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     "source": [
      "# Introduction to GPy: Gaussian Process Regression in GPy\n",
      "\n",
      "# Gaussian Process Summer School, Melbourne, Australia\n",
      "### 25th-27th February 2015\n",
      "### Neil D. Lawrence and Nicolas Durrande\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import pods\n",
      "import pylab as plt\n",
      "import GPy\n",
      "from IPython.display import display"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/pytz/__init__.py:29: UserWarning: Module pods was already imported from /Users/neil/sods/ods/pods/__init__.pyc, but /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages is being added to sys.path\n",
        "  from pkg_resources import resource_stream\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Covariance Function Parameter Estimation\n",
      "\n",
      "In this session we are going to optimize the parameters of the Gaussian process using gradient based optimization approaches. These maximize the likelihood function: which is defined as the probability of the model given the parameters, $p(\\mathbf{y}|\\mathbf{X}, \\boldsymbol{\\theta})$. \n",
      "\n",
      "First we'll load in the olympic marathon data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data = pods.datasets.olympic_marathon_men()\n",
      "x = data['X']\n",
      "y = data['Y']"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then we'll construct a Gaussian process model with an exponentiated quadratic covariance function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "k = GPy.kern.RBF(1)\n",
      "model = GPy.models.GPRegression(x, y, k)\n",
      "display(model)\n",
      "model.plot()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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        "    width: 100%;\n",
        "    padding: 3px;\n",
        "}\n",
        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -118.821194703<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
        "<b>Updates</b>: True<br>\n",
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        "<table class=\"tg\">\n",
        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
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       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x10ec85050>"
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       "output_type": "pyout",
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dPWir8+t5xzxetwN9sVEUigru/5HVwgGAj376JgBqJLhh/TOGHgWlFK+/sh4f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\n0NDQmCFopq2hoaExQ9BMW0NDQ2OG8P8BA9nmAjMAbioAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ec8a550>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rather sensibly, we've given the model an initial plot, and we can clearly see that the inital length scale is too low. This makes sense, our prior says that the length scale is 1 year, which means that athletic performance varies across very short time scales. This is perhaps unlikely, let's choose a larger lengthscale and try again.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model.rbf.lengthscale = 10.\n",
      "model.plot()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "{'dataplot': [<matplotlib.lines.Line2D at 0x10f27aed0>],\n",
        " 'gpplot': [[<matplotlib.lines.Line2D at 0x10f277a90>],\n",
        "  [<matplotlib.patches.Polygon at 0x10f277cd0>],\n",
        "  [<matplotlib.lines.Line2D at 0x10f27a350>],\n",
        "  [<matplotlib.lines.Line2D at 0x10f27a890>]]}"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9aJuSt2h7+7w8t+kAAB+5OT5LrwohzptRksu/fuFW/vG+60hPdbHvcD1/969P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\nRvr9BlWludhs8d36fnvvCf7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xWGOMo2sHCP0Rmm2F1vra8PZTwM/D22uAP2itg0CLUupN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\nTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x103c28850>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That seems better, now we can think about optimization. First though we have to consider the fact that some of the parameters are constrained (for example lengthscales and variances can only be positive). `GPy` allows the user to specify such constraints when constructing the model.\n",
      "\n",
      "### Parameter Constraints\n",
      "\n",
      "As we have seen during the lectures, the parameters values can be estimated by maximizing the likelihood of the observations. Since we don\u2019t want one of the variance to become negative during the optimization, we can constrain all parameters to be positive before running the optimisation."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model.constrain_positive('.*') # Constrains all parameters matching .* to be positive, i.e. everything"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "WARNING: reconstraining parameters GP_regression\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The warnings are because the parameters are already constrained by default, the software is warning us that they are being reconstrained.\n",
      "\n",
      "Now we can optimize the model using the \n",
      "```\n",
      "model.optimize()\n",
      "``` method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model.optimize()\n",
      "model.plot()\n",
      "display(model)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style type=\"text/css\">\n",
        ".pd{\n",
        "    font-family: \"Courier New\", Courier, monospace !important;\n",
        "    width: 100%;\n",
        "    padding: 3px;\n",
        "}\n",
        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -6.94713791215<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
        "<b>Updates</b>: True<br>\n",
        "</p>\n",
        "<style type=\"text/css\">\n",
        ".tg  {font-family:\"Courier New\", Courier, monospace !important;padding:2px 3px;word-break:normal;border-collapse:collapse;border-spacing:0;border-color:#DCDCDC;margin:0px auto;width:100%;}\n",
        ".tg td{font-family:\"Courier New\", Courier, monospace !important;font-weight:bold;color:#444;background-color:#F7FDFA;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#DCDCDC;}\n",
        ".tg th{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;color:#fff;background-color:#26ADE4;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#DCDCDC;}\n",
        ".tg .tg-left{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;text-align:left;}\n",
        ".tg .tg-right{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;text-align:right;}\n",
        "</style>\n",
        "<table class=\"tg\">\n",
        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right> 25.3995048241</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>    152.045313</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>0.048506484546</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x10ec85050>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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ereC51z+hoqYZgNm5qdx18zLm5qV5Nc6JYGhNeVUzvVYrESEWFkxPISVeJtQS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\n2mXojV8l3QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x103c28550>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The parameters obtained after optimisation can be compared with the values selected by hand above. As previously, you can modify the kernel used for building the model to investigate its influence on the model.\n",
      "\n",
      "By adding covariance functions together we can try and decompose the observation in to a longer lengthscale process and a shorter lengthscale process. Below we consider a GP that is initialised with a long lengthscale exponentiated quadratic, and a Matern $\\frac{5}{2}$ covariance to take account of shorter lengthscale effects. We also add a bias term to allow for an overall average."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 5 a) answer \n",
      "kern = GPy.kern.RBF(1, lengthscale=80) + GPy.kern.Matern52(1, lengthscale=10) + GPy.kern.Bias(1)\n",
      "model = GPy.models.GPRegression(x, y, kern)\n",
      "model.optimize()\n",
      "model.plot()# Exercise 5 d) answer\n",
      "model.log_likelihood()\n",
      "display(model)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style type=\"text/css\">\n",
        ".pd{\n",
        "    font-family: \"Courier New\", Courier, monospace !important;\n",
        "    width: 100%;\n",
        "    padding: 3px;\n",
        "}\n",
        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -5.99078279431<br>\n",
        "<b>Number of Parameters</b>: 6<br>\n",
        "<b>Updates</b>: True<br>\n",
        "</p>\n",
        "<style type=\"text/css\">\n",
        ".tg  {font-family:\"Courier New\", Courier, monospace !important;padding:2px 3px;word-break:normal;border-collapse:collapse;border-spacing:0;border-color:#DCDCDC;margin:0px auto;width:100%;}\n",
        ".tg td{font-family:\"Courier New\", Courier, monospace !important;font-weight:bold;color:#444;background-color:#F7FDFA;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#DCDCDC;}\n",
        ".tg th{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;color:#fff;background-color:#26ADE4;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#DCDCDC;}\n",
        ".tg .tg-left{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;text-align:left;}\n",
        ".tg .tg-right{font-family:\"Courier New\", Courier, monospace !important;font-weight:normal;text-align:right;}\n",
        "</style>\n",
        "<table class=\"tg\">\n",
        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>add.rbf.variance       </td><td class=tg-right>  2.58425103654</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.rbf.lengthscale    </td><td class=tg-right>  102.447254296</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.Mat52.variance     </td><td class=tg-right>0.0204066240251</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.Mat52.lengthscale  </td><td class=tg-right>  6.52778209539</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.bias.variance      </td><td class=tg-right>  17.5408872485</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>0.0368133074589</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x10fa2ce10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hEmy3khxUy6cv/4S7vvp1/vnffgrQndCfeXH9gLX98cRlGJwurCImzMaq+Rmy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\n82V1FBRVkl9URWllAwdOXOD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       "text": [
        "<matplotlib.figure.Figure at 0x10fa35110>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise 2\n",
      "\n",
      "Now model Model the data with a product of an exponentiated quadratic covariance function and a linear covariance function. Fit the covariance function parameters. Why are the variance parameters of the linear part so small? How could this be fixed?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 2 answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "DRAFT\n",
      "===\n",
      "\n",
      "## Gene Expression Example\n",
      "\n",
      "We now look at a real data example where there are multiple modes to the solution. In [Kalaitzis and Lawrence](http://www.biomedcentral.com/1471-2105/12/180) the objective was to understand when a temporal gene expression was either *noise* or had some underlying signal. To determine this Gaussian process models were fitted with and without a temporal kernel, and the likelihoods were compared. In the thousands of genes they considered, there were some where the posterior error surface for the lengthscale and the signal/noise ratio was multi modal. We will consider one of those genes. The example can also be rerun as\n",
      "```python\n",
      "GPy.examples.regression.multiple_optima()\n",
      "```\n",
      "The first thing to do is write a helper function for computint the likelihoods add different signal/noise ratios and different lengthscales. This is to allow us to visualize the error surface. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def contour_objective(x, y, log_length_scales, log_SNRs, kernel_call=GPy.kern.RBF):\n",
      "    '''Helper function to contour an objective function in a set up where there \n",
      "    is a kernel for signal corrupted by noise.'''\n",
      "    lls = []\n",
      "    num_data=y.shape[0]\n",
      "    kernel = kernel_call(1, variance=1., lengthscale=1.)\n",
      "    model = GPy.models.GPRegression(x, y, kernel=kernel)\n",
      "    y = y - y.mean()\n",
      "    for log_SNR in log_SNRs:\n",
      "        SNR = 10.**log_SNR\n",
      "        length_scale_lls = []\n",
      "        for log_length_scale in log_length_scales:\n",
      "            model['.*lengthscale'] = 10.**log_length_scale\n",
      "            model.kern['.*variance'] = SNR\n",
      "            Kinv = GPy.util.linalg.pdinv(model.kern.K(x)+np.eye(num_data))[0]\n",
      "            total_var = 1./num_data*np.dot(np.dot(y.T, Kinv), y)\n",
      "            noise_var = total_var\n",
      "            signal_var = SNR*total_var \n",
      "            model.kern['.*variance'] = signal_var\n",
      "            model.Gaussian_noise = noise_var\n",
      "            length_scale_lls.append(model.log_likelihood())\n",
      "            print SNR, 10.**log_length_scale\n",
      "            display(model)\n",
      "        lls.append(length_scale_lls)\n",
      "    \n",
      "    return np.array(lls)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we load in the data and compute the likelihood values."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data = pods.datasets.della_gatta_TRP63_gene_expression(gene_number=937)\n",
      "x = data['X']\n",
      "y = data['Y']\n",
      "y = y - y.mean()\n",
      "kern = GPy.kern.RBF(input_dim=1)\n",
      "model = GPy.models.GPRegression(x, y, kern)\n",
      "resolution = 2\n",
      "log_lengthscales = np.linspace(1, 3.5, resolution)\n",
      "log_SNRs = np.linspace(-2.5, 1., resolution)\n",
      "lls = contour_objective(x, y, log_lengthscales, log_SNRs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.00316227766017 10.0\n"
       ]
      },
      {
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        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -2.10275061954<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
        "<b>Updates</b>: True<br>\n",
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        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
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        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right>0.00025506380619</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>            10.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right> 0.0806582576232</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x1121b3810>"
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      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.00316227766017 3162.27766017\n"
       ]
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        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -2.12547857371<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
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        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right>0.000255972082804</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>    3162.27766017</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>  0.0809454799079</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
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       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x1121b3810>"
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       "text": [
        "10.0 10.0\n"
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        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -1.41450534969<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
        "<b>Updates</b>: True<br>\n",
        "</p>\n",
        "<style type=\"text/css\">\n",
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        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right> 0.0671183959588</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>            10.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>0.00671183959588</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
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       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x1121b3810>"
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        "10.0 3162.27766017\n"
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        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -4.33641210151<br>\n",
        "<b>Number of Parameters</b>: 3<br>\n",
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        "  <th><b>GP_regression.</b></th>\n",
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        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
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        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right> 0.779949847367</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>  3162.27766017</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>0.0779949847367</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
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        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -16.7147336689<br>\n",
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        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
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        "<tr><td class=tg-left>rbf.variance           </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>rbf.lengthscale        </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
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      "\n",
      "plt.contour(lengthscales, log_SNRs, lls, 20, cmap=plt.cm.jet)"
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     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "NameError",
       "evalue": "name 'lengthscales' is not defined",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-37-aaa47ce1d09d>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcontour\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlengthscales\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog_SNRs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m20\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcmap\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcm\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjet\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[0;31mNameError\u001b[0m: name 'lengthscales' is not defined"
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     "collapsed": false,
     "input": [
      "plt.plot(x, y-y.mean())"
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     "metadata": {},
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       "prompt_number": 38,
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Vy76iVAHdAqxw95vNbDCwjbtvVrib2SvAT9x9rpkNBdq6+6AGtiu6KiAIvXmO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\nQESkTCkBiIiUKSUAEZEypQQgIlKm/j97/HKURg95ZwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11212d210>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Sampling with Hamiltonian Monte Carlo"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "model.kern.lengthscale=30.\n",
      "model.kern.variance=0.5\n",
      "model.Gaussian_noise=0.01\n",
      "model.kern.lengthscale.set_prior(GPy.priors.InverseGamma.from_EV(30.,100.))\n",
      "model.kern.variance.set_prior(GPy.priors.InverseGamma.from_EV(0.5, 1.)) #Gamma.from_EV(1.,10.))\n",
      "model.Gaussian_noise.set_prior(GPy.priors.InverseGamma.from_EV(0.01,1.))\n",
      "_ = model.plot()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "WARNING: reconstraining parameters GP_regression.rbf.lengthscale\n",
        "WARNING: reconstraining parameters GP_regression.rbf.lengthscale\n",
        "WARNING: reconstraining parameters GP_regression.rbf.variance\n",
        "WARNING: reconstraining parameters GP_regression.rbf.variance\n",
        "WARNING: reconstraining parameters GP_regression.Gaussian_noise\n",
        "WARNING: reconstraining parameters GP_regression.Gaussian_noise\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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HXBMnH/3eN/n9E4/z+G+fZcM11/LWO8W8uD+f2sb3c1JcVCgbVi5hZWYiq7KSiJ3i/2lL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\nIRag/w9ewmbeFGaXpAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x111daff10>"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hmc = GPy.inference.mcmc.HMC(model, stepsize=5e-2)\n",
      "s = hmc.sample(num_samples=1000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(s)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": [
        "[<matplotlib.lines.Line2D at 0x112116250>,\n",
        " <matplotlib.lines.Line2D at 0x1121bf350>,\n",
        " <matplotlib.lines.Line2D at 0x1121bf8d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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vtEcEvodkeDJ4dsmz2BPspLhTKKsr48YFN3Lf58bkpbpgneV3tifYrVwukdIW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\n2symeMqEU6ztbTNDDmV+su9PuP3I2zv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       "text": [
        "<matplotlib.figure.Figure at 0x111dafd90>"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "labels = ['kern variance', 'kern lengthscale','noise variance']\n",
      "samples = s[300:] # cut out the burn-in period\n",
      "from scipy import stats\n",
      "xmin = samples.min()\n",
      "xmax = samples.max()\n",
      "xs = np.linspace(xmin,xmax,100)\n",
      "for i in xrange(samples.shape[1]-1):\n",
      "    kernel = stats.gaussian_kde(samples[:,i])\n",
      "    plt.plot(xs,kernel(xs),label=labels[i])\n",
      "_ = plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x111d09ad0>"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig = plt.figure(figsize=(14,4))\n",
      "ax = fig.add_subplot(131)\n",
      "_=ax.plot(samples[:,0],samples[:,1],'.')\n",
      "ax.set_xlabel(labels[0]); ax.set_ylabel(labels[1])\n",
      "ax = fig.add_subplot(132)\n",
      "_=ax.plot(samples[:,1],samples[:,2],'.')\n",
      "ax.set_xlabel(labels[1]); ax.set_ylabel(labels[2])\n",
      "ax = fig.add_subplot(133)\n",
      "_=ax.plot(samples[:,0],samples[:,2],'.')\n",
      "ax.set_xlabel(labels[0]); ax.set_ylabel(labels[2])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": [
        "<matplotlib.text.Text at 0x111d6d490>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ZLUIFb0pGEyeo1Ppk0ybXsyREiSiyRPawmZ2aONApie1eavWA7aJSlEKUml+Y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\n2hBFqPd4rEqdx2Nf6j0Way4iNpXKkq4HFtd7m5IGk02Abo6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       "text": [
        "<matplotlib.figure.Figure at 0x111aa4e10>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## More Advanced: Uncertainty Propagation\n",
      "\n",
      "Let $x$ be a random variable defined over the real numbers, $\\Re$, and $f(\\cdot)$ be a function mapping between the real numbers $\\Re \\rightarrow \\Re$. Uncertainty\n",
      "propagation is the study of the distribution of the random variable $f ( x )$.\n",
      "\n",
      "We will see in this section the advantage of using a model when only a few observations of $f$ are available. We consider here the 2-dimensional Branin test function\n",
      "defined over [\u22125, 10] \u00d7 [0, 15] and a set of 25 observations as seen in Figure 3."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Definition of the Branin test function\n",
      "def branin(X):\n",
      "    y = (X[:,1]-5.1/(4*np.pi**2)*X[:,0]**2+5*X[:,0]/np.pi-6)**2\n",
      "    y += 10*(1-1/(8*np.pi))*np.cos(X[:,0])+10\n",
      "    return(y)\n",
      "\n",
      "# Training set defined as a 5*5 grid:\n",
      "xg1 = np.linspace(-5,10,5)\n",
      "xg2 = np.linspace(0,15,5)\n",
      "X = np.zeros((xg1.size * xg2.size,2))\n",
      "for i,x1 in enumerate(xg1):\n",
      "    for j,x2 in enumerate(xg2):\n",
      "        X[i+xg1.size*j,:] = [x1,x2]\n",
      "\n",
      "Y = branin(X)[:,None]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We assume here that we are interested in the distribution of $f (U )$ where $U$ is a\n",
      "random variable with uniform distribution over the input space of $f$. We will focus on\n",
      "the computation of two quantities: $E[ f (U )]$ and $P( f (U ) > 200)$.\n",
      "\n",
      "## Computation of $E[f(U)]$\n",
      "\n",
      "The expectation of $f (U )$ is given by $\\int_x f ( x )\\text{d}x$. A basic approach to approximate this\n",
      "integral is to compute the mean of the 25 observations: `np.mean(Y)`. Since the points\n",
      "are distributed on a grid, this can be seen as the approximation of the integral by a\n",
      "rough Riemann sum. The result can be compared with the actual mean of the Branin\n",
      "function which is 54.31.\n",
      "\n",
      "Alternatively, we can fit a GP model and compute the integral of the best predictor\n",
      "by Monte Carlo sampling:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit a GP\n",
      "# Create an exponentiated quadratic plus bias covariance function\n",
      "kg = GPy.kern.RBF(input_dim=2, ARD=True)\n",
      "kb = GPy.kern.Bias(input_dim=2)\n",
      "k = kg + kb\n",
      "\n",
      "# Build a GP model\n",
      "model = GPy.models.GPRegression(X,Y,k)\n",
      "\n",
      "# fix the noise variance to something low\n",
      "model.Gaussian_noise.variance = 1e-5\n",
      "model.Gaussian_noise.variance.constrain_fixed()\n",
      "display(model)\n",
      "\n",
      "# optimize the model\n",
      "model.optimize()\n",
      "\n",
      "# Plot the resulting approximation to Brainin\n",
      "# Here you get a two-d plot becaue the function is two dimensional.\n",
      "model.plot()\n",
      "display(model.add.rbf.lengthscale)\n",
      "\n",
      "# Compute the mean of model prediction on 1e5 Monte Carlo samples\n",
      "Xp = np.random.uniform(size=(1e5,2))\n",
      "Xp[:,0] = Xp[:,0]*15-5\n",
      "Xp[:,1] = Xp[:,1]*15\n",
      "Yp = model.predict(Xp)[0]\n",
      "np.mean(Yp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "WARNING: reconstraining parameters GP_regression.Gaussian_noise.variance\n"
       ]
      },
      {
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        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: GP regression<br>\n",
        "<b>Log-likelihood</b>: -74285.8774977<br>\n",
        "<b>Number of Parameters</b>: 5<br>\n",
        "</p>\n",
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        "<tr>\n",
        "  <th><b>GP_regression.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>add.rbf.variance       </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.rbf.lengthscale    </td><td class=tg-right> (2,)</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.bias.variance      </td><td class=tg-right>  1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>Gaussian_noise.variance</td><td class=tg-right>1e-05</td><td class=tg-left>  fixed   </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<GPy.models.gp_regression.GPRegression at 0x114a4da10>"
       ]
      },
      {
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        "<table class=\"tg\">\n",
        "\n",
        "<tr>\n",
        "  <th><b>Index</b></th>\n",
        "  <th><b>GP_regression.add.rbf.lengthscale</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>[0]</td><td  class=tg-right>28.9811652911</td><td class=tg-left>+ve</td><td class=tg-left></td><td class=tg-left>N/A</td></tr>\n",
        "<tr><td class=tg-left>[1]</td><td  class=tg-right>303.000201018</td><td class=tg-left>+ve</td><td class=tg-left></td><td class=tg-left>N/A</td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "\u001b[1mGP_regression.add.rbf.lengthscale\u001b[0;0m:\n",
        "Param([  28.98116529,  303.00020102])"
       ]
      },
      {
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       "prompt_number": 36,
       "text": [
        "58.283419091796873"
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dmsGBKVJI5akPIcZFZZ+8QFCADPnL0MuMyAzXJsJliaLIepFLusPA5VKb2ORC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       "text": [
        "<matplotlib.figure.Figure at 0x1149e7fd0>"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise 6\n",
      "\n",
      "a) Has the approximation of the mean been improved by using the GP model?"
     ]
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "# Exercise 6 a) answer "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "b) One particular feature of GPs we have not use for now is their prediction variance. Can you use it to define some confidence intervals around the previous result?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 6 b) answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Computation of $P( f (U ) > 200)$\n",
      "\n",
      "In various cases it is interesting to look at the probability that $f$ is greater than a given\n",
      "threshold. For example, assume that $f$ is the response of a physical model representing\n",
      "the maximum constraint in a structure depending on some parameters of the system\n",
      "such as Young\u2019s modulus of the material (say $Y$) and the force applied on the structure\n",
      "(say $F$). If the later are uncertain, the probability of failure of the structure is given by\n",
      "$P( f (Y, F ) > \\text{f_max} )$ where $f_\\text{max}$ is the maximum acceptable constraint.\n",
      "\n",
      "### Exercise 7\n",
      "\n",
      "a) As previously, use the 25 observations to compute a rough estimate of the probability that $f (U ) > 200$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 7 a) answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "b) Compute the probability that the best predictor is greater than the threshold."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 7 b) answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "c) Compute some confidence intervals for the previous result"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 7 c) answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These two values can be compared with the actual value {$P( f (U ) > 200) = 1.23\\times 10^{\u22122}$ .\n",
      "\n",
      "We now assume that we have an extra budget of 10 evaluations of f and we want to\n",
      "use these new evaluations to improve the accuracy of the previous result.\n",
      "\n",
      "### Exercise 8\n",
      "\n",
      "a) Given the previous GP model, where is it interesting to add the new observations if we want to improve the accuracy of the estimator and reduce its variance?"
     ]
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "# Exercise 8 a) answer"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "b) Can you think about (and implement!) a procedure that updates sequentially the model with new points in order to improve the estimation of $P( f (U ) > 200)$?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Exercise 8 b) answer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}