{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "code", "collapsed": false, "input": [ "import pymc\n", "import daft\n", "import csv\n", "from scipy.sparse import coo_matrix\n", "from sklearn.decomposition import ProjectedGradientNMF\n", "from itertools import product, izip" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Welcome" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**Bayesian networks are a method for building your own machine learning models.**\n", "\n", "- With Bayesian networks, we model our data problem as a **causal network**, or a **story** involving **hidden** and **observed** variables.\n", " - We come up with **a story** that we think **explains our data**.\n", " - We **use Bayes's rule to find** _posterior_ probability distributions of the **hidden variables** given our observed variables (data).\n", " - We use the full posterior distributions for our next action (decision, recommendation, prediction).\n", "- **Advantages**:\n", " - **Confidence estimates.**\n", " - **Flexibility.** We can change the story and re-train the model.\n", " - Classical machine learning methods are **inflexible**: Code and theory only works for its specific problem.\n", " - Your real world data probably has some important \n", "Difficult to adapt to your particular problem, which may not have been studied by researchers.\n", " - **Disadvantages**:\n", " - **Slow** (unless you do a lot of math)\n", " - **You have to think.**\n", "- **For many of your data problems, you may want or need to fit a custom machine learning model.**\n", " - Some important aspect of your data probably hasn't been studied by researchers.\n", " - I don't agree with this picture: http://scikit-learn.org/stable/_static/ml_map.png\n" ] }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Bayesian Statistics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have some data $X$. How did it get there?\n", "\n", "**We don't know.** There is no one right answer, only answers that are more or less supported by the evidence.\n", "\n", "A textbook example: Somebody tests HIV positive.\n", "\n", "Story:\n", "\n", "- Grab a person off the street in New York. (0.03% of NY is diagnosed with HIV)\n", "- Give them an HIV test.\n", " - For HIV positive patients, test returns positive 99% of the time. (True positive rate.)\n", " - For HIV negative patients, test returns negative 0.1% of the time. (False positive rate.)\n", "\n", "Does the person have HIV? **We don't know**. But we can compute probabilities, given this story, using Bayes' rule.\n", "\n", "Let $+$ denote the event the 'test positive', $-$ denote the event 'test negative', and $H$ denote the event 'HIV positive'. Then using Bayes's rule,\n", "$$\\begin{eqnarray*}\n", "p(H|+) & = & \\frac{p(+|H)p(H)}{p(+)} \\\\\\\\\n", " & = & \\frac{p(+|H)P(H)}{p(+|H)p(H)+p(+|\\overline{H})p(\\overline{H})} \\\\\\\\\n", " & = & \\frac{(0.99)(0.0003)}{(0.99)(0.0003)+(0.001)(0.9997)} \\\\\\\\\n", " & \\approx & 0.229\n", "\\end{eqnarray*}$$\n" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Bayesian Statistics, More Formally" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To do Bayesian statistics, we need\n", "\n", " - A **probability model**, a story of how the data came to be.\n", " - **Hidden variables** $\\theta$, about which which we want to infer probabilities.\n", " - **Observed variables** $X$.\n", " - A **prior** on our hidden variables $p(\\theta)$: What do we believe before seeing any data?\n", " - A **likelihood** $p(X | \\theta)$ relating hidden variables to observed variables\n", "\n", "Then Bayes' rule ties them together:\n", "\n", "$$\\begin{eqnarray*}\n", "p(\\theta|X) & = & \\frac{p(X|\\theta)p(\\theta)}{p(X)} \\\\\\\\\n", "& = & \\frac{p(X|\\theta)P(\\theta)}{\\int p(X|\\theta)p(\\theta) d\\theta} \\\\\\\\\n", "\\end{eqnarray*}$$\n", "\n", "Bayesian inference is\n", "\n", " - Useful: hidden variables\n", " - Hard: the denominator is high-dimensional integral. You can't just use Simpson's rule; it will take exponential time.\n", " - We use Markov Chain Monte Carlo (MCMC) with the PyMC package." ] }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Movie Recommendations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have $N$ users and $M$ items. Given a small samples of ratings $r_{ij}$, predict $\\hat{r}_{i'j'}$ for pairs $(i', j')$ we don't observe.\n", "\n", "Our toy dataset: http://mlcomp.org/datasets/736, 190 training ratings (81 test), $N = 6, M = 91$.\n", "\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def loadData(filename):\n", " data = np.loadtxt(filename)\n", " users = data[:,0].astype(int)\n", " items = data[:,1].astype(int)\n", " ratings = data[:,2]\n", " \n", " nExamples = len(users)\n", " \n", " return (nExamples, users, items, ratings)\n", "\n", "TRAIN_FILE = '736/train'\n", "TEST_FILE = '736/test'\n", "\n", "(nTrain, users, items, ratings) = loadData(TRAIN_FILE)\n", "(nTest, testUsers, testItems, testRatings) = loadData(TEST_FILE)\n", "\n", "N = np.max(users)\n", "M = np.max(items)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Always look at our data!" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.figure()\n", "plt.hist(ratings)\n", "plt.figure()\n", "plt.hist(testRatings)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "(array([ 47., 18., 9., 2., 0., 3., 1., 0., 0., 1.]),\n", " array([ 0.75432862, 1.2940267 , 1.83372478, 2.37342286, 2.91312094,\n", " 3.45281902, 3.9925171 , 4.53221518, 5.07191326, 5.61161134,\n", " 6.15130942]),\n", " )" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the test set, it appears like we have 1-6 stars, but something weird is happening in the training set! The histogram says there are outliers, so let's take a look:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "outlierIdxs = ratings > 6\n", "print np.flatnonzero(outlierIdxs)\n", "print ratings[outlierIdxs]" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[ 83 163 165 167 168]\n", "[ 9.54361436 1379.98131862 8.76865101 2907.19215935 1380.47446215]\n" ] } ], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "I delete the outliers from my data file; much better. **Always** make a new copy; do not overwrite your original data!" ] }, { "cell_type": "code", "collapsed": false, "input": [ "CLEAN_TRAIN_FILE = '736/trainClean'\n", "(nTrain, users, items, ratings) = loadData(CLEAN_TRAIN_FILE)\n", "\n", "plt.hist(ratings)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 7, "text": [ "(array([ 95., 43., 17., 5., 5., 3., 13., 1., 1., 2.]),\n", " array([ 0.75722523, 1.23237711, 1.70752898, 2.18268086, 2.65783273,\n", " 3.13298461, 3.60813648, 4.08328836, 4.55844023, 5.03359211,\n", " 5.50874398]),\n", " )" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we've taken a look at our data, let's tell a story.\n", "\n", "The classic _matrix factorization_ method for recommendations represents users and items as $d$-dimensional vectors.\n", "\n", "- Each dimension represents some abstract \"taste\": horror? chick flick? comedy?\n", "- A rating $r_{ij}$ is just the dot product of the respective user and item vectors: $\\mathbf{u}_i ^\\top \\mathbf{v}_j$.\n", "\n", "In Bayesian land,\n", "\n", "- For user $i = 1, \\ldots, N$, draw $\\mathbf{u}_i \\sim \\mathcal{N}(0, \\tau_0 I_d)$\n", "- For item $j = 1, \\ldots, M$ draw $\\mathbf{v}_j \\sim \\mathcal{N}(0, \\tau_0 I_d)$\n", "- For $i = 1, \\ldots, N$ and $j = 1, \\ldots, M$,\n", "\n", " - Compute $z_{ij} = \\mathbf{u}_i ^\\top \\mathbf{v}_j$\n", " - Draw $r_{ij} \\sim \\mathcal{LN}(z_{ij}, \\tau_1)$.\n", "\n", "The Bayesian network looks like\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from matplotlib import rc\n", "rc(\"font\", family=\"serif\", size=36)\n", "rc(\"text\", usetex=True)\n", "\n", "pgm = daft.PGM([5, 4], grid_unit=5, node_unit=3)\n", "pgm.add_node(daft.Node(\"tau0\", r\"$\\tau_0$\", 2, 3.5, fixed=True))\n", "pgm.add_node(daft.Node(\"u\", r\"$\\mathbf{u}_i$\", 1, 2.5))\n", "pgm.add_node(daft.Node(\"v\", r\"$\\mathbf{v}_j$\", 3, 2.5))\n", "\n", "pgm.add_node(daft.Node(\"z\", r\"$z_{ij}$\", 2, 2.5))\n", "pgm.add_node(daft.Node(\"r\", r\"$r_{ij}$\", 2, 1.5, observed=True))\n", "\n", "pgm.add_node(daft.Node(\"tau1\", r\"$\\tau_1\", 0.25, 1.5, fixed=True))\n", "\n", "pgm.add_edge(\"tau0\", \"u\")\n", "pgm.add_edge(\"tau0\", \"v\")\n", "pgm.add_edge(\"u\", \"z\")\n", "pgm.add_edge(\"v\", \"z\")\n", "pgm.add_edge(\"z\", \"r\")\n", "pgm.add_edge(\"tau1\", \"r\")\n", "\n", "# [start_x, start_y, xlen, ylen]\n", "pgm.add_plate(daft.Plate([0.5, 0.5, 2, 2.5], label=r\"$i = 1, \\ldots, N$\"))\n", "pgm.add_plate(daft.Plate([1.5, 0.25, 2, 2.7], shift=0.5, label=r\"$j = 1, \\ldots, M$\"))\n", "\n", "pgm.render()\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 8, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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Q9ezZ0++ehAH+iDLtRZYvX6477rhDtWvXNh3Fa7Vq1UpNmzbVunXrTEexzdKl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\nli5dGvd9SYN/mHzwwQeW7AsAkBgcFaa3b98e9SL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"text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finding hyperparmeters is the unprincipled black magic of machine learning. I basically tried some numbers until they worked." ] }, { "cell_type": "code", "collapsed": false, "input": [ "D = 10\n", "tau0 = 5\n", "tau1 = 5" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "u = pymc.Normal('u', 0, tau0, size=(D, N))\n", "v = pymc.Normal('v', 0, tau0, size=(D, M))\n", "\n", "z = pymc.Lambda('z', lambda u=u, v=v: np.dot(u.T, v))\n", "r = np.empty(nTrain, dtype=object)\n", "\n", "for n, (i, j) in enumerate(izip(users, items)):\n", " r[n] = pymc.Lognormal('x_%d_%d' % (i, j),\n", " z[i-1, j-1], # mean\n", " tau1, # precision (confidence)\n", " observed=True,\n", " value=ratings[n])\n", "\n", "model = pymc.Model([u, v, z, r])" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Posterior Inference and Markov Chain Monte Carlo" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# There was a bug in PyMC; this is just to hack around code.\n", "if 3 in model.__dict__:\n", " del model.__dict__[3]\n", " \n", "pymc.MAP(model).fit()\n", "mcmc = pymc.MCMC(model)\n", "mcmc.sample(10000)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " \r", "[****************100%******************] 10000 of 10000 complete" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Joint probability:\n", "\n", "\n", "$$p\\left(\\left\\{ r_{ij},z_{ij}\\right\\} _{ij\\in I}\\left\\{ \\mathbf{u}_{i}\\right\\} _{i=1}^{N},\\left\\{ \\mathbf{v}\\right\\} _{j=1}^{M},\\tau_{0},\\tau_{1}\\right)=\\prod_{ij\\in I}\\mathcal{LN}(r_{ij}|\\mathbf{u}_{i}^{\\top}\\mathbf{v}_{i},\\tau_{1})\\prod_{i=1}^{N}\\mathcal{N}(\\mathbf{u}_{i},\\tau_{0}I_{d})\\prod_{j=1}^{M}\\mathcal{N}(\\mathbf{v}_{j},\\tau_{0}I_{d})$$\n", "\n", "Posterior probabilities:\n", "\n", "$$p\\left(\\left\\{ z_{ij}\\right\\} _{ij\\in I}\\left\\{ \\mathbf{u}_{i}\\right\\} _{i=1}^{N},\\left\\{ \\mathbf{v}\\right\\} _{j=1}^{M},\\tau_{0},\\tau_{1}\\vert\\left\\{ r_{ij}\\right\\} _{ij\\in I}\\right)=\\frac{\\prod_{ij\\in I}\\mathcal{LN}(r_{ij}|\\mathbf{u}_{i}^{\\top}\\mathbf{v}_{i},\\tau_{1})\\prod_{i=1}^{N}\\mathcal{N}(\\mathbf{u}_{i},\\tau_{0}I_{d})\\prod_{j=1}^{M}\\mathcal{N}(\\mathbf{v}_{j},\\tau_{0}I_{d})}{\\int\\prod_{ij\\in I}\\mathcal{LN}(r_{ij}|\\mathbf{u}_{i}^{\\top}\\mathbf{v}_{i},\\tau_{1})\\prod_{i=1}^{N}\\mathcal{N}(\\mathbf{u}_{i},\\tau_{0}I_{d})\\prod_{j=1}^{M}\\mathcal{N}(\\mathbf{v}_{j},\\tau_{0}I_{d})d\\mathbf{z}d\\mathbf{u}d\\mathbf{v}}$$\n", "\n", "Posterior probabilities are the **cornerstone of Bayesian statistics**. They give us information like\n", "\n", " - Predicted ratings,$r_{ij}$\n", " - Error bars on predictions,$\\sqrt{\\mathbf{V}[r_{ij}]}$\n", " - Shape of latent representations of movies and users,$p(\\mathbf{u}_i | \\ldots)$,$p(\\mathbf{v}_i | \\ldots)$.\n", "\n", "\n", "**Markov chain Monte Carlo:** (MCMC) Do not evaluate the posterior explicitly. Construct a _Markov chain_ whose _invariant distribution_ is the posterior.\n", "\n", "In fact, with MCMC we never know how to write down the posterior. _Eventually_, our Markov chain generates samples from it.\n", "\n", " - Markov chain: a process whose future depends only on the present, not on its own history\n", " - Computationally easy to simulate.\n", " - Analytically, rich theory, which justifies the method.\n", " \n", "Gibbs sampling: The easiest MCMC method.\n", " \n", " - Let$x_1, \\ldots, x_n$be random variables, and$\\mathbf{y}$be observed.\n", " - For$t = 1, 2, \\ldots$:\n", " - For$i = 1, \\ldots, n$:\n", " - Sample$p(x_i | x_1, \\ldots, x_{i-1}, x_{i+1}, x_i, \\mathbf{y})$\n", "\n", "Properties:\n", "\n", " - Each inner loop generates a vector$\\mathbf{x}^{(t)}_{1:n}$. These vectors form a Markov chain.\n", " - At the invariant distribution (when you run then chain for \"long enough\"), the Markov chain generates samples from the conditional$p(\\mathbf{x} | \\mathbf{y})$.\n", " - For any subset of variables$x_i, x_j, x_k$, keeping just those variables from the Markov chain is a sample from the marginal distribution$p(x_i, x_j, x_k | \\mathbf{y})$.\n", " \n", " " ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "\n", "Posterior Analysis" ] }, { "cell_type": "code", "collapsed": false, "input": [ "zsamples = mcmc.trace('z')[5000:]\n", "zmean = np.mean(zsamples, 0)\n", "\n", "meanExpZ = np.mean(np.exp(zsamples), 0)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We learned a _distribution_ over _all_ log-ratings$z_{ij}$. Let's take a peek at the first rating. The shape of the histogram captures our knowledge, including knowledge of uncertainty." ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.hist(np.exp(zsamples[:,1,1]))\n", "\n", "iFirst, jFirst, rFirst = users[1], items[1], ratings[1]\n", "print meanExpZ[iFirst,jFirst]\n", "print rFirst" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "1.36496704562\n", "1.5098059294\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Accuracy Validation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's output our predicted ratings as the _mean_ rating. Let$I$be the set of item-user pairs rated, with true ratings$r_{ij}$. Recall$z_{ui}\$ is the logarithm of the rating. The standard error metric is _root-mean squared error_ (RMSE):\n", "\n", "$$\\text{RMSE}(I) = \\sqrt{\\frac{1}{|I|}\\sum_{(i,j)\\in I}\\left(\\exp z_{ij}-r_{ij}\\right)^{2}}$$\n", "\n", "Let us calculate the RMSE for both training and test sets. In general, if the RMSE is way better for training set, then we've overfit." ] }, { "cell_type": "code", "collapsed": false, "input": [ "#trainRMSE = np.sqrt(np.mean((np.exp(rmean[users-1, items-1]) - ratings) ** 2))\n", "#testRMSE = np.sqrt(np.mean((np.exp(rmean[testUsers-1, testItems-1]) - testRatings) ** 2))\n", "trainRMSE = np.sqrt(np.mean( (meanExpZ[users-1, items-1] - ratings) ** 2))\n", "testRMSE = np.sqrt(np.mean( (meanExpZ[testUsers-1, testItems-1] - testRatings) ** 2))\n", "\n", "print trainRMSE\n", "print testRMSE" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.686684550069\n", "0.876480474553\n" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "The Competitor, Nonnegative Matrix Factorization" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Again, I jiggled the sparsity (beta0) knob; these were the best results I got.\n", "\n", "Note exactly a fair comparison, since the NMF code is far better vectorized." ] }, { "cell_type": "code", "collapsed": false, "input": [ "R = coo_matrix((ratings, (users - 1, items - 1))).todense()\n", "skm = ProjectedGradientNMF(n_components=D, sparseness='data', beta=70)\n", "W = skm.fit_transform(R.T)\n", "\n", "reconstruct = np.dot(skm.components_.T, W.T)\n", "\n", "nmfTrainRMSE = np.sqrt(np.mean( (reconstruct[users-1, items-1] - ratings) ** 2))\n", "nmfTestRMSE = np.sqrt(np.mean( (reconstruct[testUsers-1, testItems-1] - testRatings) ** 2))\n", "\n", "print nmfTrainRMSE\n", "print nmfTestRMSE\n", "\n", "#print skm.reconstruction_err_\n", "#pint np.linalg.norm(reconstruct - R)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.907722163866\n", "1.39713801489\n" ] } ], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }