{ "metadata": { "name": "", "signature": "sha256:1e50eafeced1fd718b5becc02f1afe7714d8be047bb65e70583e8c6f1e5fd65f" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Special Topics: Gaussian Processes\n", "\n", "### 16th December 2014 Neil D. Lawrence\n", "\n", "Over the last two sessions we've begun considering classification models and logistic regresssion. In particular, for naive Bayes, we considered a set of assumptions that allowed us to build a joint model of our data set. In particular for naive Bayes we specified \n", "\n", "1. Data conditional independence.\n", "2. Feature conditional independence.\n", "3. Marginal likelihood of labels was Bernoulli distributed.\n", "\n", "This allowed us to specify the joint density of our labels and our input data, $p(\\mathbf{y}, \\mathbf{X}|\\boldsymbol{\\theta})$. And we conditioned on the training data to make predictions about the test data. \n", "\n", "## Generalized Linear Models \n", "\n", "Logistic regression is part of a wider class of models known as *generalized linear models*. In these models we determine that some characteristic of the model is speicified by a function that is liniear in the parameters. So we might suggest that\n", "$$\n", "\\log \\frac{p(\\mathbf{x})}{1-p(\\mathbf{x})} = f(\\mathbf{x}; \\mathbf{w})\n", "$$\n", "where $f(\\mathbf{x}; \\mathbf{w})$ is a linear-in-the-parameters function (here the parameters are $\\mathbf{w}$, which is generally non-linear in the inputs. So far we have considered basis function models of the form\n", "$$\n", "f(\\mathbf{x}) = \\mathbf{w}^\\top \\boldsymbol{\\phi}(\\mathbf{x}).\n", "$$\n", "\n", "## Gaussian Processes\n", "\n", "When we form a Gaussian process we do something that is slightly more akin to the naive Bayes approach, but actually is closely related to the generalized linear model approach. Models where we model the entire joint distribution of our training data, $p(\\mathbf{y}, \\mathbf{X})$ are sometimes described as *generative models*. Because we can use sampling to generate data sets that represent all our assumptions. However, as we discussed in the sessions on logistic regression and naive Bayes, this can be a bad idea, because if our assumptions are wrong then we can make poor predictions. We can try to make more complex assumptions about data to alleviate the problem, but then this typically leads to challenges for tractable application of the sum and rules of probability that are needed to compute the relevant marginal and conditional densities. If we know the form of the question we wish to answer then we typically try and represent that directly, through $p(\\mathbf{y}|\\mathbf{X})$. In practice, we also have been making assumptions of conditional independence given the model parameters, \n", "$$\n", "p(\\mathbf{y}|\\mathbf{X}, \\mathbf{w}) = \\prod_{i=1}^{n} p(y_i | \\mathbf{x}_i, \\mathbf{w})\n", "$$\n", "Gaussian processes are *not* normally considered to be *generative models*, but we will be much more interested in the principles of conditioning in Gaussian processes because we will use conditioning to make predictions between our test and training data. We will avoid the data conditional indpendence assumption in favour of a richer assumption about the data, in a Gaussian process we assume data is *jointly Gaussian* with a particular mean and covariance,\n", "$$\n", "\\mathbf{y}|\\mathbf{X} \\sim \\mathcal{N}(\\mathbf{m}(\\mathbf{X}), \\mathbf{K}(\\mathbf{X})),\n", "$$\n", "where the conditioning is on the inputs $\\mathbf{X}$ which are used for computing the mean and covariance. For this reason they are known as mean and covariance functions.\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# download the software\n", "import urllib\n", "\n", "urllib.urlretrieve('https://github.com/sods/ods/archive/master.zip', 'master.zip')\n", "\n", "# unzip the software\n", "import zipfile\n", "zip = zipfile.ZipFile('./master.zip', 'r')\n", "for name in zip.namelist():\n", " zip.extract(name, '.')\n", "\n", "# add the module location to the python path. \n", "import sys\n", "sys.path.append(\"./ods-master/\") " ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The history saving thread hit an unexpected error (DatabaseError('database disk image is malformed',)).History will not be written to the database.\n" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "And next we get the plots running in line. We also also load in the `pylab` library as `plt`." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import numpy as np\n", "import scipy as sp\n", "import matplotlib.pyplot as plt\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Marginal Likelihood\n", "\n", "To understand the Gaussian process we're going to build on our understanding of the marginal likelihood for Bayesian regression. In the session on [Bayesian regression](./week7.ipynb) we sampled directly from the weight vector, $\\mathbf{w}$ and applied it to the basis matrix $\\boldsymbol{\\Phi}$ to obtain a sample from the prior and a sample from the posterior. It is often helpful to think of modeling techniques as *generative* models. To give some thought as to what the process for obtaining data from the model is. From the perspective of Gaussian processes, we want to start by thinking of basis function models, where the parameters are sampled from a prior, but move to thinking about sampling from the marginal likelihood directly.\n", "\n", "## Sampling from the Prior\n", "\n", "The first thing we'll do is to set up the parameters of the model, these include the parameters of the prior, the parameters of the basis functions and the noise level. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "# set prior variance on w\n", "alpha = 4.\n", "# set the order of the polynomial basis set\n", "degree = 5\n", "# set the noise variance\n", "sigma2 = 0.01" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we have the variance, we can sample from the prior distribution to see what form we are imposing on the functions *a priori*. \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now compute a range of values to make predictions at, spanning the *new* space of inputs," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def polynomial(x, degree, loc, scale):\n", " degrees = np.arange(degree+1)\n", " return ((x-loc)/scale)**degrees" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "now let's build the basis matrices. First we load in the data\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import pods\n", "data = pods.datasets.olympic_marathon_men()\n", "x = data['X']\n", "y = data['Y']\n" ], "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": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "loc = 1950.\n", "scale = 100.\n", "num_data = x.shape[0]\n", "num_pred_data = 100 # how many points to use for plotting predictions\n", "x_pred = np.linspace(1880, 2030, num_pred_data)[:, None] # input locations for predictions\n", "Phi_pred = polynomial(x_pred, degree=degree, loc=loc, scale=scale)\n", "Phi = polynomial(x, degree=degree, loc=loc, scale=scale)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Weight Space View\n", "\n", "To generate typical functional predictions from the model, we need a set of model parameters. We assume that the parameters are drawn independently from a Gaussian density,\n", "$$\n", "\\mathbf{w} \\sim \\mathcal{N}(\\mathbf{0}, \\alpha\\mathbf{I}),\n", "$$\n", "then we can combine this with the definition of our prediction function $f(\\mathbf{x})$,\n", "$$\n", "f(\\mathbf{x}) = \\mathbf{w}^\\top \\boldsymbol{\\phi}(\\mathbf{x}).\n", "$$\n", "We can now sample from the prior density to obtain a vector $\\mathbf{w}$ using the function `np.random.normal` and combine these parameters with our basis to create some samples of what $f(\\mathbf{x})$ looks like," ] }, { "cell_type": "code", "collapsed": false, "input": [ "num_samples = 10\n", "K = degree+1\n", "for i in xrange(num_samples):\n", " z_vec = np.random.normal(size=(K, 1))\n", " w_sample = z_vec*np.sqrt(alpha)\n", " f_sample = np.dot(Phi_pred,w_sample)\n", " plt.plot(x_pred, f_sample)\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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LLHiKLXiKwV1sYaAY3MXQXwz9hdBfbDCcaRA2v3cYJfyJztp03onOPcNqVeHJ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\nlEZGVHtLS9XnKC1VbTatogIqK+PmlGi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"text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Function Space View\n", "\n", "The process we have used to generate the samples is a two stage process. To obtain each function, we first generated a sample from the prior,\n", "$$\n", "\\mathbf{w} \\sim \\mathcal{N}(\\mathbf{0}, \\alpha \\mathbf{I})\n", "$$\n", "then if we compose our basis matrix, $\\boldsymbol{\\Phi}$ from the basis functions associated with each row then we get,\n", "$$\n", "\\mathbf{\\Phi} = \\begin{bmatrix}\\boldsymbol{\\phi}(\\mathbf{x}_1) \\\\ \\vdots \\\\ \\boldsymbol{\\phi}(\\mathbf{x}_n)\\end{bmatrix}\n", "$$\n", "then we can write down the vector of function values, as evaluated at\n", "$$\n", "\\mathbf{f} = \\begin{bmatrix} f_1 \\\\ \\vdots f_n\\end{bmatrix}\n", "$$\n", "in the form\n", "$$\n", "\\mathbf{f} = \\boldsymbol{\\Phi} \\mathbf{w}.\n", "$$\n", "\n", "Now we can use standard properties of multivariate Gaussians to write down the probability density that is implied over $\\mathbf{f}$. In particular we know that if $\\mathbf{w}$ is sampled from a multivariate normal (or multivariate Gaussian) with covariance $\\alpha \\mathbf{I}$ and zero mean, then assuming that $\\boldsymbol{\\Phi}$ is a deterministic matrix (i.e. it is not sampled from a probability density) then the vector $\\mathbf{f}$ will also be distributed according to a zero mean multivariate normal as follows,\n", "$$\n", "\\mathbf{f} \\sim \\mathcal{N}(\\mathbf{0},\\alpha \\boldsymbol{\\Phi} \\boldsymbol{\\Phi}^\\top).\n", "$$\n", "\n", "The question now is, what happens if we sample $\\mathbf{f}$ directly from this density, rather than first sampling $\\mathbf{w}$ and then multiplying by $\\boldsymbol{\\Phi}$. Let's try this. First of all we define the covariance as\n", "$$\n", "\\mathbf{K} = \\alpha \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top.\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = alpha*np.dot(Phi_pred, Phi_pred.T)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can use the `np.random.multivariate_normal` command for sampling from a multivariate normal with covariance given by $\\mathbf{K}$ and zero mean," ] }, { "cell_type": "code", "collapsed": false, "input": [ "for i in np.arange(10):\n", " f_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), f_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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E3LlzmT9/PhdffDFpaWln9sR+P7z3npjBO53w4INw661Hl41HCFLoJb0aVRX1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\no5D3LvhtXW0qMXRdpqO7w5E0TGYTneZOOs2dmMw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"text": [ "" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The samples appear very similar to those which we obtained indirectly. That is no surprise because they are effectively drawn from the same mutivariate normal density. However, when sampling $\\mathbf{f}$ directly we created the covariance for $\\mathbf{f}$. We can visualise the form of this covaraince in an image in python with a colorbar to show scale." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(K, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 10, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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TD4Dvko0MzwYOkHWKR4E3A78/6+R7qqwsM2fT3HxaxL2FtIeptIg6u0b0kdrQVT2LuLKM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\n/xD4CvBZ4KdDFS3q+3JJUp/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"text": [ "" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This image is the covariance expressed between different points on the function. In regression we normally also add independent Gaussian noise to obtain our observations $\\mathbf{y}$,\n", "$$\n", "\\mathbf{y} = \\mathbf{f} + \\boldsymbol{\\epsilon}\n", "$$\n", "where the noise is sampled from an independent Gaussian distribution with variance $\\sigma^2$,\n", "$$\n", "\\epsilon \\sim \\mathcal{N}(\\mathbf{0}, \\sigma^2 \\mathbf{I}).\n", "$$\n", "we can use properties of Gaussian variables, i.e. the fact that sum of two Gaussian variables is also Gaussian, and that it's covariance is given by the sum of the two covariances, whilst the mean is given by the sum of the means, to write down the marginal likelihood,\n", "$$\n", "\\mathbf{y} \\sim \\mathcal{N}(\\mathbf{0}, \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top + \\sigma^2\\mathbf{I}).\n", "$$\n", "Sampling directly from this density gives us the noise corrupted functions," ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = alpha*np.dot(Phi_pred, Phi_pred.T) + sigma2*np.eye(x_pred.size)\n", "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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1kfniC3jjDdi8WS0b6DR9rKVWUial4NXVi4hna6nOegY0i0bCQwkU/lSIdx9v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\n1cB9wxQ8mOCGFs3a5O4sOppNPaambIfKMkcGBbGyt7fPr7S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17IfKmpXhE+mDn51/RtCb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"text": [ "" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the effect of our noise term is to roughen the sampled functions, we can also increase the variance of the noise to see a different effect," ] }, { "cell_type": "code", "collapsed": false, "input": [ "sigma2 = 1.\n", "K = alpha*np.dot(Phi_pred, Phi_pred.T) + sigma2*np.eye(x_pred.size)\n", "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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fqEr0FkHEFx32Vx+ghdIr/3VsjB/OzBlhfOc7CLfcQnlTEO1o4zqiU6lRfv7z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9Q4nU5cLjHng4f/gHb4Fn44NcV4Nss3Vq2itbUVed9X+Hzzv5DNwrE/jvJZ6V6+80Ajv/gFfOR7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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Function Space Reflection\n", "\n", "How do you include the noise term when sampling in the weight space point of view?\n", "\n", "## Gaussian Process\n", "\n", "In our [session on Bayesian regression](./week7.ipynb) we sampled from the prior over paraemters. Through the properties of multivariate Gaussian densities this prior over parameters implies a particular density for our data observations, $\\mathbf{y}$. In this session we sampled directly from this distribution for our data, avoiding the intermediate weight-space representation. This is the approach taken by *Gaussian processes*. In a Gaussian process you specify the *covariance function* directly, rather than *implicitly* through a basis matrix and a prior over parameters. Gaussian processes have the advantage that they can be *nonparametric*, which in simple terms means that they can have *infinite* basis functions. In the lectures we introduced the *exponentiated quadratic* covariance, also known as the RBF or the Gaussian or the squared exponential covariance function. This covariance function is specified by\n", "$$\n", "k(\\mathbf{x}, \\mathbf{x}^\\prime) = \\alpha \\exp\\left( -\\frac{\\left\\Vert \\mathbf{x}-\\mathbf{x}^\\prime\\right\\Vert^2}{2\\ell^2}\\right).\n", "$$\n", "where $\\left\\Vert\\mathbf{x} - \\mathbf{x}^\\prime\\right\\Vert^2$ is the squared distance between the two input vectors \n", "$$\n", "\\left\\Vert\\mathbf{x} - \\mathbf{x}^\\prime\\right\\Vert^2 = (\\mathbf{x} - \\mathbf{x}^\\prime)^\\top (\\mathbf{x} - \\mathbf{x}^\\prime) \n", "$$\n", "Let's build a covariance matrix based on this function. First we define the form of the covariance function," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def exponentiated_quadratic(x, x_prime, variance, lengthscale):\n", " squared_distance = ((x-x_prime)**2).sum()\n", " return variance*np.exp((-0.5*squared_distance)/lengthscale**2)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use this to compute *directly* the covariance for $\\mathbf{f}$ at the points given by `x_pred`. Let's define a new function `K()` which does this," ] }, { "cell_type": "code", "collapsed": false, "input": [ "def compute_kernel(X, X2, kernel, **kwargs):\n", " K = np.zeros((X.shape[0], X2.shape[0]))\n", " for i in np.arange(X.shape[0]):\n", " for j in np.arange(X2.shape[0]):\n", " K[i, j] = kernel(X[i, :], X2[j, :], **kwargs)\n", " \n", " return K" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can image the resulting covariance," ] }, { "cell_type": "code", "collapsed": false, "input": [ "K = compute_kernel(x_pred, x_pred, exponentiated_quadratic, variance=1., lengthscale=10.)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To visualise the covariance between the points we can use the `imshow` function in matplotlib." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(K, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 16, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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G6FGuGpUk1fdIQ0dvmwh/rxwGtvZ4/18DXy6O/wVcGmu6gVCSlJvtwBbgSuDV\nwHmV979FCJZPA3YDb43dzEAoSarvZEPH6SaLx33AUWAPsLFyzZ3AbPF8F3BFrOnOEUqdZWpFWZLU\nispnG0mtgKVtx3Y+8PXhLm259cCh0vlB4HJCwOvl3wOfiN3QQChJqq+dq0avBF4JPDt2kYFQklRf\nqkD4wxn40Uzsiv3AjaXzdcBtPa57KvAe4AXAA7EbGgglMXjH5PYMlY670n0stQLSVLpvJLViEtgZ\nuU9bPG46HPPu21a9Yn7ubxNwP7AZqF50EfARwurR+wZ9pYFQklTfaKtP3ADsAFYBNwHHCKtIKV7/\nfeCxhB7hfOs29LuZgVCSlJu9wNrKaztKz/9dcQzFQChJqs/qE5KWl0y3Y4vNF0KaSveR1ApIU+k+\nmloBw1e6nyjNF/5C5DMdZyCUJNXXzvSJoRgIJUn1GQglLW957EIz7tQKSFPgN5ZaAYvYhab0n/+X\nWMW3I5d2mYFQklTfaNMnknLTbUlSp9kjlCTVZ/qEpO7INLUCklS6j6ZWQJJK97HUChh+O7bydf+I\nNeyrfqcAA6EkKQVXjUqSOi3jQOhiGUlSp9kjlFRTHjmGkFGl+0iOYfU7YzmGE48/9d5qHs8f0CDT\nJyRJypM9QklSfaZPSBJ0PbUC0lS6j6VWwBIr3T/mwupdVTAQSpLqy3jVqIFQklRfxoHQxTKSpE6z\nRyipQaZWlCVJrah8NrodWzmz4pxeLUzI9AlJkvJkj1CSVJ/pE5I0yKCxs/YMlY670n0stQIWUem+\nPDR6ft9mdZ6BUJJUX8arRg2EkqT6Mg6ELpaRJHWaPUJJY5Lpdmyx+UJIU+k+kloBi6h0X7rnxBP7\nNiUN0yckScqTPUJJUn2mT0hSXXnsQjPu1AoYfheaudLnzvyHvl+dxtzgS9rKoVFJUqcZCCVJnWYg\nlCR1mnOEkloo09QKSFLpPppaAUNvx1Z+b02kWV1nj1CS1GkGQklSp40oEB4Zzddk6ci4G9BiR8bd\ngJY7Mu4GtNi9425ABz3S0NHTJsLI72Fga4/3nwLcCfwUeN2glo9ojvAIMDWar8rOEfxt+jmCv03M\nEbrz+yx2zvDrwEVL/7plWOl+VaQpGdoObAGOAruBncCx0vs/JATI3xjmZi6WkSQlMLLyE5PF477i\ncQ+wEdhVuuYHxXHVMDd0jlCSlJP1wKHS+UHg8jo3nKjVnMFmgCsa/g5J0nD2AtMN3HcOZhPd6rPA\nHaXzd8IZ5W24AAAEl0lEQVTCWHUl8HvAy4vz64ALgLf2uNnbgB8D74p9Y9NDo9MN31+StKw8tzjm\nvbN6wX7gxtL5OuC2Ot/oHKEkKYGRzRHOdz03AfcDm4Ftfa4datSz6aFRSdLyNwffa+jW/xhOj1VX\nAO8hLIa9qTi2FO/tKD60n7Dg9ufAg8BlhGHS0xgIJUl1jToQJtX0qtFBSY9dciHwGUKC0wzwiuL1\nNcCthC7+x4BzxtG4FlkBHAA+UZz7+wRnA+8HvklYJbcRf5uyVwF/C3wBeHfxmr/PSI00oT6ppgPh\nfNLjlcCrgfMa/r42ewR4LWFi96XAOwj/R72e8H/US4DvEFZAddlrCP/Qz5f59PcJthF+h6cWxyH8\nbeY9FngLYa5oPXAp8Hz8fTSkJgNhOenxKKeSHrvqe8CXiufHCD3D9cAG4L2E/SFuodu/0ROBFwF/\nyamhEH+f4ErgDwhbRp0gLBjwtwl+QvjfyyRwJnAW8AD+PiN2oqGjeU0GwuRJj8vIkwk9w8+z8Hc6\nRPg/b1f9KfAGwuT2PH+f8AfCauBm4G7gTYR/8P1tgp8Qen9HCH9wfo7wO/n7aCjuLDN6a4APEYZJ\nf4wLluZdDXyfMD9Y/k38fUIQvBT4CCE3dx3wW/jbzDuf8EfCZYTNV59F+N+Tv89IOUfYy37CDuDz\n1gF3Nfh9OVhF+MfsA4RJfAi/09ri+drivIueDbwE+DvCBrrPI/xO/j5wH6GcwicIvZ+dwAvwt5m3\ngfBvy32EzZY/TMjI9vcZKYdGeyknPU4RJrLvbvD72m6CMF/xNU6taoPwm1xLGOq6lu7+sfAWwsra\nXwZ+G/g08Dv4+8w7TJjjegxhI+Hb8beZ91ngmYRFM2cALySsSfD30VCaHhq9gZDceDvw5ywsk9E1\nzwFeSejpHCiOFxCGdC4i/MV/ASFJVKdWjfr7BK8nrML+ImHBzAfxt5l3nLAK+6OETSq/TEhV8vcZ\nqXyHRh1DlyTVNRf+/mjC06DhWOVeo5KkBEa212hyrhqVJHWaPUJJUgKjmc9rgoFQkpSAQ6OSJGXJ\nHqEkKYF8h0btEUqSOs0eoSQpAecIJUnKkj1CSVIC+c4RGgglSQk4NCpJUpbsEUqSEsh3aNQeoSSp\n0+wRSpISsEcoSVKW7BFKkhLId9WogVCSlIBDo5IkZckeoSQpgXyHRu0RSpI6zR6hJCkB5wglScqS\nPUJJUgL5zhEaCCVJCTg0KklSluwRSpISyHdo1B6hJCk3m4BvAIeBrX2u+UPgW8AXgKfEbmaPUJKU\nwEjnCLcDW4CjwG5gJ3Cs9P4G4LnAM4HnA38CXN3vZvYIJUk5mSwe9xEC4R5gY+WajcBfAz8iBMm1\nsRsaCCVJCZxo6DjNeuBQ6fwgcHnlmg3F6/N+AFzcr+UOjUqSEnh7Uzd+cAmfmSiOsrl+F9sjlCTV\nNdHgcW7lu/azcPHLOuCuyjV3A5eVzs8nLJyRJGlZOEBYOTpFGCY9r/L+BuAO4HHAK4BPjrJxkiQ1\n7QpC+sR9wH8sXttSHPPeCfwdIX0iulhGkiRJkiRJkiRJkiRJkiRJkiRJkiQtX/8fn+UmQsLVgQ4A\nAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can sample functions from the marginal likelihood." ] }, { "cell_type": "code", "collapsed": false, "input": [ "for i in xrange(10):\n", " y_sample = np.random.multivariate_normal(mean=np.zeros(x_pred.size), cov=K)\n", " plt.plot(x_pred.flatten(), y_sample.flatten())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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MpKYCRkaAu7t21j9yBJg4sUxLZGdn4+DBgzAxMcGIESNw/vx5pKSkqL2eRCZB40ON4RTj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iba1qR7OqebjH7WUCm29riDtuWtjO67/PlRR25z+lVHiXyeUamuIxERERERERERERER\nERERERERERERERERERGR1PofSXipsKWGF48AAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise: Moving Parameters\n", "\n", "Have a play with the parameters for this covariance function (the lengthscale and the variance) and see what effects the parameters have on the types of functions you observe." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gaussian Process\n", "\n", "The Gaussian process perspective takes the marginal likelihood of the data to be a joint Gaussian density with a covariance given by $\\mathbf{K}$. So the model likelihood is of the form,\n", "$$\n", "p(\\mathbf{y}|\\mathbf{X}) = \\frac{1}{(2\\pi)^{\\frac{n}{2}}|\\mathbf{K}|^{\\frac{1}{2}}} \\exp\\left(-\\frac{1}{2}\\mathbf{y}^\\top \\left(\\mathbf{K}+\\sigma^2 \\mathbf{I}\\right)^{-1}\\mathbf{y}\\right)\n", "$$\n", "where the input data, $\\mathbf{X}$, influences the density through the covariance matrix, $\\mathbf{K}$ whose elements are computed through the covariance function, $k(\\mathbf{x}, \\mathbf{x}^\\prime)$.\n", "\n", "This means that the negative log likelihood (the objective function) is given by,\n", "$$\n", "E(\\boldsymbol{\\theta}) = \\frac{1}{2} \\log |\\mathbf{K}| + \\frac{1}{2} \\mathbf{y}^\\top \\left(\\mathbf{K} + \\sigma^2\\mathbf{I}\\right)^{-1}\\mathbf{y}\n", "$$\n", "where the *parameters* of the model are also embedded in the covariance function, they include the parameters of the kernel (such as lengthscale and variance), and the noise variance, $\\sigma^2$.\n", "\n", "Let's create a class in python for storing these variables." ] }, { "cell_type": "code", "collapsed": false, "input": [ "class GP():\n", " def __init__(self, X, y, sigma2, kernel, **kwargs):\n", " self.K = compute_kernel(X, X, kernel, **kwargs)\n", " self.X = X\n", " self.y = y\n", " self.sigma2 = sigma2\n", " self.kernel = kernel\n", " self.kernel_args = kwargs\n", " self.update_inverse()\n", " \n", " def update_inverse(self):\n", " # Preompute the inverse covariance and some quantities of interest\n", " ## NOTE: This is not the correct *numerical* way to compute this! It is for ease of use.\n", " self.Kinv = np.linalg.inv(self.K+self.sigma2*np.eye(self.K.shape[0]))\n", " # the log determinant of the covariance matrix.\n", " self.logdetK = np.linalg.det(self.K+self.sigma2*np.eye(self.K.shape[0]))\n", " # The matrix inner product of the inverse covariance\n", " self.Kinvy = np.dot(self.Kinv, self.y)\n", " self.yKinvy = (self.y*self.Kinvy).sum()\n", "\n", " \n", " def log_likelihood(self):\n", " # use the pre-computes to return the likelihood\n", " return -0.5*(self.K.shape[0]*np.log(2*np.pi) + self.logdetK + self.yKinvy)\n", " \n", " def objective(self):\n", " # use the pre-computes to return the objective function \n", " return -self.log_likelihood()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Making Predictions\n", "\n", "We now have a probability density that represents functions. How do we make predictions with this density? The density is known as a process because it is *consistent*. By consistency, here, we mean that the model makes predictions for $\\mathbf{f}$ that are unaffected by future values of $\\mathbf{f}^*$ that are currently unobserved (such as test points). If we think of $\\mathbf{f}^*$ as test points, we can still write down a joint probability density over the training observations, $\\mathbf{f}$ and the test observations, $\\mathbf{f}^*$. This joint probability density will be Gaussian, with a covariance matrix given by our covariance function, $k(\\mathbf{x}_i, \\mathbf{x}_j)$. \n", "$$\n", "\\begin{bmatrix}\\mathbf{f} \\\\ \\mathbf{f}^*\\end{bmatrix} \\sim \\mathcal{N}\\left(\\mathbf{0}, \\begin{bmatrix} \\mathbf{K} & \\mathbf{K}_\\ast \\\\ \\mathbf{K}_\\ast^\\top & \\mathbf{K}_{\\ast,\\ast}\\end{bmatrix}\\right)\n", "$$\n", "where here $\\mathbf{K}$ is the covariance computed between all the training points, $\\mathbf{K}_\\ast$ is the covariance matrix computed between the training points and the test points and $\\mathbf{K}_{\\ast,\\ast}$ is the covariance matrix computed betwen all the tests points and themselves. To be clear, let's compute these now for our example, using `x` and `y` for the training data (although `y` doesn't enter the covariance) and `x_pred` as the test locations." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# set covariance function parameters\n", "variance = 16.0\n", "lengthscale = 8\n", "# set noise variance\n", "sigma2 = 0.05\n", "\n", "K = compute_kernel(x, x, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "K_star = compute_kernel(x, x_pred, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "K_starstar = compute_kernel(x_pred, x_pred, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we use this structure to visualise the covariance between test data and training data. This structure is how information is passed between trest and training data. Unlike the maximum likelihood formalisms we've been considering so far, the structure expresses *correlation* between our different data points. However, just like the [naive Bayes approach](./week9.ipynb) we now have a *joint density* between some variables of interest. In particular we have the joint density over $p(\\mathbf{f}, \\mathbf{f}^*)$. The joint density is *Gaussian* and *zero mean*. It is specified entirely by the *covariance matrix*, $\\mathbf{K}$. That covariance matrix is, in turn, defined by a covariance function. Now we will visualise the form of that covariance in the form of the matrix,\n", "$$\n", "\\begin{bmatrix} \\mathbf{K} & \\mathbf{K}_\\ast \\\\ \\mathbf{K}_\\ast^\\top & \\mathbf{K}_{\\ast,\\ast}\\end{bmatrix}\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(np.vstack([np.hstack([K, K_star]), np.hstack([K_star.T, K_starstar])]), interpolation='none')\n", "# Add lines for separating training and test data\n", "ax.axvline(x.shape[0]-1, color='w')\n", "ax.axhline(x.shape[0]-1, color='w')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 20, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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m+Wi7MNuTfz7Sbf9jqt83b4WPB85fEMf7He9PsZReUfSmOVu9z0l8O/8hZewt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\n496z7hB9vy3wN6A/8FPgE2A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"text": [ "" ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are four blocks to this color plot. The upper left block is the covariance of the training data with itself, $\\mathbf{K}$. We see some structure here due to the missing data from the first and second world wars. Alongside this covariance (to the right and below) we see the cross covariance between the training and the test data ($\\mathbf{K}_*$ and $\\mathbf{K}_*^\\top$). This is giving us the covariation between our training and our test data. Finally the lower right block The banded structure we now observe is because some of the training points are near to some of the test points. This is how we obtain 'communication' between our training data and our test data. If there is no structure in $\\mathbf{K}_*$ then our belief about the test data simply matches our prior.\n", "\n", "## Conditional Density\n", "\n", "Just as in naive Bayes, we first defined the joint density (although there it was over both the labels and the inputs, $p(\\mathbf{y}, \\mathbf{X})$ and now we need to define *conditional* distributions that answer particular questions of interest. In particular we might be interested in finding out the values of the function for the prediction function at the test data given those at the training data, $p(\\mathbf{f}_*|\\mathbf{f})$. Or if we include noise in the training observations then we are interested in the conditional density for the prediction function at the test locations given the training observations, $p(\\mathbf{f}^*|\\mathbf{y})$. \n", "\n", "As ever all the various questions we could ask about this density can be answered using the *sum rule* and the *product rule*. For the multivariate normal density the mathematics involved is that of *linear algebra*, with a particular emphasis on the *partitioned inverse* or [*block matrix inverse*](http://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion), but they are beyond the scope of this course, so you don't need to worry about remembering them or rederiving them. We are simply writing them here because it is this *conditional* density that is necessary for making predictions.\n", "\n", "The conditional density is also a multivariate normal,\n", "$$\n", "\\mathbf{f}^* | \\mathbf{f} \\sim \\mathcal{N}(\\boldsymbol{\\mu}_f,\\mathbf{C}_f)\n", "$$\n", "with a mean given by\n", "$$\n", "\\boldsymbol{\\mu}_f = \\mathbf{K}_*^\\top \\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1} \\mathbf{y}\n", "$$\n", "and a covariance given by \n", "$$\n", "\\mathbf{C}_f = \\mathbf{K}_{*,*} - \\mathbf{K}_*^\\top \\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1} \\mathbf{K}_\\ast.\n", "$$\n", "Let's compute what those posterior predictions are for the olympic marathon data." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def posterior_f(self, X_test):\n", " K_star = compute_kernel(self.X, X_test, self.kernel, **self.kernel_args)\n", " A = np.dot(self.Kinv, K_star)\n", " mu_f = np.dot(A.T, y)\n", " C_f = K_starstar - np.dot(A.T, K_star)\n", " return mu_f, C_f\n", "\n", "# attach the new method to class GP():\n", "GP.posterior_f = posterior_f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "model = GP(x, y, sigma2, exponentiated_quadratic, variance=variance, lengthscale=lengthscale)\n", "mu_f, C_f = model.posterior_f(x_pred)\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "where for convenience we've defined\n", "\n", "$$\\mathbf{A} = \\left[\\mathbf{K} + \\sigma^2\\mathbf{I}\\right]^{-1}\\mathbf{K}_*.$$ \n", "\n", "We can visualize the covariance of the *conditional*," ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots(figsize=(8,8))\n", "im = ax.imshow(C_f, interpolation='none')\n", "fig.colorbar(im)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 23, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 23 }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we can plot the mean of the conditional" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(x, y, 'rx')\n", "plt.plot(x_pred, mu_f, 'b-')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 24, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 24 }, { "cell_type": "markdown", "metadata": {}, "source": [ "as well as the associated error bars. These are given (similarly to the Bayesian parametric model from the last lab) by the standard deviations of the marginal posterior densities. The marginal posterior variances are given by the diagonal elements of the posterior covariance," ] }, { "cell_type": "code", "collapsed": false, "input": [ "var_f = np.diag(C_f)[:, None]\n", "std_f = np.sqrt(var_f)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 25 }, { "cell_type": "markdown", "metadata": {}, "source": [ "They can be added to the underlying mean function to give the error bars," ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.plot(x, y, 'rx')\n", "plt.plot(x_pred, mu_f, 'b-')\n", "plt.plot(x_pred, mu_f+2*std_f, 'b--')\n", "plt.plot(x_pred, mu_f-2*std_f, 'b--')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 26, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 26 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This gives us a prediction from the Gaussian process. Remember machine learning is \n", "$$\n", "\\text{data} + \\text{model} \\rightarrow \\text{prediction}.\n", "$$\n", "Here our data is from the olympics, and our model is a Gaussian process with two parameters. The assumptions about the world are encoded entirely into our Gaussian process covariance. The GP covariance assumes that the function is highly smooth, and that correlation falls off with distance (scaled according to the length scale, $\\ell$). The model sustains the uncertainty about the function, this means we see an increase in the size of the error bars during periods like the 1st and 2nd World Wars when no olympic marathon was held. \n", "\n", "## Exercises\n", "\n", "Now try changing the parameters of the covariance function (and the noise) to see how the predictions change.\n", "\n", "Now try sampling from this conditional density to see what your predictions look like. What happens if you sample from the conditional density in regions a long way into the future or the past? How does this compare with the results from the polynomial model?\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Importance of the Covariance Function\n", "\n", "The covariance function encapsulates our assumptions about the data. The equations for the distribution of the prediction function, given the training observations, are highly sensitive to the covariation between the test locations and the training locations as expressed by the matrix $\\mathbf{K}_*$. We defined a matrix $\\mathbf{A}$ which allowed us to express our conditional mean in the form,\n", "$$\n", "\\boldsymbol{\\mu}_f = \\mathbf{A}^\\top \\mathbf{y},\n", "$$\n", "where $\\mathbf{y}$ were our *training observations*. In other words our mean predictions are always a linear weighted combination of our *training data*. The weights are given by computing the covariation between the training and the test data ($\\mathbf{K}_*$) and scaling it by the inverse covariance of the training data observations, $\\left[\\mathbf{K} + \\sigma^2 \\mathbf{I}\\right]^{-1}$. This inverse is the main computational object that needs to be resolved for a Gaussian process. It has a computational burden which is $O(n^3)$ and a storage burden which is $O(n^2)$. This makes working with Gaussian processes computationally intensive for the situation where $n>10,000$. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo('ewJ3AxKclOg')" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", " \n", " " ], "metadata": {}, "output_type": "pyout", "prompt_number": 27, "text": [ "" ] } ], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Improving the Numerics\n", "\n", "In practice we shouldn't be using matrix inverse directly to solve the GP system. One more stable way is to compute the *Cholesky decomposition* of the kernel matrix. The log determinant of the covariance can also be derived from the Cholesky decomposition." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def update_inverse(self):\n", " # Perform Cholesky decomposition on matrix\n", " self.R = sp.linalg.cholesky(self.K + self.sigma2*self.K.shape[0])\n", " # compute the log determinant from Cholesky decomposition\n", " self.logdetK = 2*np.log(np.diag(self.R)).sum()\n", " # compute y^\\top K^{-1}y from Cholesky factor\n", " self.Rinvy = sp.linalg.solve_triangular(self.R, self.y)\n", " self.yKinvy = (self.Rinvy**2).sum()\n", " \n", " # compute the inverse of the upper triangular Cholesky factor\n", " self.Rinv = sp.linalg.solve_triangular(self.R, np.eye(self.K.shape[0]))\n", " self.Kinv = np.dot(self.Rinv, self.Rinv.T)\n", "\n", "GP.update_inverse = update_inverse" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 29 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Capacity Control\n", "\n", "Gaussian processes are sometimes seen as part of a wider family of methods known as kernel methods. Kernel methods are also based around covariance functions, but in the field they are known as Mercer kernels. Mercer kernels have interpretations as inner products in potentially infinite dimensional Hilbert spaces. This interpretation arises because, if we take $\\alpha=1$, then the kernel can be expressed as\n", "$$\n", "\\mathbf{K} = \\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^\\top \n", "$$\n", "which imples the elements of the kernel are given by,\n", "$$\n", "k(\\mathbf{x}, \\mathbf{x}^\\prime) = \\boldsymbol{\\phi}(\\mathbf{x})^\\top \\boldsymbol{\\phi}(\\mathbf{x}^\\prime).\n", "$$\n", "So we see that the kernel function is developed from an inner product between the basis functions. Mercer's theorem tells us that any valid *positive definite function* can be expressed as this inner product but with the caveat that the inner product could be *infinite length*. This idea has been used quite widely to *kernelize* algorithms that depend on inner products. The kernel functions are equivalent to covariance functions and they are parameterized accordingly. In the kernel modeling community it is generally accepted that kernel parameter estimation is a difficult problem and the normal solution is to cross validate to obtain parameters. This can cause difficulties when a large number of kernel parameters need to be estimated. In Gaussian process modelling kernel parameter estimation (in the simplest case proceeds) by maximum likelihood. This involves taking gradients of the likelihood with respect to the parameters of the covariance function. \n", "\n", "## Gradients of the Likelihood\n", "\n", "The easiest conceptual way to obtain the gradients is a two step process. The first step involves taking the gradient of the likelihood with respect to the covariance function, the second step involves considering the gradient of the covariance function with respect to its parameters.\n", "\n", "## Overall Process Scale\n", "\n", "In general we won't be able to find parameters of the covariance function through fixed point equations, we will need to do graident based optimization.\n", "\n", "## Capacity Control and Data Fit\n", "\n", "The objective function can be decomposed into two terms, a capacity control term, and a data fit term. The capacity control term is the log determinant of the covariance. The data fit term is the matrix inner product between the data and the inverse covariance." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }