Mixture Models.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Histograms of data often reveal that they do not follow any standard probability distribution. Sometimes we have explanatory variables (or covariates) to account for the different values, and normally distributed errors are adequate, as in normal regression. However, if we only have the data values themselves and no covariates, we might have to fit a non-standard distribution to the data. One way to do this is by mixing standard distributions.\n",
    "\n",
    "Mixture distributions are just a weighted combination of probability distribtuions. For example, we could take an exponential distribution with mean 1 and normal distribution with mean 3 and variance 1 (although typically the two mixture components would have the same support; here the exponential component has to be non-negative and the normal component can be positive or negative). Suppose we give them weights: 0.4 for the exponential distribution and 0.6 for the normal distribution. We could write the PDF for this distribution as\n",
    "<br>\n",
    "$$p(y) = 0.4 . exp(-y) . I_{(y>=0)} + 0.6 . \\frac{1}{\\sqrt{2\\pi}} exp(-\\frac{1}{2}(y-3)^{2})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The PDF of this mixture distribution would look like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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yA0mAKhwRQIDaZAaDDFPyh8vExwkz4nAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"40/60 mixture of exponential and normal distributions\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width = 6, repr.plot.height=5)\n",
    "\n",
    "curve( 0.4*dexp(x, 1.0) + 0.6*dnorm(x, 3.0, 1.0), \n",
    "       from=-2.0, to=7.0, \n",
    "       ylab=\"density\", \n",
    "       xlab=\"y\", \n",
    "       main=\"40/60 mixture of exponential and normal distributions\", \n",
    "       lwd=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could think of these two distributions as governing two distinct populations, one following the exponential distribution and the other following the normal distribution.\n",
    "\n",
    "Let’s draw the weighted PDFs for each population."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BChbdiFDmH0Cp+7bRDaXZEEQ8zGJnSQ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\nRK5CYII=",
      "text/plain": [
       "Plot with title \"40/60 mixture of exponential and normal distributions\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "curve( 0.4*dexp(x, 1.0) + 0.6*dnorm(x, 3.0, 1.0), \n",
    "       from=-2.0, to=7.0, \n",
    "       ylab=\"density\", \n",
    "       xlab=\"y\", \n",
    "       main=\"40/60 mixture of exponential and normal distributions\", \n",
    "       lwd=2)\n",
    "\n",
    "curve( 0.4*dexp(x, 1.0), \n",
    "       from=-2.0, \n",
    "       to=7.0, \n",
    "       col=\"red\", \n",
    "       lty=2, \n",
    "       add=TRUE)\n",
    "\n",
    "curve( 0.6*dnorm(x, 3.0, 1.0), \n",
    "       from=-2.0, to=7.0, \n",
    "       col=\"blue\", \n",
    "       lty=2, \n",
    "       add=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general form for a discrete mixture of distributions is as follows:\n",
    "\n",
    "$$p(y) = \\sum_{j=1}^{J}w_{j}.f_{j}(y)$$\n",
    "\n",
    "where the $\\omega$’s are positive weights that add up to 1 (they are probabilities) and each of the J $f_{j}(y)$ functions is a PDF for some distribution. In the example above, the weights were 0.4 and 0.6, $f_{1}$ was an exponential PDF and $f_{2}$ was a normal PDF.\n",
    "\n",
    "One way to simulate from a mixture distribution is with a hierarchical model. We first simulate an indicator for which “population” the next observation will come from using the weights $\\omega$. Let’s call this $z_{i}$. In the example above, $z_{i}$ would take the value 1 (indicating the exponential distribution) with probability 0.4 and 2 (indicating the normal distribution) with probability 0.6. Next, simulate the observation yi from the distribution corresponding to $z_{i}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(117)\n",
    "n = 1000\n",
    "z = numeric(n)\n",
    "y = numeric(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bkNcoIm6P2Yu0MfDRdhSNAEvR9zd2hg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\nCkFDCkFDCkFDCkFDCkFDCkFDyh9qS9CaSr9G2QAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Histogram of y\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for(i in 1:n)\n",
    "{\n",
    "    # returns a 1 with probability 0.4, or a 2 with probability 0.6\n",
    "    z[i] = sample.int(2, 1, prob = c(0.4, 0.6))\n",
    "    if(z[i] == 1)\n",
    "    {\n",
    "        y[i] = rexp(1, rate = 1.0)\n",
    "    }\n",
    "    else\n",
    "    {\n",
    "        y[i] = rnorm(1, mean = 3.0, sd = 1.0)\n",
    "    }\n",
    "}\n",
    "\n",
    "hist(y, breaks = 40)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayesian Inference for mixture models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we fit a mixture model to data, we usually only have the y values and do not know which “population” they belong to. Because the z variables are unobserved, they are called latent variables. We can treat them as parameters in a hierarchical model and perform Bayesian inference for them. The hierarchial model might look like this:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y_{i} \\, | \\, z_{i},\\theta \\, = \\, f_{z_{i}}(y \\, | \\, \\theta \\,) \\,\\,  i = 1,...,n $$<br>\n",
    "$$Pr(z_{i} | j,w)  = w_{j}, j = 1,...,J$$<br>\n",
    "$$ w ~ p(w)$$<br>\n",
    "$$ \\theta ~ p(\\theta)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where we might use a Dirichlet prior (see the review of distributions in the supplementary material) for the weight vector ω and conjugate priors for the population-specific parameters in θ. With this model, we could obtain posterior distributions for z (population membership of the observations), ω (population weights), and θ (population-specific parameters in fj). Next, we will look at how to fit a mixture of two normal distributions in JAGS."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "200"
      ],
      "text/latex": [
       "200"
      ],
      "text/markdown": [
       "200"
      ],
      "text/plain": [
       "[1] 200"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dat = read.csv(\"mixture.csv\", header = FALSE)\n",
    "y = dat$V1\n",
    "(n = length(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Plot with title \"Histogram of y\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(y, breaks = 40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PR5wKFvEsQFHIj0eYDQJIgLOBDp8+At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\nr9mmZtrUAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"density.default(x = y)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It appears that we have two populations, but we do not know which population each observation belongs to. We can learn this, along with the mixture weights and population-specific parameters with a Bayesian hierarchical model.\n",
    "\n",
    "We will use a mixture of two normal distributions with variance 1 and different (and unknown) means."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"package 'rjags' was built under R version 3.4.4\"Loading required package: coda\n",
      "Warning message:\n",
      "\"package 'coda' was built under R version 3.4.4\"Linked to JAGS 4.3.0\n",
      "Loaded modules: basemod,bugs\n"
     ]
    }
   ],
   "source": [
    "library(\"rjags\")\n",
    "library(\"coda\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_string = \" model {\n",
    "    for(i in 1:length(y)){\n",
    "        y[i] ~ dnorm(mu[z[i]], prec)\n",
    "        z[i] ~ dcat(omega)\n",
    "    }\n",
    "    \n",
    "    mu[1] ~ dnorm(-1.0, 1.0/1e2)\n",
    "    mu[2] ~ dnorm(1.0, 1.0/1e2) T(mu[1],) # ensures mu[1] < mu[2]\n",
    "    \n",
    "    prec ~ dgamma(1.0/2.0, 1.0*1.0/2.0)\n",
    "    sig = sqrt(1.0/prec)\n",
    "\n",
    "    omega ~ ddirich(c(1.0, 1.0))\n",
    "} \""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_jags = list(y=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Compiling model graph\n",
      "   Resolving undeclared variables\n",
      "   Allocating nodes\n",
      "Graph information:\n",
      "   Observed stochastic nodes: 200\n",
      "   Unobserved stochastic nodes: 204\n",
      "   Total graph size: 614\n",
      "\n",
      "Initializing model\n",
      "\n"
     ]
    }
   ],
   "source": [
    "params = c(\"mu\", \"sig\", \"omega\", \"z[1]\", \"z[31]\", \"z[49]\", \"z[6]\") # Select some z's to monitor\n",
    "\n",
    "mod = jags.model(textConnection(mod_string), \n",
    "                 data=data_jags, \n",
    "                 n.chains=3)\n",
    "update(mod, 1e3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_sim = coda.samples(model=mod,\n",
    "                        variable.names=params,\n",
    "                        n.iter=5e4)\n",
    "\n",
    "mod_csim = as.mcmc(do.call(rbind, mod_sim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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cDf6N8/X+OOC8zPzC7rzlTAo1//nSgGHI5e71+yci42qnI/P0M/hk77ExZat+/n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\nAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Density of omega[2]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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YRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRB\nYABhEBhAGAQGEAaBAYRBYABhEBhAmNkCG9iSHH0AhJkvcI4q4EXY+0cHgaVh7x8dBJZmwb3P\nJF2C1/O5HbCpWkxV13XlvrQP2+/2q2mXVE27qGmfsQ8b+9Uubdq1Gruqe8Kubx+3X+wa9j/X\niF3knnM/ukZsu3Vbgtuao3HrtavYNtva3A913divriUTCqv9xuwrAsbUfvvhv/bn9uWm9mU1\n4cnGbtrV0L5xt9SV4NozVSir8vXY1eyTvu6wXb89W7lb1+0q90L7c2PX9juirvweCT/VcetV\npvgmx3zxoxqL76CNyT0CV0HgKvZg1/EGAkdfrgUOT9Xdi50WTTUWODzXC9y/wgscW2mfGwns\nWwrq9gL3DASufGn25f6AEAXufe/fS9TT+zYQ2LfpCu+a7cobvFe/CS/wqC4TV40C+/2WM77U\nfNWEUKs3mSUEHnjmNKrcEGXc6NVbbAW2S+sgcF2bYaceCGy8wPWg4fpaYFN7d6owjltXO4Gj\newOB6zCex8HSDbZ17V3xm/cCuzYbfxwKAtf+bfgiosDGRH0bN+5WflSv43GhP9Q03dvwS03l\nj06mqeNbCAKbqpsy1AaB01GrN5lFRuCxwMkjcBT5hRF41MpA4OpC4GbhEdj0bQaBzdMRuH4w\nAhtG4NdQqzeZ3AKHPld73xLOgU30o/YjbiewP+G8dw7shvYw3lVV5059LbD7wfjC/NzaRK39\nS4fnwHZV+7fb754D11U3oQjDY+VP3N3TJp692jHVn9Qbt11Td0N4GL9jSVFgE/ZOdw7sfgq7\nZAfnwItvMC9q9SbDCNzBCDwhXzUh1OpNhnNgzoFT8lUTQq3eZLgK3cFV6An5qgmhVm8yi5wD\ncx84Duvl3QdefIN5Uas3mWlv0N4Qv3NXnBG4gBF4er5qQqjVm8ykN2jiijfW5hxY/xw4IV81\nIdTqTWYJgauxwMkjMFehO4HDTwtfhUZgXXILzH1gwfvACKwLI3AHIzAC67HERSzOgcXOgbmI\npQv3gTsOfBV6er5qQqjVmwz3gTtbuA88Jd/FN5gXtXqTSXuDw7Vv/4vpMHIwAndtKo3Az/NV\nE0Kt3mT4LHQcLI98Djw9XzUh1OpNhn+N1HHcq9AJ+aoJoVZvMoucA3MfWOs+8KNfYIfA+4YR\nuOO4I7C5enB3C2pCqNWbzLQPcjz4HX+cA+ufAyOwLhM/yDH5qdDxZo3A8cVchY6rLnwVGoF1\nmfgG76/GfWD9+8AIrAufxOo47gjMRSxduA8cB8sDnwMn5KsmhFq9yXAVuuO4V6ET8lUTQq3e\nZLgP3Ll/4FFYfCIAAB22SURBVPvACfkuvsG8qNWbDCNwByPwhHzVhFCrNxnOgTkHTslXTQi1\nepPhKnTHga9CT89XTQi1epPh3wN3thz3PnBKvotvMC9q9SbDCNzBCDwhXzUh1OpNhnPgOFhy\nDnwRLP+gX4HX3+CD38jBVWiuQu8EtXqT4T5w5z73gafku/gG86JWbzLcB+5gBJ6Qr5oQavUm\nwzkw58Ap+aoJoVZvMlyF7uAq9IR81YRQqzcZ7gN3tnAfeEq+i28wL2r1JsMI3MEIPCFfNSHU\n6k2Gc+A4WHIOPCVfNSHU6k2Gq9AdXIWekK+aEGr1JpPyBm+uy33gYu4DT8o35wZXQK3eZCa9\nwcRfK8sILDYCJ+SrJoRavclMe4M+2SlH6NijBwJzDrz7c+Dp+aoJoVZvMlPfoJtQTmggdLxZ\nI3B8MVeh46rLX4Wemq+aEGr1JjP9Dd7+raPl3geuD3YfeGK++Ta4Cmr1JpP7IlYYORiBuzY1\nRuCp+aoJoVZvMtwHjoPlkc+Bp+erJoRavclwH7jjuFehE/JVE0Kt3mTS3uBw7Tu/sSH0S+4D\n690HnpRv3g0ujlq9yTACdzACT8hXTQi1epPhHJhz4JR81YRQqzcZ/jVSx4GvQvPXCWXh3wN3\nthz3PrC5enA/3ywbXA+1epNZ5LPQjMBaIzAC6zLxs9CTn4o9eiAw58B7PwdGYF0mvsH7q3EV\nWv8qNALrssg5MPeBxe4DcxFLFu4Ddxx3BE7IV00ItXqT4T4w58Ap+aoJoVZvMtwH7jjuVeiE\nfNWEUKs3Ge4Dd7Yc9z5wSr6LbzAvavUmwwjcwQg8IV81IdTqTYZz4DhYcg48JV81IdTqTYar\n0B1chZ6Qr5oQavUmw33gzv0D3wdOyHfxDeZFrd5kGIE7GIEn5KsmhFq9ybz+Bm9/eif26IHA\nnANLngPzGzkk4Cp0B1ehJ+SrJoRavclwH7izhfvAU/JdfIN5Uas3GUbgDkbgCfmqCaFWbzLc\nB46DJefAU/JVE0Kt3mS4Ct3BVegJ+aoJoVZvMtwH7tznPvCUfBffYF7U6k2GEbiDEXhCvmpC\nqNWbDOfAnAOn5KsmhFq9yUx7g9N/5UroeLNG4PhirkLHVZe+Cs2v1JFl0hs0Vw/uNsB9YMH7\nwCn5ZtngeqjVm0xugcPIwQjctSkwAiOwLksIzDmw2DkwAuuyyAjMVWitq9AIrEvui1jcB1a8\nD8xFLFm4D9xx3BE4IV81IdTqTYb7wJwDp+SrJoSRI/UNvrz27Q1WsDAz4suQ7zad+kjMiGx2\n5rA27P2jg8DSsPePDgJLs/xVaNg38wWGLcnRBx7eBybulXk1u2mpJbaeuQ6xZnZVzKQtTBB4\n0ULyo1ZvcsETP8iRXkgi++rsCPx0VRHU6l1I4OX3w746OwI/XVUEtXqXEnhx9tXZDyZwwkWs\nvfSXqajVi8A7amZXxWRiV8VMQK1eBN5RM7sqJhO7KmYCavUi8I6a2VUxmdhVMRNQqxeBd9TM\nrorJxK6KmYBavbICA8ALIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMJsJ7DYc\n/gnbzW+TGnnw+vWb8W8qRzEmRzH52XjzL6BW7kK/UmcBfH/0Fdz8NqmRB69fvxn/pma3Yp7W\ntBUbb/4FxI43r+zgjd6h74xZemkO8zI1g8D7wkhVG9AQuClRYLd2lukAAmdCq1oHAr/QjMk0\nduYQ2OQpZgEQeA2OJnCuzp7lVDrXCJxpVpEZBF6B1IrVBd7RQJ5H4EzFLAECL09yweICm8GX\njZ0Jv9dxH8UsAgIvTnq92gKb/usuZq1MoXeGVrWvlLupwHM/rWAyfehhdx/kyFJMfvggx7K8\n8ueLtN4hAIxAYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRB\nYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhjiHww3d5jF0AZXKM3mvuvdFjvH0o\nl2P0YASGQjlGDzZN95cNmvA3SuKfdOj+nEJYrel+Atg/x+ipUdPwsP/x4psZrQqwe47RUS8E\nvlp6/Q1AgmN01n6iHEffpn949Y0pNMhwjJ46nhdfDMbXAjdH2S8gzzE66vUU+uGpcHOU/QLy\nHKOjBl27i83u4UhiMxCcKTTIQE8FEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaB\nAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhA\nGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRB\nYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhShLYdKS+8uvdmD9XbTWf7eK3f37B\nycSlsDo+1ffP5+s1zfef+88XmLNKnVN4XeBT+5rzVVv/+cb+2p/PRizYooi5fjxdz/93jwJz\nVqlzKq/t+OtX2SV/zH/tAd0d1d9jojLBFoXf661o31PXnfqcfM4qdU6l2/+fp7em+dcecs8/\nTZg7uQdv5u0rrvxz9k/3o3Y3gncB/mfeXHNqwRZF2Ov/3PAZI2x1PodB+eNkTh9+PR/gya1/\nCmGVnLNKnVMZBPSnDdx/t4dXP3nyD8Jx/Pfksv59EOyf9vk2+ffvRi3Yogh7/deK2UUYsjq7\nlPz8Ogr8Yezp8meYcheds0qdU+n2/9/mtz0CfzU/dsm5jfLbP/huH4TToA/74ByC73HphyVt\n9KfvccOrvRXoiHt9FGHr7q89RttHv81X1NFNhe1w2ryFI3XROavUOZVu//+6719/30cTpdNg\ngtX+8BuO6sO4PsxHM1jyYd7HDS/8BuAGQ4G7CH3GdlE7p/77GZ936745y31yZeesUudURvv/\n63R5pjO6Su2/X1y59LkOlowTlQm2KIZT6IvZr/3682aV/hxkac+Wz+Zz+OpCc1apcyqj/f9m\n/n7/Xgg8WPfWkflvOG9yVydPv5ct6gRbFGGvf/qZ82ih//rz7xxzdAva1L/jRKvsnFXqnMpo\n/xvzY+89+BOgL3te1D2w3Dg3itc9wtnWOax02TCsi9/rnyd7UttFOBxYw7WO/mB9Htw1Ljpn\nlTqnMtr/7uqkPf76K5f/4iXMcB/pJ16dvDHFdjMz//zPdcOwLsMPcnQR9gKfB1eh7Tlyq+KX\nsUdvT9E5q9Q5lfH+b+dVHz822a83c3KflbMP/osrx/uDd4Ltn79qGFbFpxI+ShkjHEyhB/eB\n7ThtPy35Fi9hNWXnrFLn2twLUCbYg/N7Mk8/OW2Rz1mlzrXpr1U/Xwq7o79X+HxF7ZxV6lwb\n+WCPzcm8/zxfqykgZ5U6AeAGCAwgDAIDCIPAAMIgMIAwCAwgDAIDCIPAAMIgMIAwCAwgDAID\nCIPAAMIgMIAwCAwgDAIDCIPAAMIgMIAwCAwgDAIDCIPAAMIgMIAwCAwgzGyBDWxJjj4AwswX\nOEcV8CLs/aODwNKw948OAkvD3j86r/eA22dhFSxMpvgOwpbXJ14j9Q3O3kPjH7fu3uWTN77S\nkdtBCFw6eeMrHbkdhMClkze+0pHbQQhcOnnjKx25HYTApZM3vtKR20EIXDp54ysduR2EwKWT\nN77SkdtBCFw6eeMrHbkdtIzA9v7ynZvMCLwys+I+HHI7aBGBTVzxxtoIvDKz4j4ccjsIgUtn\nVtwJvPjRvp0hVz4Cl86suNORM2CMXPkIXDqz4k5HzoAxcuVzEat0ZsWdjpwBY+TK5zZS6eSN\nLzVfNeTKR+DSyRtfar5qyJW/rMDDtW9fpdy6e5fPrLjTkTNgjFz5jMClkze+p43JGTBGrnwE\nLp288flWHvw+FzkDxsiVj8Clkze+0My9m4SCBoyRKx+BSydvfF1DBoF3wTIf5Jg+xdq6e5fP\nrLgfh3xzcb4tbIFc+Qt9kGPyU1t37/KZFXd6yhcLF/6dqvnJuYPWYKEp9P3VEHhlZsWdzqXA\ni28wL2r1cg5cPHnjS81XTQi1ehG4ePLGdz/M23NQNSHU6kXg4skbX2q+akKo1YvAxZM3vtR8\n1YRQqxeBiydvfKGZ+5dsEXhdELh08sY3buX5fX41IdTqReDiyRvfuBUE3hwELp288Y1bQeDN\nQeDSyRvfuBUE3hwELp288YVmuIi1F9YT+Hbo9Qu81I9HH3jN7Mj1BnZEpvgmx/zwx92jVu/m\nI/B1h1tCriteOW4scFRZo9S88SXnu/gG86JW7+YCL9bvIZA3vtR81YRQqxeBiydvfKn5qgmh\nVi8CF0/e+FLzVRNCrV4ELp688aXmqyaEWr0IXDx540vNV00ItXoRuHjyxvcg2Fu3reSEUKsX\ngYsnb3yp+aoJoVbvogLfXBeBV2ZW3Okg8LosIvCdjwHdamDr7l0+s+JOB4HXZZkRePpv7t+6\ne5fPrLjTQeB1WWoKPfU392/dvctnVtzpIPC6LHcOfHMCjcCrMyvudBB4XbiIVTqz4r6XrLl7\nkoTA68JtpNLJG1/firndHAKvCwKXTt74Bq3cvsqBwOuyrMDDtW/fWdq6e5fPrLgfxmoQeHsY\ngUsnb3yjVibc51cTQq1eBC6evPGNm0HgzUHg0skbX2q+akKo1YvAxZM3vtR81YRQq5fPQhfP\nrLifBX0j9Idx7x61epf6LPTkp7bu3uUzK+50EHhdFppC318NgVdmVtzpIPC6cA5cOnnjC83c\nPUNC4JVB4NLJG9+4FW4jbQ4Cl07e+MatIPDmIHDp5I1v3AoCbw4Cl07e+MatIPDmIHDp5I0v\nNMNFrL2AwKWTN77UfNWEUKsXgYsnb3yp+aoJoVYvAhdP3vhS81UTQq1eBC6evPGl5qsmhFq9\nKwrMb+TYhkzxTY754Y+7R61eRuDiyRtfar5qQqjVi8DFkze+B8Hyzwm3AIFLJ298qfmqCaFW\nLwIXT974UvNVE0KtXgQunrzxpearJoRavQhcPHnjS81XTQi1ehG4ePLGl5qvmhBq9SJw8eSN\nLzVfNSHU6l1I4On/WmXr7l0+s+Kena+aEGr1LiNwwr8X3bp7l8+suGfnqyaEWr0IXDyz4p6d\nr5oQavUicPHMint2vmpCqNWLwMUzK+7Z+aoJoVYvF7GKZ1bcs/NVE0KtXm4jFU/e+FLzVRNC\nrV4ELp688aXmqyaEWr0LCzxcm3/Qvw2z4p6dr4GFmRHZ7Mxhbdj7RweBpWHvHx0Elmb5q9Cw\nb+YLDFuSow88vA9M3CvzanbTUktsPXMdYs3sqphJW5gg8KKF5Eet3qU+yJFeSCL76uwI/HRV\nEdTqXeo20uL7YV+dHYGfriqCWr3rf5AjE/vq7AcTOOEi1l76y1TU6kXgHTWzq2IysatiJqBW\nLwLvqJldFZOJXRUzAbV6EXhHzeyqmEzsqpgJqNWLwDtqZlfFZGJXxUxArV5ZgQHgBRAYQBgE\nBhAGgQGEQWAAYRAYQBgEBhAGgQGEQWAAYRAYQJjNBHYbDv+E7ea3SY08eP36zfg3laMYk6OY\n/Gy8+RdQK3ehX6mzAL4/+gpufpvUyIPXr9+Mf1OzWzFPa9qKjTf/AmLHm1d28Ebv0HfGLL00\nh3mZmkHgfWGkqg1oCNyUKLBbO8t0AIEzoVWtA4FfaMZkGjtzCGzyFLMACLwGRxM4V2fPciqd\nawTONKvIDAKvQGrF6gLvaCDPI3CmYpYAgZcnuWBxgc3gy8bOhN/ruI9iFgGBFye9Xm2BTf91\nF7NWptA7Q6vaV8rdVOC5n1YwmT70sLsPcmQpJj98kGNZXvnzRVrvEABGIDCAMAgMIAwCAwiD\nwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgM\nIAwCAwiDwADCIDCAMMcQ+OG7PMYugDI5Ru81997oMd4+lMsxejACQ6EcowebpvvLBk34GyXx\nTzp0f04hrNZ0PwHsn2P01KhpeNj/ePHNjFYF2D3H6KgXAl8tvf4GIMExOms/UY6jb9M/vPrG\nFBpkOEZPHc+LLwbja4Gbo+wXkOcYHfV6Cv3wVLg5yn4BeY7RUYOu3cVm93AksRkIzhQaZKCn\nAgiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCA\nMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiD\nwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgMIAwCAwiDwADCIDCAMAgM\nIAwCAwiDwADCIDCAMCUJbDpSX/n1bsyfq7bs99+T/fbdPv8xWArr4lN9/3y+XpvVn/vPF5iz\nSp1TeF3gU/ua81Vb9vsf1yd8/5EKtihirh9P1/P/3aPAnFXqnMprO/76VWHJp0vybP7aL/+9\n3D7Mw+/1/4z5nrru1Ofkc1apcyphxxvzeXprmn/tIff804S5k3vwZt6+4so/Z/90P2p3I3j4\n+XTqf3CTL5lgiyLs9X9u+IwRtjqfw6D8cTInP/UNCZ7c+qcQVsk5q9Q5lUFAf9rA/fcwNzrH\nB+E4/ntyWf/eD/ZsvgbBmkYo2KIIe/3XitlFGLI6u/mvn19HgT+MPV3+DFPuonNWqXMqXUB/\nm9/2CPzV/Pi50YcN3j74bh+E06AP++Acgu9x6bslX+0K9sEfu86HVrBFEff6KMLW3V97jLaP\nfpvOQH8y206/mrdwpC46Z5U6p9IJ/Ou+f/19H13YOA0mWO0Pv+GoPozrI1yHtCucYn/oro3J\nBFsUQ4G7CH3GdlE7p/77GZ936745y9/9q4rOWaXOqXQC269fp/GZzmDq1K90ceXS5xqO9V/h\nqe8/5v3TxqwTbFEMp9AXs1/79efNKv05yNKeLZ/N5/DVheasUudURgK/mb/fvxcCD9a9dWT+\nG86b/KuGuv9qXdwoiu5a8fkqS//159855ugWtKl/x4lW2Tmr1DmVkcDG/Nh7D/4E6MueF3UP\nLDfOjeJ1j3Gwf9pD9O8fd0CXCbYo/F7/PNmT2i7C4cAarnUML0v1d42LzlmlzqmMBHZXJ+3x\n15/c/ItnOeE+0k+8Onljij06zv91C98H7cOqRMWsdV2EvcDnwVVoe458ttel7NHbU3TOKnVO\nZSRwm+zp48cm+/VmTv/sEvvgv7hyvD/4LNjm78m8/Rs2DKviUwkfpYwRDqbQg/vAdpy2c+C3\neAmrKTtnlTrX5l6AMsEenN+TefrJaYt8zip1rk1/UeP5Utgd/b3C5ytq56xS59rIB3tsTub9\n5/laTQE5q9QJADdAYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaBAYRBYABhEBhAGAQGEAaB\nAYRBYABhEBhAGAQGEAaBAYRBYABhEBjcr5BR+jUy0ENm4DqBiQ9ACiIDBBaGyACBhZkdmYEt\nydEHEFiY+QLnqAJeJNPeN1zEUgWBpWHvHx0Eloa9f3Re7wG3z8IqWJhM8d3O9Gm+amxwWWIm\nqW9w9h4a/7h19y6fvPGl5quGXPkIXDp540vNVw258hG4dPLGF5q5P32TM2CMXPkIXDp54xu3\ncusvbWbZwmbIlY/ApZM3vnErCLw5CFw6eeMbt4LAm4PApZM3vnErCLw5CFw6eeMLzXARay8s\nI7C5/1lZBF6ZWXGnI2fAGLnyFxH40b9WQeCVmRV3OnIGjJErH4FLZ1bc6cgZMEaufAQunVlx\npyNnwBi58hG4dGbFnY6cAWPkyuciVunMijsdOQPGyJXPbaTSyRtfar5qyJWPwKWTN77UfNWQ\nK39ZgZ//g++tu3f5zIo7HTkDxsiVzwhcOnnjS81XDbnyEbh08saXmq8acuUjcOnkjS81XzXk\nykfg0skbX2q+asiVv8wHOR78yjwEXplZcSfw4m9J3Bly5S/0QY7JT23dvctnVtzpyBkwRq78\nhabQ91dD4JWZFXc6cgaMkSufc+DSyRtfar5qyJWPwKWTN77UfNWQKx+BSydvfKn5qiFXPgKX\nTt74UvNVQ658BC6dvPGl5quGXPkIXDp540vNVw258hG4dPLGl5qvGnLlI3Dp5I0vNV815MpH\n4NLJG19qvmrIlY/ApZM3vtR81ZArfz2B+Y0c25ApvskxL76FRZErnxG4dPLG97QxOQPGyJWP\nwKWTNz7fyvR/LqqGXPkIXDp54wvNmLuNyRkwRq58BC6dvPF1DRkE3gUIXDp54xs0dfuXb8gZ\nMEaufAQunbzxPW1MzoAxcuUjcOnkjS81XzXkykfg0skbX2q+asiVv6TAU6ZYW3fv8pkVd0LC\n/FbKTVhEYH6t7I6YFXc6cgaMkSt/mRF4+n3Crbt3+cyKOx05A8bIlb/UFHrqfcKtu3f5zIr7\nbrp3p8tyBoyRK3+5c+Bp9wm37t7lMyvuO9E+aE7OgDFy5XMRq3Rmxf0kWATeHG4jlU7e+Mat\nIPDmIHDp5I1v3AoCb86yAj+/T7h19y6fWXHfy5WLWHuBEbh08saXmq8acuUjcOnkjS81XzXk\nykfg0skbX2q+asiVj8Clkze+1HzVkCufz0KXzqy405EzYIxc+Qt9FnryU1t37/KZFXc6cgaM\nkSt/oSn0/dUQeGVmxZ2OnAFj5MrnHLh08saXmq8acuUjcOnkjS81XzXkykfg0skbX2q+asiV\nj8Clkze+1HzVkCsfgUsnb3yp+aohVz4Cl07e+FLzVUOufAQunbzxPQj25md3jBrL7aBlQODS\nyRtfar5qQqjVi8DFkze+1HzVhFCrF4GLJ298qfmqCaFW74oC3z7J2Lp7l0+m+CbH/PDH3aNW\nLyNw8eSNLzVfNSHU6kXg4skbX2q+akKo1YvAxZM3vtR81YRQqxeBiydvfKn5qgmhVi8CF0/e\n+FLzVRNCrV4ELp688aXmqyaEWr0IXDx540vNV00ItXoRuHjyxpear5oQavUuJPCDT4Yj8MrM\nint2vmpCqNW7jMDm6sHdBrbu3uUzK+7Z+aoJoVYvAhfPrLhn56smhFq9CFw8s+Kena+aEGr1\nInDxzIp7dr5qQqjVy0Ws4pkV9+x81YRQq5fbSMWTN77UfNWEUKsXgYsnb3yp+aoJoVbvwgIP\n1+Yf9G/DrLhn57vU756DyIzIZmcOa8PePzoILA17/+ggsDTLX4WGfTNfYNiSHH3g4X1g4l6Z\nV7Obllpi65nrEGtmV8VM2sIEgRctJD9q9S71QY70QhLZV2dH4KeriqBW71K3kRbfD/vq7Aj8\ndFUR1Opd/4McmdhXZz+YwAkXsfbSX6aiVi8C76iZXRWTiV0VMwG1ehF4R83sqphM7KqYCajV\ni8A7amZXxWRiV8VMQK1eBN5RM7sqJhO7KmYCavXKCgwAL4DAAMIgMIAwCAwgDAIDCIPAAMIg\nMIAwCAwgDAIDCIPAAMJsJrDbcPgnbDe/TWrkwevXb8a/qRzFmBzF5Gfjzb+AWrkL/UqdBfD9\n0Vdw89ukRh68fv1m/Jua3Yp5WtNWbLz5FxA73ryygzd6h74zZumlOczL1AwC7wsjVW1AQ+Cm\nRIHd2lmmAwicCa1qHQj8QjMm09iZQ2CTp5gFQOA1OJrAuTp7llPpXCNwpllFZhB4BVIrVhd4\nRwN5HoEzFbMECLw8yQWLC2wGXzZ2Jvxex30UswgIvDjp9WoLbPqvu5i1MoXeGVrVvlLupgLP\n/bSCyfShh919kCNLMfnhgxzL8sqfL9J6hwAwAoEBhEFgAGEQGEAYBAYQBoEBhEFgAGEQGEAY\nBAYQBoEBhEFgAGEQGEAYBAYQBoEBhEFgAGEQGEAYBAYQBoEBhEFgAGEQGEAYBAYQBoEBhDmG\nwA/f5TF2AZTJMXqvufdGj/H2oVyO0YMRGArlGD3YNN1fNmjC3yiJf9Kh+3MKYbWm+wlg/xyj\np0ZNw8P+x4tvZrQqwO45Rke9EPhq6fU3AAmO0Vn7iXIcfZv+4dU3ptAgwzF66nhefDEYXwvc\nHGW/gDzH6KjXU+iHp8LNUfYLyHOMjhp07S42u4cjic1AcKbQIAM9FUAYBAYQBoEBhEFgAGEQ\nGEAYBAYQBoEBhEFgAGEQGEAYBAYQBoEBhEFgAGEQGEAYBAYQBoEBhEFgAGEQGEAYBAYQBoEB\nhEFgAGH+D2KT4rD3QOMjAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Density of z[49]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QKGMAFDmIAhTMAQJmAIEzCECRjCBAxh\nAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAh\nTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMw\nhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEC\nhjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFM\nwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCE\nCRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKG\nMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzA\nECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJ\nGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYw\nAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQ\nJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkY\nwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjAB\nQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAm\nYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjC\nBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFD\nmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZg\nCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIE\nDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOY\ngCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAI\nEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQM\nYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iA\nIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgT\nMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAhTMAQJmAIEzCECRjCBAxh\nAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGEChjABQ5iAIUzAECZgCBMwhAkYwgQMYQKGMAFDmIAh\nTMAQJmAIEzCECRjCBAxhAoYwAUOYgCFMwBAmYAgTMIQJGMIEDGH/D4/J6NXPt6ePAAAAAElF\nTkSuQmCC",
      "text/plain": [
       "Plot with title \"Density of z[6]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## convergence diagnostics\n",
    "options(repr.plot.height=8, repr.plot.width=8)\n",
    "\n",
    "plot(mod_sim, ask=TRUE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>mu[1]</th><th scope=col>mu[2]</th><th scope=col>omega[1]</th><th scope=col>omega[2]</th><th scope=col>sig</th><th scope=col>z[1]</th><th scope=col>z[31]</th><th scope=col>z[49]</th><th scope=col>z[6]</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>Lag 0</th><td>1.000000000  </td><td> 1.0000000000</td><td> 1.000000000 </td><td> 1.000000000 </td><td>1.000000000  </td><td> 1.000000000 </td><td>1.0000000000 </td><td> 1.0000000000</td><td>NaN          </td></tr>\n",
       "\t<tr><th scope=row>Lag 1</th><td>0.536099339  </td><td> 0.3384676314</td><td> 0.295383596 </td><td> 0.295383596 </td><td>0.403606508  </td><td> 0.005039109 </td><td>0.0422154937 </td><td> 0.0286155098</td><td>NaN          </td></tr>\n",
       "\t<tr><th scope=row>Lag 5</th><td>0.087021215  </td><td> 0.0459193965</td><td> 0.043485679 </td><td> 0.043485679 </td><td>0.025646252  </td><td>-0.001876947 </td><td>0.0057659168 </td><td> 0.0046831708</td><td>NaN          </td></tr>\n",
       "\t<tr><th scope=row>Lag 10</th><td>0.004709923  </td><td> 0.0008958866</td><td>-0.001487281 </td><td>-0.001487281 </td><td>0.003159687  </td><td>-0.001110068 </td><td>0.0027979663 </td><td> 0.0006938540</td><td>NaN          </td></tr>\n",
       "\t<tr><th scope=row>Lag 50</th><td>0.007356007  </td><td>-0.0009672351</td><td> 0.005605020 </td><td> 0.005605020 </td><td>0.003668588  </td><td>-0.004174102 </td><td>0.0006498841 </td><td>-0.0002232669</td><td>NaN          </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllllll}\n",
       "  & mu{[}1{]} & mu{[}2{]} & omega{[}1{]} & omega{[}2{]} & sig & z{[}1{]} & z{[}31{]} & z{[}49{]} & z{[}6{]}\\\\\n",
       "\\hline\n",
       "\tLag 0 & 1.000000000   &  1.0000000000 &  1.000000000  &  1.000000000  & 1.000000000   &  1.000000000  & 1.0000000000  &  1.0000000000 & NaN          \\\\\n",
       "\tLag 1 & 0.536099339   &  0.3384676314 &  0.295383596  &  0.295383596  & 0.403606508   &  0.005039109  & 0.0422154937  &  0.0286155098 & NaN          \\\\\n",
       "\tLag 5 & 0.087021215   &  0.0459193965 &  0.043485679  &  0.043485679  & 0.025646252   & -0.001876947  & 0.0057659168  &  0.0046831708 & NaN          \\\\\n",
       "\tLag 10 & 0.004709923   &  0.0008958866 & -0.001487281  & -0.001487281  & 0.003159687   & -0.001110068  & 0.0027979663  &  0.0006938540 & NaN          \\\\\n",
       "\tLag 50 & 0.007356007   & -0.0009672351 &  0.005605020  &  0.005605020  & 0.003668588   & -0.004174102  & 0.0006498841  & -0.0002232669 & NaN          \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | mu[1] | mu[2] | omega[1] | omega[2] | sig | z[1] | z[31] | z[49] | z[6] |\n",
       "|---|---|---|---|---|---|---|---|---|---|\n",
       "| Lag 0 | 1.000000000   |  1.0000000000 |  1.000000000  |  1.000000000  | 1.000000000   |  1.000000000  | 1.0000000000  |  1.0000000000 | NaN           |\n",
       "| Lag 1 | 0.536099339   |  0.3384676314 |  0.295383596  |  0.295383596  | 0.403606508   |  0.005039109  | 0.0422154937  |  0.0286155098 | NaN           |\n",
       "| Lag 5 | 0.087021215   |  0.0459193965 |  0.043485679  |  0.043485679  | 0.025646252   | -0.001876947  | 0.0057659168  |  0.0046831708 | NaN           |\n",
       "| Lag 10 | 0.004709923   |  0.0008958866 | -0.001487281  | -0.001487281  | 0.003159687   | -0.001110068  | 0.0027979663  |  0.0006938540 | NaN           |\n",
       "| Lag 50 | 0.007356007   | -0.0009672351 |  0.005605020  |  0.005605020  | 0.003668588   | -0.004174102  | 0.0006498841  | -0.0002232669 | NaN           |\n",
       "\n"
      ],
      "text/plain": [
       "       mu[1]       mu[2]         omega[1]     omega[2]     sig        \n",
       "Lag 0  1.000000000  1.0000000000  1.000000000  1.000000000 1.000000000\n",
       "Lag 1  0.536099339  0.3384676314  0.295383596  0.295383596 0.403606508\n",
       "Lag 5  0.087021215  0.0459193965  0.043485679  0.043485679 0.025646252\n",
       "Lag 10 0.004709923  0.0008958866 -0.001487281 -0.001487281 0.003159687\n",
       "Lag 50 0.007356007 -0.0009672351  0.005605020  0.005605020 0.003668588\n",
       "       z[1]         z[31]        z[49]         z[6]\n",
       "Lag 0   1.000000000 1.0000000000  1.0000000000 NaN \n",
       "Lag 1   0.005039109 0.0422154937  0.0286155098 NaN \n",
       "Lag 5  -0.001876947 0.0057659168  0.0046831708 NaN \n",
       "Lag 10 -0.001110068 0.0027979663  0.0006938540 NaN \n",
       "Lag 50 -0.004174102 0.0006498841 -0.0002232669 NaN "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>mu[1]</dt>\n",
       "\t\t<dd>38739.5225751533</dd>\n",
       "\t<dt>mu[2]</dt>\n",
       "\t\t<dd>55459.7300056614</dd>\n",
       "\t<dt>omega[1]</dt>\n",
       "\t\t<dd>60308.3345714942</dd>\n",
       "\t<dt>omega[2]</dt>\n",
       "\t\t<dd>60308.3345714942</dd>\n",
       "\t<dt>sig</dt>\n",
       "\t\t<dd>58498.968046033</dd>\n",
       "\t<dt>z[1]</dt>\n",
       "\t\t<dd>143540.955024925</dd>\n",
       "\t<dt>z[31]</dt>\n",
       "\t\t<dd>128250.05341117</dd>\n",
       "\t<dt>z[49]</dt>\n",
       "\t\t<dd>131235.782668954</dd>\n",
       "\t<dt>z[6]</dt>\n",
       "\t\t<dd>100000.000000018</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[mu\\{{[}\\}1\\{{]}\\}] 38739.5225751533\n",
       "\\item[mu\\{{[}\\}2\\{{]}\\}] 55459.7300056614\n",
       "\\item[omega\\{{[}\\}1\\{{]}\\}] 60308.3345714942\n",
       "\\item[omega\\{{[}\\}2\\{{]}\\}] 60308.3345714942\n",
       "\\item[sig] 58498.968046033\n",
       "\\item[z\\{{[}\\}1\\{{]}\\}] 143540.955024925\n",
       "\\item[z\\{{[}\\}31\\{{]}\\}] 128250.05341117\n",
       "\\item[z\\{{[}\\}49\\{{]}\\}] 131235.782668954\n",
       "\\item[z\\{{[}\\}6\\{{]}\\}] 100000.000000018\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "mu[1]\n",
       ":   38739.5225751533mu[2]\n",
       ":   55459.7300056614omega[1]\n",
       ":   60308.3345714942omega[2]\n",
       ":   60308.3345714942sig\n",
       ":   58498.968046033z[1]\n",
       ":   143540.955024925z[31]\n",
       ":   128250.05341117z[49]\n",
       ":   131235.782668954z[6]\n",
       ":   100000.000000018\n",
       "\n"
      ],
      "text/plain": [
       "    mu[1]     mu[2]  omega[1]  omega[2]       sig      z[1]     z[31]     z[49] \n",
       " 38739.52  55459.73  60308.33  60308.33  58498.97 143540.96 128250.05 131235.78 \n",
       "     z[6] \n",
       "100000.00 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "autocorr.diag(mod_sim)\n",
    "effectiveSize(mod_sim)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Iterations = 2001:52000\n",
       "Thinning interval = 1 \n",
       "Number of chains = 3 \n",
       "Sample size per chain = 50000 \n",
       "\n",
       "1. Empirical mean and standard deviation for each variable,\n",
       "   plus standard error of the mean:\n",
       "\n",
       "            Mean       SD  Naive SE Time-series SE\n",
       "mu[1]    -2.1253 0.167052 4.313e-04      8.487e-04\n",
       "mu[2]     1.4896 0.125886 3.250e-04      5.346e-04\n",
       "omega[1]  0.3872 0.040697 1.051e-04      1.657e-04\n",
       "omega[2]  0.6128 0.040697 1.051e-04      1.657e-04\n",
       "sig       1.1353 0.073732 1.904e-04      3.049e-04\n",
       "z[1]      1.0087 0.093079 2.403e-04      2.457e-04\n",
       "z[31]     1.5748 0.494378 1.276e-03      1.381e-03\n",
       "z[49]     1.8044 0.396699 1.024e-03      1.095e-03\n",
       "z[6]      2.0000 0.004472 1.155e-05      1.155e-05\n",
       "\n",
       "2. Quantiles for each variable:\n",
       "\n",
       "            2.5%     25%     50%     75%   97.5%\n",
       "mu[1]    -2.4518 -2.2371 -2.1259 -2.0136 -1.7952\n",
       "mu[2]     1.2373  1.4072  1.4911  1.5741  1.7331\n",
       "omega[1]  0.3086  0.3594  0.3867  0.4145  0.4682\n",
       "omega[2]  0.5318  0.5855  0.6133  0.6406  0.6914\n",
       "sig       1.0058  1.0838  1.1298  1.1808  1.2952\n",
       "z[1]      1.0000  1.0000  1.0000  1.0000  1.0000\n",
       "z[31]     1.0000  1.0000  2.0000  2.0000  2.0000\n",
       "z[49]     1.0000  2.0000  2.0000  2.0000  2.0000\n",
       "z[6]      2.0000  2.0000  2.0000  2.0000  2.0000\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(mod_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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RVkfe/ml1o73J7BD+HiNGz2V7n5/u5O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\nzP8BvAONSKrh58wAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Density of sig\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "par(mfrow=c(3,2))\n",
    "densplot(mod_csim[,c(\"mu[1]\", \"mu[2]\", \"omega[1]\", \"omega[2]\", \"sig\")])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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u/tc+7324ur4zKrnwPbnD1W9rfsX8C1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\n",
      "text/plain": [
       "Plot with title \"Density of z[6]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## for the z's\n",
    "options(repr.plot.height = 5, repr.ploy.width=8)\n",
    "par(mfrow=c(2,2))\n",
    "\n",
    "densplot(mod_csim[,c(\"z[1]\", \"z[31]\", \"z[49]\", \"z[6]\")])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "      1       2 \n",
       "0.99126 0.00874 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(mod_csim[,\"z[1]\"]) / nrow(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "      1       2 \n",
       "0.42522 0.57478 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(mod_csim[,\"z[31]\"]) / nrow(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "        1         2 \n",
       "0.1956467 0.8043533 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(mod_csim[,\"z[49]\"]) / nrow(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "      1       2 \n",
       "0.00002 0.99998 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(mod_csim[,\"z[6]\"]) / nrow(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-2.26617489046013</li>\n",
       "\t<li>-0.370266623399382</li>\n",
       "\t<li>0.0365563954229409</li>\n",
       "\t<li>3.75480802092135</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -2.26617489046013\n",
       "\\item -0.370266623399382\n",
       "\\item 0.0365563954229409\n",
       "\\item 3.75480802092135\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -2.26617489046013\n",
       "2. -0.370266623399382\n",
       "3. 0.0365563954229409\n",
       "4. 3.75480802092135\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -2.2661749 -0.3702666  0.0365564  3.7548080"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y[c(1, 31, 49, 6)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we look back to the y values associated with these z variables we monitored, we see that y1 is clearly in Population 1’s territory, y31 is ambiguous, y49 is ambiguous but is closer to Population 2’s territory, and y6 is clearly in Population 2’s territory. The posterior distributions for the z variables closely reflect our assessment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ignore this part."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Poisson Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "dat = read.csv(\"callers.csv\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>calls</th><th scope=col>days_active</th><th scope=col>isgroup2</th><th scope=col>age</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>2 </td><td>32</td><td>0 </td><td>27</td></tr>\n",
       "\t<tr><td>4 </td><td>81</td><td>0 </td><td>32</td></tr>\n",
       "\t<tr><td>0 </td><td>41</td><td>0 </td><td>22</td></tr>\n",
       "\t<tr><td>1 </td><td>36</td><td>0 </td><td>28</td></tr>\n",
       "\t<tr><td>0 </td><td>55</td><td>0 </td><td>31</td></tr>\n",
       "\t<tr><td>0 </td><td>25</td><td>0 </td><td>33</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " calls & days\\_active & isgroup2 & age\\\\\n",
       "\\hline\n",
       "\t 2  & 32 & 0  & 27\\\\\n",
       "\t 4  & 81 & 0  & 32\\\\\n",
       "\t 0  & 41 & 0  & 22\\\\\n",
       "\t 1  & 36 & 0  & 28\\\\\n",
       "\t 0  & 55 & 0  & 31\\\\\n",
       "\t 0  & 25 & 0  & 33\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| calls | days_active | isgroup2 | age |\n",
       "|---|---|---|---|\n",
       "| 2  | 32 | 0  | 27 |\n",
       "| 4  | 81 | 0  | 32 |\n",
       "| 0  | 41 | 0  | 22 |\n",
       "| 1  | 36 | 0  | 28 |\n",
       "| 0  | 55 | 0  | 31 |\n",
       "| 0  | 25 | 0  | 33 |\n",
       "\n"
      ],
      "text/plain": [
       "  calls days_active isgroup2 age\n",
       "1 2     32          0        27 \n",
       "2 4     81          0        32 \n",
       "3 0     41          0        22 \n",
       "4 1     36          0        28 \n",
       "5 0     55          0        31 \n",
       "6 0     25          0        33 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(dat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_str = \" model {\n",
    "    # Likelihood\n",
    "    for (i in 1:length(calls)) {\n",
    "        calls[i] ~ dpois( days_active[i] * lam[i] )\n",
    "        log(lam[i]) = b[1] + b[2]*age[i] + b[3]*isgroup2[i]\n",
    "    }\n",
    "\n",
    "    # Priors\n",
    "    for(j in 1:3){\n",
    "        b[j] ~ dnorm(0, 1/1e4)\n",
    "    }\n",
    "} \""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Compiling model graph\n",
      "   Resolving undeclared variables\n",
      "   Allocating nodes\n",
      "Graph information:\n",
      "   Observed stochastic nodes: 224\n",
      "   Unobserved stochastic nodes: 3\n",
      "   Total graph size: 1218\n",
      "\n",
      "Initializing model\n",
      "\n"
     ]
    }
   ],
   "source": [
    "mod = jags.model(textConnection(mod_str), data = as.list(dat), n.chains = 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "update(mod, 5e3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "params = c(\"b\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_sim = coda.samples(mod, variable.names = params, n.iter = 1e4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_csim = as.mcmc(do.call(rbind, mod_sim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>b[1]</dt>\n",
       "\t\t<dd>-2.63400033259069</dd>\n",
       "\t<dt>b[2]</dt>\n",
       "\t\t<dd>-0.0629560179414695</dd>\n",
       "\t<dt>b[3]</dt>\n",
       "\t\t<dd>1.54700626379753</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[b\\{{[}\\}1\\{{]}\\}] -2.63400033259069\n",
       "\\item[b\\{{[}\\}2\\{{]}\\}] -0.0629560179414695\n",
       "\\item[b\\{{[}\\}3\\{{]}\\}] 1.54700626379753\n",
       "\\end{description*}\n"
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      "text/markdown": [
       "b[1]\n",
       ":   -2.63400033259069b[2]\n",
       ":   -0.0629560179414695b[3]\n",
       ":   1.54700626379753\n",
       "\n"
      ],
      "text/plain": [
       "       b[1]        b[2]        b[3] \n",
       "-2.63400033 -0.06295602  1.54700626 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "colMeans(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate the probability that a person of age 29 who is in group2, calls atleast 3 times a day.\n",
    "\n",
    "lamb = exp(mod_csim[,1] + mod_csim[,2]*29 + mod_csim[,3]*1) * 30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.229633333333333"
      ],
      "text/latex": [
       "0.229633333333333"
      ],
      "text/markdown": [
       "0.229633333333333"
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      "text/plain": [
       "[1] 0.2296333"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum(rpois(30000, lamb) >= 3.0)/30000"
   ]
  }
 ],
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