{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Metropolis Sampling\n",
    "\n",
    "#### Marcel Lüthi, Departement of Mathematics and Computer Science, University of Basel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook we will present an implementation of the Metropolis algorithm and experiment with it. As a target distribution we will use a bivariate normal distribution. This simple setup is ideally suited for exploring the properties of the algorithm: We know how the true target distribution looks like, can change its shape, and we can visualize the samples. \n",
    "\n",
    "The intuition that we develop while experimenting with this simple examples, will help us to understand the more complicated cases that we will explore later in this course. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Preparation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we start, we need to download the plotting library [EvilPlot](https://cibotech.github.io/evilplot/) and make it available in the Jupyter-Notebook. It may take some time when you execute the following cells for the first time. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[32mimport \u001b[39m\u001b[36m$ivy.$                                   \n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mcom.cibo.evilplot.plot._\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mcom.cibo.evilplot.plot.renderers.PointRenderer\n",
       "\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mcom.cibo.evilplot.plot.aesthetics.DefaultTheme._\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mcom.cibo.evilplot.numeric.Point\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mcom.cibo.evilplot.colors._\n",
       "\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mbreeze.linalg.{DenseVector, DenseMatrix}\n",
       "\u001b[39m\r\n",
       "\u001b[32mimport \u001b[39m\u001b[36mbreeze.stats.distributions.MultivariateGaussian \n",
       "\n",
       "\u001b[39m\r\n",
       "defined \u001b[32mclass\u001b[39m \u001b[36mShowJupyter\u001b[39m"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "import $ivy.`io.github.cibotech::evilplot:0.8.1`\n",
    "import com.cibo.evilplot.plot._\n",
    "import com.cibo.evilplot.plot.renderers.PointRenderer\n",
    "\n",
    "import com.cibo.evilplot.plot.aesthetics.DefaultTheme._\n",
    "import com.cibo.evilplot.numeric.Point\n",
    "import com.cibo.evilplot.colors._\n",
    "\n",
    "import breeze.linalg.{DenseVector, DenseMatrix}\n",
    "import breeze.stats.distributions.MultivariateGaussian \n",
    "\n",
    "implicit class ShowJupyter(plot: com.cibo.evilplot.geometry.Drawable) {\n",
    "  def show() : Image = {\n",
    "    Image.fromRenderedImage(plot.asBufferedImage, Image.PNG)  \n",
    "  }\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also initialize a global random number generator, which we use whenever we need a new random number in our algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mrng\u001b[39m: \u001b[32mscala\u001b[39m.\u001b[32mutil\u001b[39m.\u001b[32mRandom\u001b[39m = scala.util.Random@34a1985c"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "// We keep random number generator around as global state\n",
    "val rng = new scala.util.Random()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Metropolis algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Metropolis algorithm works by simulating a random path through the states on which the target distribution is defined. Starting from a given state, a possible new state is proposed using a proposal function. The new state is then evaluated given the target distribution and rejected or accepted depending on the likelihood ratio between the old and new state.\n",
    "\n",
    "This motivates the following definitions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mtype\u001b[39m \u001b[36mState\u001b[39m"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type State = DenseVector[Double]  // The state is represented as a vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mtype\u001b[39m \u001b[36mProposal\u001b[39m"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type Proposal = State => State    // The proposal function produces a new state from a given state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mtype\u001b[39m \u001b[36mDistributionEvaluator\u001b[39m"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type DistributionEvaluator = State => Double // The distribution evaluator evaluates a probability of each state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to understand and be able to visualize how the Metropolis algorithm walks through the state-space, we introduce a logger. The logger keeps track of the sequence of states, together with the information, which new state was proposed in each step:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mclass\u001b[39m \u001b[36mLogger\u001b[39m"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "class Logger {\n",
    "    case class StepInfo(state : State, proposedState : State) {\n",
    "        def accepted : Boolean = state == proposedState\n",
    "    }\n",
    "\n",
    "    private val stepSeq = collection.mutable.Buffer[StepInfo]()\n",
    "    \n",
    "    def steps : Seq[StepInfo] = stepSeq.toSeq\n",
    "    \n",
    "    def logStep(state : State, proposedState : State) : Unit = stepSeq.append(StepInfo(state, proposedState))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With these definitions, we are ready to implement the Metropolis sampler. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mfunction\u001b[39m \u001b[36mmetropolisSampler\u001b[39m"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "\n",
    "def metropolisSampler(p : DistributionEvaluator, q : Proposal, initialState : State, logger : Logger) : Iterator[State] = {\n",
    "\n",
    "    // Simulates one step\n",
    "    def nextStep(currentState : State) : State = {\n",
    "        \n",
    "        // propose a new state from the given state\n",
    "        val proposedState = q(currentState) \n",
    "        \n",
    "        // accept based on the ratio of probabilities between \n",
    "        // the new and the old state\n",
    "        val r = rng.nextDouble()\n",
    "        val alpha  = scala.math.min(1.0, p(proposedState) / p(currentState));\n",
    "        val nextState = if (r < alpha) proposedState else currentState\n",
    "        logger.logStep(nextState, proposedState)\n",
    "        nextState\n",
    "    }\n",
    "    \n",
    "    // create an iterator starting from the initial state\n",
    "    Iterator.iterate(initialState)(nextStep)\n",
    "}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is it - this simple code is a complete implementation of the celebrated Metropolis algorithm! "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Toy example: sampling from a bivariate normal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now run the algorithm on a toy example, where we sample from a bivariate normal distribution. \n",
    "We start by defining the target probability distribution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mbivariateNormal\u001b[39m: \u001b[32mMultivariateGaussian\u001b[39m = \u001b[33mMultivariateGaussian\u001b[39m(\n",
       "  DenseVector(9.0, 10.0),\n",
       "  2.0  0.5  \n",
       "0.5  1.0  \n",
       ")"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val bivariateNormal = MultivariateGaussian(\n",
    "    mean = DenseVector(9.0, 10.0), \n",
    "    covariance = DenseMatrix((2.0, 0.5), (0.5, 1.0))\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot some samples from the distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     },
     "metadata": {
      "height": "2400",
      "width": "3231"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\u001b[36msamples\u001b[39m: \u001b[32mIndexedSeq\u001b[39m[\u001b[32mDenseVector\u001b[39m[\u001b[32mDouble\u001b[39m]] = \u001b[33mVector\u001b[39m(\n",
       "  DenseVector(7.950063059653292, 9.665357409794472),\n",
       "  DenseVector(7.897476790412558, 9.197888519410425),\n",
       "  DenseVector(6.210966685420566, 9.93020822662213),\n",
       "  DenseVector(8.192650396042627, 9.397569656684123),\n",
       "  DenseVector(8.336046271557342, 10.28007826687203),\n",
       "  DenseVector(8.590456087934829, 10.604253910618507),\n",
       "  DenseVector(10.742780009679185, 9.97268152768115),\n",
       "  DenseVector(8.243260612290735, 8.345027389250161),\n",
       "  DenseVector(8.772700290567586, 10.779154820458052),\n",
       "  DenseVector(8.89060983679405, 9.141380544285607),\n",
       "  DenseVector(9.575247444854442, 11.176331926542087),\n",
       "  DenseVector(9.420488498784927, 9.768752730729066),\n",
       "  DenseVector(7.450690764697665, 10.747945203145385),\n",
       "  DenseVector(9.130535550533367, 10.848343191504156),\n",
       "  DenseVector(9.248312306711691, 10.833523108927844),\n",
       "  DenseVector(10.398547752687266, 11.19834270465996),\n",
       "  DenseVector(7.977392156846392, 10.180300442365315),\n",
       "  DenseVector(8.87026699526358, 9.505827787246515),\n",
       "  DenseVector(8.391855425238063, 9.1013800778294),\n",
       "  DenseVector(8.718448241569766, 10.059654354269696),\n",
       "  DenseVector(8.182811836545667, 9.127419062841977),\n",
       "  DenseVector(7.522046764414963, 8.837648197788488),\n",
       "  DenseVector(8.366007965721682, 8.641553647723578),\n",
       "  DenseVector(7.582405824003544, 8.21372724419766),\n",
       "  DenseVector(6.466502909744587, 10.519600882120297),\n",
       "  DenseVector(10.89851720652511, 8.460125068073468),\n",
       "  DenseVector(11.370071765385482, 11.545188548738976),\n",
       "  DenseVector(8.73120207808771, 8.849972901815843),\n",
       "  DenseVector(7.321254703338562, 10.534840852820686),\n",
       "  DenseVector(9.67655185728469, 9.372375052841612),\n",
       "  DenseVector(7.067027164189062, 9.346475828194185),\n",
       "  DenseVector(9.256796610174717, 8.89612716519381),\n",
       "  DenseVector(10.16640003320734, 10.759403325636672),\n",
       "  DenseVector(11.404955599592181, 10.433078521829183),\n",
       "  DenseVector(7.232789375007407, 10.530078177368562),\n",
       "  DenseVector(8.576056217192264, 10.640261281061594),\n",
       "  DenseVector(8.838894356703719, 11.097272025529975),\n",
       "  DenseVector(8.612431988952201, 9.225312915579183),\n",
       "..."
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val samples = for (_ <- 0 until 100) yield bivariateNormal.sample()\n",
    "ScatterPlot(    \n",
    "    samples.map(s => Point(s(0), s(1)))    \n",
    ").xAxis()\n",
    " .yAxis()\n",
    " .frame()\n",
    " .xLabel(\"x\")\n",
    " .xbounds(0, 20)\n",
    " .yLabel(\"y\")\n",
    " .ybounds(0, 20)\n",
    " .rightLegend()\n",
    " .render()\n",
    " .show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also need to define a proposal generator. We define a simple random walk proposal, which chooses the new direction and step length randomly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mfunction\u001b[39m \u001b[36mrandomWalkProposal\u001b[39m"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def randomWalkProposal(x : State) : State = {\n",
    "    val stepLength = 0.5\n",
    "    \n",
    "    val step = DenseVector(rng.nextGaussian() * stepLength, rng.nextGaussian() * stepLength)\n",
    "    x + step\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can draw samples using our sampler. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mlogger\u001b[39m: \u001b[32mLogger\u001b[39m = ammonite.$sess.cmd27$Helper$Logger@1428f2d3\r\n",
       "\u001b[36msamples\u001b[39m: \u001b[32mSeq\u001b[39m[\u001b[32mState\u001b[39m] = \u001b[33mList\u001b[39m(\n",
       "  DenseVector(10.0, 10.0),\n",
       "  DenseVector(9.720643015666251, 9.708472943256385),\n",
       "  DenseVector(9.998134353362762, 10.006841187977123),\n",
       "  DenseVector(10.237948537418532, 10.971645154560111),\n",
       "  DenseVector(9.819643666968798, 10.819400876162973),\n",
       "  DenseVector(9.840398761481561, 11.027326740439163),\n",
       "  DenseVector(9.342556894605615, 10.850817105130197),\n",
       "  DenseVector(10.17022244894715, 10.630751992854817),\n",
       "  DenseVector(10.332602351809195, 9.93399996773307),\n",
       "  DenseVector(10.310662447412549, 10.342190308832237),\n",
       "  DenseVector(10.319540619957818, 10.120953241811405),\n",
       "  DenseVector(10.80636188910151, 10.46997008396131),\n",
       "  DenseVector(10.201127157611145, 11.031051385564199),\n",
       "  DenseVector(10.340306593391059, 9.295506509111483),\n",
       "  DenseVector(10.12415641544559, 9.306995653846224),\n",
       "  DenseVector(10.12415641544559, 9.306995653846224),\n",
       "  DenseVector(10.12415641544559, 9.306995653846224),\n",
       "  DenseVector(10.432626266146578, 10.168951370556062),\n",
       "  DenseVector(10.432626266146578, 10.168951370556062),\n",
       "  DenseVector(9.464697815618106, 9.668210122512415),\n",
       "  DenseVector(9.560270122912694, 10.092292781541644),\n",
       "  DenseVector(9.529050861129745, 9.597895628050676),\n",
       "  DenseVector(9.46772933619005, 9.520881377074707),\n",
       "  DenseVector(10.276135235393053, 9.456376788567347),\n",
       "  DenseVector(10.27364467318171, 9.616746008306068),\n",
       "  DenseVector(9.87479780392965, 10.547434187671335),\n",
       "  DenseVector(9.561350665632311, 10.769140952872622),\n",
       "  DenseVector(9.561350665632311, 10.769140952872622),\n",
       "  DenseVector(8.887118238654468, 10.618118072661842),\n",
       "  DenseVector(8.375465599708871, 9.97640655127039),\n",
       "  DenseVector(8.241176034456126, 10.168522847702429),\n",
       "  DenseVector(8.630096038976866, 10.067092689786765),\n",
       "  DenseVector(8.85288470445321, 10.202495488649314),\n",
       "  DenseVector(8.251616569722923, 10.286334057878408),\n",
       "  DenseVector(7.8124533328398655, 10.31764175044807),\n",
       "  DenseVector(8.025697559028835, 9.839905086650246),\n",
       "  DenseVector(7.952883255778813, 9.647423746082575),\n",
       "  DenseVector(8.454397672701985, 10.541076878847173),\n",
       "..."
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val logger = new Logger()\n",
    "val samples = metropolisSampler(bivariateNormal.pdf, randomWalkProposal, DenseVector(10, 10), logger).take(500).toSeq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing the mean and the covariance from the samples shows, that the samples approximate the target distribution rather well. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mmean\u001b[39m: \u001b[32mDenseVector\u001b[39m[\u001b[32mDouble\u001b[39m] = DenseVector(8.677396841080336, 10.262956957766043)\r\n",
       "\u001b[36mcov\u001b[39m: \u001b[32mDenseMatrix\u001b[39m[\u001b[32mDouble\u001b[39m] = 1.3069347024111155  0.4019547020016448  \n",
       "0.4019547020016448  1.0351467562971002  "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val mean = samples.reduce(_ + _) * (1.0 / samples.length)\n",
    "val cov = samples.map(x => (x - mean) * (x - mean).t).reduce(_ + _) * (1.0 / samples.length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also check this also visually by plotting the samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mfunction\u001b[39m \u001b[36mplotSamples\u001b[39m"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotSamples(samples : Seq[State]) : Image = {\n",
    "\n",
    "    ScatterPlot(\n",
    "        samples.map(s => Point(s(0), s(1)))\n",
    "    )\n",
    "    .xAxis()\n",
    "    .xbounds(0, 20)\n",
    "    .yAxis()\n",
    "    .ybounds(0, 20)\n",
    "    .frame()\n",
    "    .xLabel(\"x\")\n",
    "    .yLabel(\"y\")\n",
    "    .rightLegend()\n",
    "    .render()\n",
    "    .show()\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     },
     "metadata": {
      "height": "2400",
      "width": "3231"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotSamples(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get a deeper understanding of how the metropolis algorithm works, it is interesting to visualize not only the accepted samples, but also those which are rejected, and possibly the path that was taken. This is achieved by the following plot function, which visualizes the accepted and rejected samples in different colors and also can plot lines to show the path that was explored:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "defined \u001b[32mfunction\u001b[39m \u001b[36mplotLoggedSamples\u001b[39m"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotLoggedSamples(logger : Logger, plotLines : Boolean = false) : Image = {\n",
    "\n",
    "    val acceptedPlot = ScatterPlot.series(\n",
    "        logger.steps.filter(s => s.accepted).map(s => Point(s.state(0), s.state(1))),\n",
    "        \"accepted\", \n",
    "        HTMLNamedColors.blue, \n",
    "        pointSize = Some(3)\n",
    "    )\n",
    "    val rejectedPlot = ScatterPlot.series(\n",
    "        logger.steps.filterNot(s => s.accepted).map(s => Point(s.proposedState(0), s.proposedState(1))),\n",
    "        \"rejected\", \n",
    "        HTMLNamedColors.red ,\n",
    "        pointSize = Some(3)  \n",
    "    )\n",
    "    val linePlot =  LinePlot(\n",
    "        logger.steps.flatMap(s => \n",
    "                        if (!s.accepted) {        \n",
    "                            Seq(Point(s.proposedState(0), s.proposedState(1)), Point(s.state(0), s.state(1)))\n",
    "                        } else {\n",
    "                            Seq(Point(s.state(0), s.state(1)))\n",
    "                        })\n",
    "    )\n",
    "    \n",
    "    var plots = Seq(acceptedPlot)\n",
    "    plots = plots :+ rejectedPlot\n",
    "    if (plotLines) plots = plots :+ linePlot\n",
    "    \n",
    "    Overlay.fromSeq(plots)\n",
    "    .xAxis()\n",
    "    .xbounds(0, 20)\n",
    "    .yAxis()\n",
    "    .ybounds(0, 20)\n",
    "    .frame()\n",
    "    .xLabel(\"x\")\n",
    "    .yLabel(\"y\")\n",
    "    .rightLegend()\n",
    "    .render()\n",
    "    .show()\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot our samples and start experimenting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
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    "#### Exercises\n",
    "\n",
    "* We have started sampling from close to the mean of the target distribution. What happens when we start at the Point (0,0)?\n",
    "    * Visualize the resulting samples using the plot functions\n",
    "    * How could we mitigate the situation?\n",
    "* Play with different step-length in the proposal.\n",
    "    * What happens to the acceptance and rejection rate? \n",
    "    * How well is the target distribution approximated after a fixed number of samples\n",
    "* Experiment with different target distributions. \n",
    "    * Make the variance larger and smaller\n",
    "    * Change the correlation \n",
    "    * Assign to all samples whose x value is larger then 5 the probability 0. \n",
    "        * Do you need to normalize the pdf? why, why not?\n",
    "* Define a function that computes the exected value of a given function, using the samples\n",
    "* How could you get samples from the marginal distributions?"
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