 1859.9,1395.13 1859.9,1395.13 1853.03,1395.13 1853.03,1395.13 1846.15,1395.13 1846.15,1395.13 1839.28,1395.13 \n",
       "  1839.28,1395.13 1832.41,1395.13 1832.41,1395.13 1825.53,1395.13 1825.53,1395.13 1818.66,1395.13 1818.66,1395.13 1811.79,1395.13 1811.79,1395.13 1804.91,1395.13 \n",
       "  1804.91,1395.13 1798.04,1395.13 1798.04,1395.13 1791.16,1395.13 1791.16,1395.13 1784.29,1395.13 1784.29,1395.13 1777.42,1395.13 1777.42,1395.13 1770.54,1395.13 \n",
       "  1770.54,1395.13 1763.67,1395.13 1763.67,1395.13 1756.8,1395.13 1756.8,1395.13 1749.92,1395.13 1749.92,1395.13 1743.05,1395.13 1743.05,1395.13 1736.17,1395.13 \n",
       "  1736.17,1395.13 1729.3,1395.13 1729.3,1395.13 1722.43,1395.13 1722.43,1395.13 1715.55,1395.13 1715.55,1395.13 1708.68,1395.13 1708.68,1395.13 1701.8,1395.13 \n",
       "  1701.8,1395.13 1694.93,1395.13 1694.93,1395.13 1688.06,1395.13 1688.06,1395.13 1681.18,1395.13 1681.18,1395.13 1674.31,1395.13 1674.31,1395.13 1667.44,1395.13 \n",
       "  1667.44,1395.13 1660.56,1395.13 1660.56,1395.13 1653.69,1395.13 1653.69,1395.13 1646.81,1395.13 1646.81,1395.13 1639.94,1395.13 1639.94,1395.13 1633.07,1395.13 \n",
       "  1633.07,1395.13 1626.19,1395.13 1626.19,1395.13 1619.32,1395.13 1619.32,1395.13 1612.44,1395.13 1612.44,1395.13 1605.57,1395.13 1605.57,1395.13 1598.7,1395.13 \n",
       "  1598.7,1395.13 1591.82,1395.13 1591.82,1395.13 1584.95,1395.13 1584.95,1395.13 1578.08,1395.13 1578.08,1395.13 1571.2,1395.13 1571.2,1395.13 1564.33,1395.13 \n",
       "  1564.33,1395.13 1557.45,1395.13 1557.45,1395.13 1550.58,1395.13 1550.58,1395.13 1543.71,1395.13 1543.71,1395.13 1536.83,1395.13 1536.83,1395.13 1529.96,1395.13 \n",
       "  1529.96,1395.13 1523.09,1395.13 1523.09,1395.13 1516.21,1395.13 1516.21,1395.13 1509.34,1395.13 1509.34,1395.13 1502.46,1395.13 1502.46,1395.13 1495.59,1395.13 \n",
       "  1495.59,1395.13 1488.72,1395.13 1488.72,1395.13 1481.84,1395.13 1481.84,1395.13 1474.97,1395.13 1474.97,1395.13 1468.09,1395.13 1468.09,1395.13 1461.22,1395.13 \n",
       "  1461.22,1395.13 1454.35,1395.13 1454.35,1395.13 1447.47,1395.13 1447.47,1395.13 1440.6,1395.13 1440.6,1395.13 1433.73,1395.13 1433.73,1395.13 1426.85,1395.13 \n",
       "  1426.85,1395.13 1419.98,1395.13 1419.98,1395.13 1413.1,1395.13 1413.1,1395.13 1406.23,1395.13 1406.23,1395.13 1399.36,1395.13 1399.36,1395.13 1392.48,1395.13 \n",
       "  1392.48,1395.13 1385.61,1395.13 1385.61,1395.13 1378.73,1395.13 1378.73,1395.13 1371.86,1395.13 1371.86,1395.13 1364.99,1395.13 1364.99,1395.13 1358.11,1395.13 \n",
       "  1358.11,1395.13 1351.24,1395.13 1351.24,1395.13 1344.37,1395.13 1344.37,1395.13 1337.49,1395.13 1337.49,1395.13 1330.62,1395.13 1330.62,1395.13 1323.74,1395.13 \n",
       "  1323.74,1395.13 1316.87,1395.13 1316.87,1395.13 1310,1395.13 1310,1395.13 1303.12,1395.13 1303.12,1395.13 1296.25,1395.13 1296.25,1395.13 1289.37,1395.13 \n",
       "  1289.37,1395.13 1282.5,1395.13 1282.5,1395.13 1275.63,1395.13 1275.63,1395.13 1268.75,1395.13 1268.75,1395.13 1261.88,1395.13 1261.88,1395.13 1255.01,1395.13 \n",
       "  1255.01,1395.13 1248.13,1395.13 1248.13,1395.13 1241.26,1395.13 1241.26,1395.13 1234.38,1395.13 1234.38,1395.13 1227.51,1395.13 1227.51,1395.13 1220.64,1395.13 \n",
       "  1220.64,1395.13 1213.76,1395.13 1213.76,1395.13 1206.89,1395.13 1206.89,1395.13 1200.02,1395.13 1200.02,1395.13 1193.14,1395.13 1193.14,1395.13 1186.27,1395.13 \n",
       "  1186.27,1395.13 1179.39,1395.13 1179.39,1395.13 1172.52,1395.13 1172.52,1395.13 1165.65,1395.13 1165.65,1395.13 1158.77,1395.13 1158.77,1395.13 1151.9,1395.13 \n",
       "  1151.9,1395.13 1145.02,1395.13 1145.02,1395.13 1138.15,1395.13 1138.15,1395.13 1131.28,1395.13 1131.28,1395.13 1124.4,1395.13 1124.4,1395.13 1117.53,1395.13 \n",
       "  1117.53,1395.13 1110.66,1395.13 1110.66,1395.13 1103.78,1395.13 1103.78,1395.13 1096.91,1395.13 1096.91,1395.13 1090.03,1395.13 1090.03,1395.13 1083.16,1395.13 \n",
       "  1083.16,1395.13 1076.29,1395.13 1076.29,1395.13 1069.41,1395.13 1069.41,1395.13 1062.54,1395.13 1062.54,1395.13 1055.66,1395.13 1055.66,1395.13 1048.79,1395.13 \n",
       "  1048.79,1395.13 1041.92,1395.13 1041.92,1395.13 1035.04,1395.13 1035.04,1395.13 1028.17,1395.13 1028.17,1395.13 1021.3,1395.13 1021.3,1395.13 1014.42,1395.13 \n",
       "  1014.42,1395.13 1007.55,1395.13 1007.55,1395.13 1000.67,1395.13 1000.67,1395.13 993.8,1395.13 993.8,1395.13 986.927,1395.13 986.927,1395.13 980.053,1395.13 \n",
       "  980.053,1395.13 973.179,1395.13 973.179,1395.13 966.305,1395.13 966.305,1395.13 959.431,1395.13 959.431,1395.13 952.557,1395.13 952.557,1395.13 945.684,1395.13 \n",
       "  945.684,1395.13 938.81,1395.13 938.81,1395.13 931.936,1395.13 931.936,1395.13 925.062,1395.13 925.062,1395.13 918.188,1395.13 918.188,1395.13 911.315,1395.13 \n",
       "  911.315,1395.13 904.441,1395.13 904.441,1395.13 897.567,1395.13 897.567,1395.13 890.693,1395.13 890.693,1395.13 883.819,1395.13 883.819,1395.13 876.945,1395.13 \n",
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     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(posterior_B2, samples_B2, :λb, xlabel = par_names[1], ylabel = \"P($(par_names[1]))\")\n",
    "plot!(likelipost_B2, likelipost_samples_B2, :λb, seriestype=:stephist, linecolor=:blue,linewidth=1.5, localmode=false, label=\"likelihood\")\n",
    "plot!(prior_B2, :λb, linewidth=1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Signal + Background\n",
    "Measurement with source: $N_{S+B}=12$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SignalAndBackground(5.0)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "struct SignalAndBackground<:AbstractDensity\n",
    "    k::Float64 # observed counts\n",
    "end\n",
    "\n",
    "function BAT.density_logval(target::SignalAndBackground, params::Union{NamedTuple,AbstractVector{<:Real}})\n",
    "    return logpdf(Poisson(params[1][1]), target.k) + logpdf(Poisson(params[1][1] + params[2][1]), target.k)  # poisson log-likelihood\n",
    "end\n",
    "\n",
    "kSB = 5\n",
    "likelihood_SB = SignalAndBackground(kSB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "hist_B2 = fit(Histogram, flatview(samples_B2.params)[1, :], FrequencyWeights(samples_B2.weight), nbins = 400, closed = :left)\n",
    "B2 = BAT.HistogramAsUvDistribution(hist_B2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "prior_SB = NamedPrior(\n",
    "    λb = B2,\n",
    "    λs = 0..30\n",
    ")\n",
    "\n",
    "posterior_SB = PosteriorDensity(likelihood_SB, prior_SB);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Trying to generate 8 viable MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:160\n",
      "┌ Info: Selected 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:269\n",
      "┌ Info: Begin tuning of 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:52\n",
      "┌ Info: MCMC Tuning cycle 1 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 2 finished, 8 chains, 8 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC tuning of 8 chains successful after 2 cycle(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:88\n"
     ]
    }
   ],
   "source": [
    "samples_SB, sampleids_SB, stats_SB, chains_SB = rand(MCMCSpec(algorithm, posterior_SB), nsamples, nchains);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(samples_SB)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error propagation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### a) use $ϵ \\propto $ Normaldistribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### b) Binomial analysis of calibration measurement with known source (neglect background because of high signal rate)\n",
    "assume: expected calibration source rate $N_\\text{expected} = 1000$ and measured by the detector $N_\\text{measured} = 123$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BinomialModel(100.0, 13.0)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "struct BinomialModel<:AbstractDensity\n",
    "    n::Float64 # n trials\n",
    "    k::Float64 # k succes\n",
    "end\n",
    "\n",
    "function BAT.density_logval(target::BinomialModel, params::Union{NamedTuple,AbstractVector{<:Real}})\n",
    "    return logpdf(Binomial(target.n, params[1][1]), target.k) # poisson log-likelihood\n",
    "end\n",
    "\n",
    "likelihood_binomial = BinomialModel(100, 13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "prior_binomial = NamedPrior(\n",
    "    p = 0..1\n",
    ")\n",
    "\n",
    "posterior_binomial = PosteriorDensity(likelihood_binomial, prior_binomial);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Binomial(n::Real, p) is deprecated. Please use Binomial(n::Integer, p) instead.\n",
      "│   caller = density_logval at In[70]:7 [inlined]\n",
      "└ @ Core ./In[70]:7\n",
      "┌ Info: Trying to generate 8 viable MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:160\n",
      "┌ Warning: Binomial(n::Real, p) is deprecated. Please use Binomial(n::Integer, p) instead.\n",
      "│   caller = density_logval at In[70]:7 [inlined]\n",
      "└ @ Core ./In[70]:7\n",
      "┌ Info: Selected 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:269\n",
      "┌ Info: Begin tuning of 8 MCMC chain(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:52\n",
      "┌ Info: MCMC Tuning cycle 1 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 2 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 3 finished, 8 chains, 0 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC Tuning cycle 4 finished, 8 chains, 8 tuned, 8 converged.\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:84\n",
      "┌ Info: MCMC tuning of 8 chains successful after 4 cycle(s).\n",
      "└ @ BAT /home/cornelius/.julia/packages/BAT/rhyTy/src/samplers/mcmc/mcmc_tuner.jl:88\n"
     ]
    }
   ],
   "source": [
    "samples_binomial, sampleids_binomial, stats_binomial, chains_binomial = rand(MCMCSpec(algorithm, posterior_binomial), nsamples, nchains);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
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