lecture15_latent-variables-independence-clustering/lecture15_final_latent-variables-independence-clustering.ipynb

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    "$$ \\LaTeX \\text{ command declarations here.}\n",
    "\\newcommand{\\R}{\\mathbb{R}}\n",
    "\\renewcommand{\\vec}[1]{\\mathbf{#1}}\n",
    "\\newcommand{\\X}{\\mathcal{X}}\n",
    "\\newcommand{\\D}{\\mathcal{D}}\n",
    "\\newcommand{\\G}{\\mathcal{G}}\n",
    "\\newcommand{\\Parents}{\\mathrm{Parents}}\n",
    "\\newcommand{\\NonDesc}{\\mathrm{NonDesc}}\n",
    "\\newcommand{\\I}{\\mathcal{I}}\n",
    "\\newcommand{\\dsep}{\\text{d-sep}}\n",
    "$$"
   ]
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   "execution_count": 1,
   "metadata": {
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   "source": [
    "from __future__ import division\n",
    "\n",
    "# scientific\n",
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt;\n",
    "import numpy as np;\n",
    "import sklearn as skl;\n",
    "import sklearn.datasets;\n",
    "import sklearn.cluster;\n",
    "\n",
    "# ipython\n",
    "import IPython;\n",
    "\n",
    "# python\n",
    "import os;\n",
    "\n",
    "#####################################################\n",
    "\n",
    "# image processing\n",
    "import PIL;\n",
    "\n",
    "# trim and scale images\n",
    "def trim(im, percent=100):\n",
    "    print(\"trim:\", percent);\n",
    "    bg = PIL.Image.new(im.mode, im.size, im.getpixel((0,0)))\n",
    "    diff = PIL.ImageChops.difference(im, bg)\n",
    "    diff = PIL.ImageChops.add(diff, diff, 2.0, -100)\n",
    "    bbox = diff.getbbox()\n",
    "    if bbox:\n",
    "        x = im.crop(bbox)\n",
    "        return x.resize(((x.size[0]*percent)//100, (x.size[1]*percent)//100), PIL.Image.ANTIALIAS);\n",
    "\n",
    "\n",
    "#####################################################\n",
    "\n",
    "# daft (rendering PGMs)\n",
    "import daft;\n",
    "\n",
    "# set to FALSE to load PGMs from static images\n",
    "RENDER_PGMS = False;\n",
    "\n",
    "# decorator for pgm rendering\n",
    "def pgm_render(pgm_func):\n",
    "    def render_func(path, percent=100, render=None, *args, **kwargs):\n",
    "        print(\"render_func:\", percent);\n",
    "        # render\n",
    "        render = render if (render is not None) else RENDER_PGMS;\n",
    "        \n",
    "        if render:\n",
    "            print(\"rendering\");\n",
    "            # render\n",
    "            pgm = pgm_func(*args, **kwargs);\n",
    "            pgm.render();\n",
    "            pgm.figure.savefig(path, dpi=300);\n",
    "            \n",
    "            # trim\n",
    "            img = trim(PIL.Image.open(path), percent);\n",
    "            img.save(path, 'PNG');\n",
    "        else:\n",
    "            print(\"not rendering\");\n",
    "        \n",
    "        # error\n",
    "        if not os.path.isfile(path):\n",
    "            raise \"Error:  Graphical model image %s not found.  You may need to set RENDER_PGMS=True.\";\n",
    "        \n",
    "        # display\n",
    "        return IPython.display.Image(filename=path);#trim(PIL.Image.open(path), percent);\n",
    "    \n",
    "    return render_func;\n",
    "\n",
    "######################################################"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "# EECS 545:  Machine Learning\n",
    "\n",
    "## Lecture 15:  Latent Variables, d-Separation, K-Means\n",
    "\n",
    "- Instructor:  **Jacob Abernethy**\n",
    "- Date:  March 14, 2016\n",
    "\n",
    "*Lecture Exposition:*  Saket & Ben"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "## References\n",
    "\n",
    "- **[MLAPP]** Murphy, Kevin. [*Machine Learning:  A Probabilistic Perspective*](https://mitpress.mit.edu/books/machine-learning-0).  2012.\n",
    "- **[PRML]** Bishop, Christopher.  [*Pattern Recognition and Machine Learning*](http://research.microsoft.com/en-us/um/people/cmbishop/prml/).  2006.\n",
    "- **[Koller & Friedman 2009]** Koller, Daphne and Nir Friedman.  [*Probabilistic Graphical Models*](https://mitpress.mit.edu/books/probabilistic-graphical-models).  2009."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
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   },
   "source": [
    "## Outline\n",
    "\n",
    "- Probabilistic Graphical Models\n",
    "    - Latent Variable Models\n",
    "    - d-separation in Bayesian Networks\n",
    "- Clustering & Mixture Models\n",
    "    - K-Means Clustering"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Latent Variable Models\n",
    "\n",
    "> Uses material from **[MLAPP]** §10.1-10.4, §11.1-11.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "### Latent Variable Models\n",
    "\n",
    "In general, the goal of probabilistic modeling is to \n",
    "> Use what we know to make *inferences* about what we don't know.\n",
    "\n",
    "Graphical models provide a natural framework for this problem.\n",
    "- Assume unobserved variables are correlated due to the influence of unobserved **latent variables**.\n",
    "- Latent variables encode beliefs about the generative process.\n",
    "\n",
    "In a graphical model, we will often **shade in** the observed variables to distinguish them from hidden variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Gaussian Mixture Models\n",
    "\n",
    "This dataset is hard to explain with a single distribution.\n",
    "- Underlying density is complicated overall...\n",
    "- But it's clearly three Gaussians!"
   ]
  },
  {
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   "execution_count": 2,
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    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
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    {
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GrdVuCJ1n0orUpAViZ8+ayzCOjdWpUmlEJjhLIhhRlfS9dvs7SztIRG23vLxM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\nB6KRDPbzICZCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7ff5c4d15278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X, y = skl.datasets.make_blobs(1000, cluster_std=[1.0, 2.5, 0.5], random_state=170)\n",
    "plt.scatter(X[:,0], X[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Mixture Models\n",
    "\n",
    "Instead, introduce a latent **cluster label** $z_j \\in [K]$ for each datapoint $x_j$,\n",
    "$$\n",
    "\\begin{align}\n",
    "z_j &\\sim \\mathrm{Cat}(\\pi_1, \\dots, \\pi_K) \n",
    "    & \\forall\\, j=1,\\dots,N \\\\\n",
    "x_j \\mid z_j &\\sim \\mathcal{N}(\\mu_{z_j}, \\Sigma_{z_j})\n",
    "    & \\forall\\, j=1,\\dots,N \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "This allows us to explain a complicated density as a **mixture** of simpler densities:\n",
    "$$\n",
    "P(x | \\mu, \\Sigma) = \\sum_{k=1}^K \\pi_k \\mathcal{N}(x | \\mu_k, \\Sigma_k)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Mixture Models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_gmm():\n",
    "    pgm = daft.PGM([4,4], origin=[-2,-1], node_unit=0.8, grid_unit=2.0);\n",
    "    # nodes\n",
    "    pgm.add_node(daft.Node(\"pi\", r\"$\\pi$\", 0, 1));\n",
    "    pgm.add_node(daft.Node(\"z\", r\"$Z_j$\", 0.7, 1));\n",
    "    pgm.add_node(daft.Node(\"x\", r\"$X_j$\", 1.3, 1, observed=True));\n",
    "    pgm.add_node(daft.Node(\"mu\", r\"$\\mu$\", 0.7, 0.3));\n",
    "    pgm.add_node(daft.Node(\"sigma\", r\"$\\Sigma$\", 1.3, 0.3));\n",
    "    \n",
    "    # edges\n",
    "    pgm.add_edge(\"pi\", \"z\", head_length=0.08);\n",
    "    pgm.add_edge(\"z\", \"x\", head_length=0.08);\n",
    "    pgm.add_edge(\"mu\", \"x\", head_length=0.08);\n",
    "    pgm.add_edge(\"sigma\", \"x\", head_length=0.08);\n",
    "    \n",
    "    pgm.add_plate(daft.Plate([0.4,0.8,1.3,0.5], label=r\"$\\qquad\\qquad\\qquad\\;\\; N$\",\n",
    "    shift=-0.1))\n",
    "    \n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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rNkJ8AOmCgIAAloMH365JqVSK/v374/379wB+rF1NmTIFU6ZMyfV7X79+xdSp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\n9y7kcjlEIhFatWqlbWn5QlU/FzR+jguEdt1kU69ePSxfvpw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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_gmm(\"images/pgm/pgm-gmm.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Hidden Markov Models\n",
    "\n",
    "Noisy observations $X_k$ generated from hidden Markov chain $Y_k$.\n",
    "$$\n",
    "P(\\vec{X}, \\vec{Y}) = P(Y_1) P(X_1 \\mid Y_1) \\prod_{k=2}^N \\left(P(Y_k \\mid Y_{k-1}) P(X_k \\mid Y_k)\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_hmm():\n",
    "    pgm = daft.PGM([7, 7], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"Y1\", r\"$Y_1$\", 1, 3.5))\n",
    "    pgm.add_node(daft.Node(\"Y2\", r\"$Y_2$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"Y3\", r\"$\\dots$\", 3, 3.5, plot_params={'ec':'none'}))\n",
    "    pgm.add_node(daft.Node(\"Y4\", r\"$Y_N$\", 4, 3.5))\n",
    "\n",
    "    pgm.add_node(daft.Node(\"x1\", r\"$X_1$\", 1, 2.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"x2\", r\"$X_2$\", 2, 2.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"x3\", r\"$\\dots$\", 3, 2.5, plot_params={'ec':'none'}))\n",
    "    pgm.add_node(daft.Node(\"x4\", r\"$X_N$\", 4, 2.5, observed=True))\n",
    "\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"Y1\", \"Y2\", head_length=0.08)\n",
    "    pgm.add_edge(\"Y2\", \"Y3\", head_length=0.08)\n",
    "    pgm.add_edge(\"Y3\", \"Y4\", head_length=0.08)\n",
    "\n",
    "    pgm.add_edge(\"Y1\", \"x1\", head_length=0.08)\n",
    "    pgm.add_edge(\"Y2\", \"x2\", head_length=0.08)\n",
    "    pgm.add_edge(\"Y4\", \"x4\", head_length=0.08)\n",
    "    \n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vk/rLR0VF4eDBgwAArVaLzp07Q6FQYNu2bVbZtc9Q+gpY3km2Tk+fPkXLli3R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\nHzTLzZbIsxpCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_hmm(\"images/pgm/hmm.png\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Unsupervised Learning\n",
    "\n",
    "Latent variables are fundamental to unsupervised and deep learning.\n",
    "- Serve as a **bottleneck**\n",
    "- Compute a **compressed** representation of data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_unsupervised():\n",
    "    pgm = daft.PGM([6, 6], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"d1\", r\"$Z_1$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"di\", r\"$Z_2$\", 3, 3.5))\n",
    "    pgm.add_node(daft.Node(\"dn\", r\"$Z_3$\", 4, 3.5))\n",
    "\n",
    "    pgm.add_node(daft.Node(\"f1\", r\"$X_1$\", 1, 2.50, observed=True))\n",
    "    pgm.add_node(daft.Node(\"fi-1\", r\"$X_2$\", 2, 2.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"fi\", r\"$X_3$\", 3, 2.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"fi+1\", r\"$X_4$\", 4, 2.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"fm\", r\"$X_N$\", 5, 2.5, observed=True))\n",
    "\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"d1\", \"f1\", head_length=0.08)\n",
    "    pgm.add_edge(\"d1\", \"fi-1\", head_length=0.08)\n",
    "    pgm.add_edge(\"d1\", \"fi\", head_length=0.08)\n",
    "    pgm.add_edge(\"d1\", \"fi+1\", head_length=0.08)\n",
    "    pgm.add_edge(\"d1\", \"fm\", head_length=0.08)\n",
    "\n",
    "    pgm.add_edge(\"di\", \"f1\", head_length=0.08)\n",
    "    pgm.add_edge(\"di\", \"fi-1\", head_length=0.08)\n",
    "    pgm.add_edge(\"di\", \"fi\", head_length=0.08)\n",
    "    pgm.add_edge(\"di\", \"fi+1\", head_length=0.08)\n",
    "    pgm.add_edge(\"di\", \"fm\", head_length=0.08)\n",
    "\n",
    "    pgm.add_edge(\"dn\", \"f1\", head_length=0.08)\n",
    "    pgm.add_edge(\"dn\", \"fi-1\", head_length=0.08)\n",
    "    pgm.add_edge(\"dn\", \"fi\", head_length=0.08)\n",
    "    pgm.add_edge(\"dn\", \"fi+1\", head_length=0.08)\n",
    "    pgm.add_edge(\"dn\", \"fm\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Ni4ujiTcVRL+/NmzYUDaTxN65cwf+/v548803i112+/btmDhxYlFfrVKlCrZv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\nPaxYLIZIJCp8v2ZnZyMrKwtyubxC97Dm5ub4+eef8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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_unsupervised(\"images/pgm/unsupervised.png\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Other Latent Variable Models\n",
    "\n",
    "Many other models in machine learning involve latent variables:\n",
    "\n",
    "- Neural Networks / Multilayer Perceptrons\n",
    "- Restricted Boltzmann Machines\n",
    "- Deep Belief Networks\n",
    "- Probabilistic PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Latent Variable Models:  Complexity\n",
    "\n",
    "Latent variable models exhibit **emergent complexity**.\n",
    "- Although each conditional distribution is simple,\n",
    "- The joint distribution is capable of modeling complex interactions.\n",
    "\n",
    "However, latent variables make learning difficult.\n",
    "- Inference is challening in models with latent variables.\n",
    "- They can introduce new dependencies between observed variables."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Bayesian Networks\n",
    "## Part II:  Inference, Learning, and d-Separation\n",
    "\n",
    "> Uses material from **[Koller & Friedman 2009]** Chapter 3, **[MLAPP]** Chapter 10, and **[PRML]** §8.2.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  Terminology\n",
    "\n",
    "Typically, our models will have\n",
    "- Observed variables $X$\n",
    "- Hidden variables $Z$\n",
    "- Parameters $\\theta$\n",
    "\n",
    "Occasionally, we will distinguish between **inference** and **learning**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  Inference\n",
    "\n",
    "**Inference**:  Estimate hidden variables $Z$ from observed variables $X$.\n",
    "$$\n",
    "P(Z | X,\\theta) = \\frac{P(X,Z | \\theta)}{P(X|\\theta)}\n",
    "$$\n",
    "\n",
    "- Denominator $P(X|\\theta)$ is sometimes called the probability of the **evidence**.\n",
    "- Occasionally we care only about a subset of the hidden variables, and marginalize out the rest."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  Learning\n",
    "\n",
    "**Learning:** Estimate parameters $\\theta$ from observed data $X$.\n",
    "$$\n",
    "P(\\theta \\mid X) = \\sum_{z \\in Z} P(\\theta, z \\mid X) = \\sum_{z \\in Z} P(\\theta \\mid z, X) P(z \\mid X)\n",
    "$$\n",
    "\n",
    "To Bayesians, parameters *are* hidden variables, so inference and learning are equivalent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  Probability Queries\n",
    "\n",
    "In general, it is useful to compute $P(A|B)$ for arbitrary collections $A$ and $B$ of variables.\n",
    "- Both inference and learning take this form.\n",
    "\n",
    "To accomplish this, we must understand the **independence structure** of any given graphical model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Review:  Local Independencies\n",
    "\n",
    "Every Bayesian Network $\\G$ encodes a set $\\I_\\ell(\\G)$ of **local independence assumptions**:\n",
    "\n",
    "> For each variable $X_k$, we have $(X_k \\perp \\NonDesc_\\G(X_k) \\mid \\Parents_\\G(X_k))$\n",
    "\n",
    "Every node $X_k$ is conditionally independent of its nondescendants given its parents.\n",
    "\n",
    "> For arbitrary sets of variables, when does $(A \\perp B \\mid C)$ hold?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Review:  I-Maps\n",
    "\n",
    "If $P$ satisfies the independence assertions made by $\\G$, we say that\n",
    "- $\\G$ is an **I-Map** for $P$\n",
    "- or that $P$ **satisfies** $\\G$.\n",
    "\n",
    "Any distribution satisfying $\\G$ shares common structure.\n",
    "- We will exploit this structure in our algorithms\n",
    "- This is what makes graphical models so **powerful**!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Review:  Factorization Theorem\n",
    "\n",
    "Last time, we proved that for any $P$ satisfying $\\G$,\n",
    "$$\n",
    "P(X_1, \\dots, X_N) = \\prod_{k=1}^N P(X_k \\mid \\Parents_\\G(X_k))\n",
    "$$\n",
    "\n",
    "If we understand independence structure, we can factorize arbitrary conditional distributions:\n",
    "$$\n",
    "P(A_1, \\dots, A_n \\mid B_1, \\dots, B_m) = \\;?\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "###  Question 1:  Is $(A \\perp B)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question1():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"c\", \"a\", head_length=0.08)\n",
    "    pgm.add_edge(\"c\", \"b\", head_length=0.08)\n",
    "\n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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PZOOjRo1CZmamAhERVQ7Lp6Qp7FqjTux2Q56CM0XSFHatUSd2uyFPwaRImsGu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\nIB21+3/6pXbqNnpsqwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question1(\"images/pgm/question1.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer 1:  No!\n",
    "\n",
    "No!  $A$ and $B$ are not marginally independent.\n",
    "- Note $C$ is not shaded, so we don't observe it.\n",
    "\n",
    "In general, \n",
    "\n",
    "$$\n",
    "P(A,B) = \\sum_{c \\in C} P(A,B,c) = \\sum_{c \\in C} P(A|C)P(B|C)P(C) \\neq P(A)P(B)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Question 2:  Is $(A \\perp B \\mid C)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question2():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5, \n",
    "                           observed=True))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"c\", \"a\", head_length=0.08)\n",
    "    pgm.add_edge(\"c\", \"b\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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U9SYlJeHvv/9Gr169nLI+d7P+cVO1alWFIiEqGJMiqY71wTIzM1OhSBzz8MMPS5Zv3rzp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\nSUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question2(\"images/pgm/question2.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer 2:  Yes!\n",
    "\n",
    "Yes!  $(A \\perp B | C)$ follows from the local independence properties of Bayesian networks.\n",
    "\n",
    "> Every variable is conditionally independent of its nondescendants given its parents.\n",
    "\n",
    "Observing $C$ *blocks* the path of influence from $A$ to $B$. Or, using factorization theorem:\n",
    "$$\n",
    "\\begin{align}\n",
    "P(A,B|C) & = \\frac{P(A,B,C)}{P(C)}  \\\\\n",
    "& = \\frac{P(C)P(A|C)P(B|C)}{P(C)} \\\\\n",
    "& = P(A|C)P(B|C)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Question 3:  Is $(A \\perp B)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question3():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"a\", \"c\", head_length=0.08)\n",
    "    pgm.add_edge(\"c\", \"b\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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see5kux79/PwUioSoeEyKpDq2LcG0dpK3bcPv1NRUlyz34sWL2LRpE8aNG+eS5bmbJzRl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\n7WRYWBhq1apV5oN01O7/AVdIbBUJ7wagAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question3(\"images/pgm/question3.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer 3:  No!\n",
    "\n",
    "Again, $C$ is not given, so $A$ and $B$ are dependent.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Question 4:  Is $(A \\perp B \\mid C)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question4():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"a\", \"c\", head_length=0.08)\n",
    "    pgm.add_edge(\"c\", \"b\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Y7YY8BZMiacamTZvw999/y9pPnjzJi8YVVrZsWaxZs0bWPnPmTJw/f979ARE5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\nihWLPUlH7f4fHta+D/jIwtAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question4(\"images/pgm/question4.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer 4:  Yes!\n",
    "\n",
    "Again, observing $C$ *blocks* influence from $A$ to $B$.\n",
    "\n",
    "> Every variable is conditionally independent of its nondescendants given its parents."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Question 5:  Is $(A \\perp B)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question5():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"a\", \"c\", head_length=0.08)\n",
    "    pgm.add_edge(\"b\", \"c\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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W69HHx0ehSIgKxqRIqmPdEkxrJ3lbN/xOSEhwynIvX76MDRs2YNy4cU5Znqu5Q1MGcn9M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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question5(\"images/pgm/question5.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer 5:  Yes!\n",
    "\n",
    "Using the factorization rule,\n",
    "$$\n",
    "P(A,B,C) = P(A)P(B)P(C\\mid A,B)\n",
    "$$\n",
    "Therefore, marginalizing out $C$,\n",
    "$$\n",
    "\\begin{align}\n",
    "P(A,B) & = \\sum_{c \\in C} P(A,B,c)  \\\\\n",
    "& = \\sum_{c \\in C} P(A)P(B) P(c \\mid A,B) \\\\\n",
    "& = P(A)P(B) \\sum_{c \\in C} P(c \\mid A,B) = P(A)P(B)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Question 6:  Is $P(A \\perp B \\mid C)$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_question6():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"c\", r\"$C$\", 2, 3.5, observed=True))\n",
    "    pgm.add_node(daft.Node(\"a\", r\"$A$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"b\", r\"$B$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"a\", \"c\", head_length=0.08)\n",
    "    pgm.add_edge(\"b\", \"c\", head_length=0.08)\n",
    "    \n",
    "    return pgm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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545TtJiUl4a+//kKPHj2csj1Xs/5xU7VqVYUiIcofkyKpjvXBMjMzU6FIHPPYY49JllNS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\n/tj5F198YfMCnAfd340bN7r9ZP/Y2FgRERGR705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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_question6(\"images/pgm/question6.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Answer:  No!\n",
    "\n",
    "$A$ can influence $B$ via $C$.  \n",
    "$$\n",
    "P(A,B | C) = \\frac{P(A,B,C)}{P(C)} = \\frac{P(A)P(B)P(C|A,B)}{P(C)}\n",
    "$$\n",
    "\n",
    "This does not factorize in general to $P(A|C)P(B|C)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Battery, Fuel, and Gauge\n",
    "\n",
    "Consider three binary random variables\n",
    "- Battery $B$ is either charged $(B=1)$ or dead, $(B=0)$\n",
    "- Fuel tank $F$ is either full $(F=1)$ or empty, $(F=0)$\n",
    "- Fuel gauge $G$ either indicates full $(G=1)$ or empty, $(G=0)$\n",
    "\n",
    "Assume $(B \\perp F)$ with priors\n",
    "- $P(B = 1) = 0.9$\n",
    "- $P(F = 1) = 0.9$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Battery, Fuel, and Gauge\n",
    "\n",
    "Given the state of the fuel tank and the battery, the fuel gauge reads full with probabilities: \n",
    " - $p(G = 1 \\mid B = 1, F = 1) = 0.8$\n",
    " - $p(G = 1 \\mid B = 1, F = 0) = 0.2$\n",
    " - $p(G = 1 \\mid B = 0, F = 1) = 0.2$\n",
    " - $p(G = 1 \\mid B = 0, F = 0) = 0.1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Battery, Fuel, and Gauge\n",
    "\n",
    "Without any observations, the probability of an empty fuel tank is\n",
    "$$\n",
    "P(F=0) = 1 - P(F = 1) = 0.1\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_bfg_1():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"G\", r\"$G$\", 2, 3.5))\n",
    "    pgm.add_node(daft.Node(\"B\", r\"$B$\", 1.3, 2.5))\n",
    "    pgm.add_node(daft.Node(\"F\", r\"$F$\", 2.7, 2.5))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"B\", \"G\", head_length=0.08)\n",
    "    pgm.add_edge(\"F\", \"G\", head_length=0.08)\n",
    "    \n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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KSY/TTz/9FK+99prseRkZGahdu7bT4yNyFSZFchtCCJsXwO3cuTPi4uJk49nZ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\n/T9quAGKpUVmfAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_bfg_1(\"images/pgm/bfg-1.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Empty Gauge\n",
    "\n",
    "Now, suppose the gauge reads $G=0$.  We have\n",
    "$$\n",
    "P(G=0) = \\sum \\limits_{B \\in \\{0, 1\\}} \\sum \\limits_{F \\in \\{0, 1\\}} \n",
    " P(G = 0 \\mid B, F) P(B) P(F) = 0.315\n",
    "$$\n",
    "\n",
    "> Verify this!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Emtpy Gauge"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_bfg_2():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"G\", r\"$G$\", 2, 3.5, offset=(0, 20), observed=True))\n",
    "    pgm.add_node(daft.Node(\"B\", r\"$B$\", 1.3, 2.5, offset=(0, 20)))\n",
    "    pgm.add_node(daft.Node(\"F\", r\"$F$\", 2.7, 2.5, offset=(0, 20)))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"B\", \"G\", head_length=0.08)\n",
    "    pgm.add_edge(\"F\", \"G\", head_length=0.08)\n",
    "    \n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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InBWHT8mtJCQkmJ0isGXLFgwaNMi4vHv3bowfPx5XrlyBRqOBTqezOmlVRGZm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\nekybNg27d+8GcOtqzri4ONx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ff/4pdR2zPUUhBHr06IH9+/cb2+rVq4dz587Bz8/PhiGT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\nodVqERgYKDtOhoaGolatWhW+SMfZ/X+My/my03qLMgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_bfg_2(\"images/pgm/bfg-2.png\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Empty Gauge\n",
    "\n",
    "Now, we also have \n",
    "$$\n",
    "p(G = 0 \\mid F = 0)\n",
    "= \\sum \\limits_{B \\in \\{0, 1\\}} p(G = 0 \\mid B, F = 0) p(B) = 0.81\n",
    "$$ \n",
    "\n",
    "Applying Bayes' Rule,\n",
    "$$\n",
    "\\begin{align}\n",
    "p(F = 0 \\mid G = 0) \n",
    "&= \\frac{p(G = 0 \\mid F = 0) p(F = 0)}{p(G = 0)} \\\\\n",
    "&\\approx 0.257 > p(F = 0) = 0.10\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Observing an empty gauge makes it more likely that the tank is empty!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Emtpy Gauge, Dead Battery\n",
    "\n",
    "Now, suppose we *also* observe a dead battery $B =0$.  Then,\n",
    "$$\n",
    "\\begin{align}\n",
    "p(F = 0 \\mid G = 0, B = 0) \n",
    "&= \\frac{p(G = 0 \\mid B = 0, F = 0) p(F = 0)}{\\sum_{F \\in \\{0, 1\\}} p(G = 0 \\mid B = 0, F) p(F)} \\\\\n",
    "&\\approx 0.111\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Emtpy Gauge, Dead Battery"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "@pgm_render\n",
    "def pgm_bfg_3():\n",
    "    pgm = daft.PGM([4, 4], origin=[0, 0])\n",
    "\n",
    "    # Nodes\n",
    "    pgm.add_node(daft.Node(\"G\", r\"$G$\", 2, 3.5, offset=(0, 20), observed=True))\n",
    "    pgm.add_node(daft.Node(\"B\", r\"$B$\", 1.3, 2.5, offset=(0, 20), observed=True))\n",
    "    pgm.add_node(daft.Node(\"F\", r\"$F$\", 2.7, 2.5, offset=(0, 20)))\n",
    "\n",
    "    # Add in the edges.\n",
    "    pgm.add_edge(\"B\", \"G\", head_length=0.08)\n",
    "    pgm.add_edge(\"F\", \"G\", head_length=0.08)\n",
    "    \n",
    "    return pgm;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lgkajKXXRKomkpCRERUXh0qVLuHnzJrKzs5GdnY28vDzY2trC1tYWDg4OqFOn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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%capture\n",
    "pgm_bfg_3(\"images/pgm/bfg-3.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Empty Gauge, Dead Battery\n",
    "\n",
    "This is the **explaining away** phenomenon.\n",
    "- The probability of an empty tank has decreased from $0.257$ to $0.111$ after observing the dead battery!\n",
    "- A dead battery *explans* why the guage indicates an empty tank, reducing the probability that the tank is *really* empty.\n",
    "\n",
    "Conditioning on a common child makes its parents dependent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Break time\n",
    "<img src=\"images/boxing_cat.gif\"/>\n",
    "\n",
    "* Will extend deadline for project proposals by a few days.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  d-Separation\n",
    "\n",
    "We say a trail $X \\leftrightharpoons Z \\leftrightharpoons Y$ is **active** when influence can flow from $X$ to $Y$ via $Z$.\n",
    "- **Causal Trail:** $X \\rightarrow Z \\rightarrow Y$ is active iff $Z$ is hidden.\n",
    "- **Evidential Trail:** $X \\leftarrow Z \\leftarrow Y$ is active iff $Z$ is hidden.\n",
    "- **Common Cause:** $X \\leftarrow Z \\rightarrow Y$ is active iff $Z$ is hidden.\n",
    "- **Common Effect:** $X \\rightarrow Z \\leftarrow Y$ is active iff $Z$ or a descendant of $Z$ is observed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  d-Separation\n",
    "\n",
    "A longer trail $X_1 \\leftrightharpoons X_2 \\leftrightharpoons \\cdots \\leftrightharpoons X_n$ is **active** if influence can flow from $X_1$ to $X_n$ along the trail.\n",
    "\n",
    "- Requires that every $X_{k-1} \\leftrightharpoons X_k \\leftrightharpoons X_{k+1}$ is active.\n",
    "\n",
    "In general, $X_1 \\leftrightharpoons X_2 \\leftrightharpoons \\cdots \\leftrightharpoons X_n$ is **active** given observed variables $Y$ if\n",
    "- Whenever we have a v-structure $X_{k-1} \\rightarrow X_k \\leftarrow X_{k+1}$ then $X_k$ or one of its descendants is in $Y$.\n",
    "- No other node along the trail is in $Y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks: d-Separation\n",
    "\n",
    "Some graphs may have more than one trail between two given nodes.\n",
    "- One node may inflence another if there exists an active trail between them.\n",
    "\n",
    "Let $\\mathbf{X}$, $\\mathbf{Y}$, and $\\mathbf{Z}$ be sets of nodes in $\\G$.  We say $\\mathbf{X}$ and $\\mathbf{Y}$ are **d-separated** given $\\mathbf{Z}$, denoted $\\dsep_\\G( \\mathbf{X};\\mathbf{Y} \\mid \\mathbf{Z})$ if there is no active trail between any nodes $X \\in \\mathbf{X}$ and $Y \\in \\mathbf{Y}$ given $\\mathbf{Z}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Bayesian Networks:  d-Separation\n",
    "\n",
    "**Theorem:** (Koller & Friedman 3.5) For almost all distributions $P$ that factorize over $\\G$, we have\n",
    "$$\n",
    "\\dsep_\\G(X; Y \\mid Z) \\iff (X \\perp Y \\mid Z)\n",
    "$$\n",
    "\n",
    "That is, except on a set of measure zero in the space of distributions, the concept of d-separation in $\\G$ exactly captures the concept of conditional independence.\n",
    "- To check if variables are independent, check for d-separation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Challenging quiz question:\n",
    "\n",
    "<img src=\"images/piazza_question.png\" width=600px/>\n",
    "\n",
    "* Notice that there is an active path from $B$ to $E$:\n",
    "$$B \\to \\boxed{D} \\leftarrow A \\to C \\to E$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Clustering & K-Means\n",
    "\n",
    "> Uses material from **[PRML]** §9.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "### Clustering:  Introduction\n",
    "\n",
    "**Goal:** Partition data $\\X = \\{ x_1, \\dots, x_n \\} \\subset \\R^d$ into $K$ disjoint **clusters**.\n",
    "- Points within a cluster should be more similar to each other than to points in other clusters.\n",
    "- Estimate **cluster centers** $\\mu_k \\in \\R^d$ for $k=1,\\dots,K$\n",
    "- Estimate **cluster assignments** $z_j \\in \\{1,\\dots,K\\}$ for each point $x_j$\n",
    "\n",
    "Usually, we fix $K$ beforehand!  Use model selection to overcome this limitation."
   ]
  },
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   "source": [
    "### K-Means Clustering\n",
    "\n",
    "The **K-Means** algorithm takes a simple, non-probabilistic approach.\n",
    "- First, pick random cluster centers $\\mu_k$.\n",
    "\n",
    "Then, repeat until convergence:\n",
    "\n",
    "<div style=\"margin:10px; padding: 10px; border: 2px solid black\">\n",
    "**E-Step:** Assign $x_j$ to the nearest cluster center $\\mu_k$,\n",
    "$$\n",
    "z_j = \\arg\\min_k || x_j - \\mu_k ||^2\n",
    "$$\n",
    "\n",
    "**M-Step:** Re-estimate cluster centers by averaging over assignments:\n",
    "$$\n",
    "\\mu_k = \\frac{1}{ \\# \\{ j \\mid z_j =k \\}} \\sum_{j=1}^N x_j \\mathbb{I}(z_j = k)\n",
    "$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: Initialization\n",
    "\n",
    "> Images taken from Bishop, **[PRML]**\n",
    "\n",
    "<img src=\"images/kmeans-1.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: E-Step\n",
    "\n",
    "<img src=\"images/kmeans-2.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: M-Step\n",
    "\n",
    "<img src=\"images/kmeans-3.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: E-Step\n",
    "\n",
    "<img src=\"images/kmeans-4.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: M-Step\n",
    "\n",
    "<img src=\"images/kmeans-5.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: E-Step\n",
    "\n",
    "<img src=\"images/kmeans-6.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: M-Step\n",
    "\n",
    "<img src=\"images/kmeans-7.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: E-Step\n",
    "\n",
    "<img src=\"images/kmeans-8.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering: M-Step\n",
    "\n",
    "<img src=\"images/kmeans-9.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means:  Cluster Geometry\n",
    "\n",
    "Clusters are convex *nearest-neighbor* regions or **Vornoi Cells**.\n",
    "- Piecewise-linear boundaries\n",
    "\n",
    "K-means will fail to identify non-convex clusters.\n",
    "- However, **kernelized K-means** is possible!\n",
    "\n",
    "<img src=\"images/voronoi.jpg\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### K-Means Clustering:  Analysis\n",
    "\n",
    "*Exercise:* Show that the K-Means algorithm finds a local minimum of the **distortion measure**, given by\n",
    "$$\n",
    "J(\\mu_1, \\ldots, \\mu_k; z_1, \\ldots, z_N) = \\sum_{j=1}^N \\sum_{k=1}^K \\mathbb{I}(z_j=k) || x_j - \\mu_k ||^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
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   "source": [
    "### K-Means:  Variants\n",
    "\n",
    "K-Means is simple and easy to extend:\n",
    "\n",
    "- **K-Means++:** Intelligently pick initial cluster centers\n",
    "- **K-Means--:** Handle outliers\n",
    "- **Nonparametric K-Means:** Automatically select number of clusters\n",
    "- **Kernelized K-Means:** Non-convex clusters\n",
    "\n",
    "Next lecture, we will cover **probabilistic clustering** through the use of mixture models."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Variant:  [K-Means++](http://ilpubs.stanford.edu:8090/778/1/2006-13.pdf)\n",
    "\n",
    "Instead of initializing cluster centers randomly,\n",
    "\n",
    "<div style=\"margin:10px; padding: 10px; border: 2px solid black\">\n",
    "<ol>\n",
    "<li> Choose the first cluster center to be a random datapoint.\n",
    "<li>Repeat until $K$ cluster centers have been selected:\n",
    "    <ol>\n",
    "    <li>For each datapoint $x_j$, compute distance $D(x_j)$ to nearest cluster.\n",
    "    <li>Choose data point $x_j$ at random to be the new cluster center, with probability proportional to $D(x_j)^2$.\n",
    "    </ol>\n",
    "<li> Run K-means as usual. \n",
    "</ol>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Variant:  [K-Means--](http://pmg.it.usyd.edu.au/outliers.pdf)\n",
    "\n",
    "Vanilla K-Means is sensitive to outliers.  Instead, assume there are $\\ell$ outliers, then\n",
    "\n",
    "<div style=\"margin:10px; padding: 10px; border: 2px solid black\">\n",
    "<ol>\n",
    "<li> Choose initial cluster centers as usual.\n",
    "<li> For $k=1,2,\\dots$:\n",
    "    <ol>\n",
    "    <li> For each datapoint $x_j$, compute distance $D(x_j)$ to nearest cluster.\n",
    "    <li> Set $L_k$ to be the $\\ell$ datapoints farthest from any cluster.\n",
    "    <li> Perform the E and M steps as usual on $X \\setminus L_k$.\n",
    "    </ol>\n",
    "<li> Return most recent outlier estimate $L_k$ in addition to usual cluster data.\n",
    "</ol>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Variant: [Nonparametric K-Means](http://www.cs.berkeley.edu/~jordan/papers/kulis-jordan-icml12.pdf)\n",
    "\n",
    "Automatically add new clusters via a **maximum cluster radius** $\\lambda$.\n",
    "\n",
    "<div style=\"margin:10px; padding: 10px; border: 2px solid black\">\n",
    "<ol>\n",
    "<li> Init $K=1$ and set $\\mu_1$ to be the mean over datapoints.\n",
    "<li> Repeat until convergence:\n",
    "    <ol>\n",
    "    <li> For each point $x_j$,\n",
    "        <ol>\n",
    "        <li> Compute distance $D(x_j)$ to nearest cluster.\n",
    "        <li> If $D(x_j) > \\lambda$, increment $K$ and create a new cluster centered at $x_j$.\n",
    "        <li> Otherwise, assign $x_j$ to a cluster as usual.\n",
    "        </ol>\n",
    "    <li> Re-estimate cluster centers as usual.\n",
    "    </ol>\n",
    "</ol>\n",
    "</div>"
   ]
  }
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