{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook is an interactive version of the [companion webpage](http://edwardlib.org/iclr2017) for the article, Deep Probabilistic Programming [(Tran et al., 2017)](https://arxiv.org/abs/1701.03757). See Edward's [API](http://edwardlib.org/api/) for details on how to interact with data, models, inference, and criticism.\n", "\n", "The code snippets assume the following versions.\n", "bash\n", "pip install edward==1.3.1\n", "pip install tensorflow==1.1.0 # alternatively, tensorflow-gpu==1.1.0\n", "pip install keras==2.0.0\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Section 3. Compositional Representations for Probabilistic Models\n", "\n", "__Figure 1__. Beta-Bernoulli program." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Bernoulli, Beta\n", "\n", "theta = Beta(1.0, 1.0)\n", "x = Bernoulli(tf.ones(50) * theta)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For an example of it in use, see\n", "[examples/beta_bernoulli.py](https://github.com/blei-lab/edward/blob/master/examples/beta_bernoulli.py)\n", "in the Github repository.\n", "\n", "__Figure 2__. Variational auto-encoder for a data set of 28 x 28 pixel images\n", "(Kingma & Welling, 2014; Rezende, Mohamed, & Wierstra, 2014)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Bernoulli, Normal\n", "from keras.layers import Dense\n", "\n", "N = 55000 # number of data points\n", "d = 50 # latent dimension\n", "\n", "# Probabilistic model\n", "z = Normal(loc=tf.zeros([N, d]), scale=tf.ones([N, d]))\n", "h = Dense(256, activation='relu')(z)\n", "x = Bernoulli(logits=Dense(28 * 28, activation=None)(h))\n", "\n", "# Variational model\n", "qx = tf.placeholder(tf.float32, [N, 28 * 28])\n", "qh = Dense(256, activation='relu')(qx)\n", "qz = Normal(loc=Dense(d, activation=None)(qh),\n", " scale=Dense(d, activation='softplus')(qh))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For an example of it in use, see\n", "[examples/vae.py](https://github.com/blei-lab/edward/blob/master/examples/vae.py)\n", "in the Github repository.\n", "\n", "__Figure 3__. Bayesian recurrent neural network (Radford M Neal, 2012).\n", "The program has an unspecified number of time steps; it uses a\n", "symbolic for loop (tf.scan)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import edward as ed\n", "import tensorflow as tf\n", "from edward.models import Normal\n", "\n", "H = 50 # number of hidden units\n", "D = 10 # number of features\n", "\n", "def rnn_cell(hprev, xt):\n", " return tf.tanh(ed.dot(hprev, Wh) + ed.dot(xt, Wx) + bh)\n", "\n", "Wh = Normal(loc=tf.zeros([H, H]), scale=tf.ones([H, H]))\n", "Wx = Normal(loc=tf.zeros([D, H]), scale=tf.ones([D, H]))\n", "Wy = Normal(loc=tf.zeros([H, 1]), scale=tf.ones([H, 1]))\n", "bh = Normal(loc=tf.zeros(H), scale=tf.ones(H))\n", "by = Normal(loc=tf.zeros(1), scale=tf.ones(1))\n", "\n", "x = tf.placeholder(tf.float32, [None, D])\n", "h = tf.scan(rnn_cell, x, initializer=tf.zeros(H))\n", "y = Normal(loc=tf.matmul(h, Wy) + by, scale=1.0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Section 4. Compositional Representations for Inference\n", "\n", "__Figure 5__. Hierarchical model (Gelman & Hill, 2006).\n", " It is a mixture of Gaussians over\n", " $D$-dimensional data $\\{x_n\\}\\in\\mathbb{R}^{N\\times D}$. There are\n", " $K$ latent cluster means $\\beta\\in\\mathbb{R}^{K\\times D}$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Categorical, Normal\n", "\n", "N = 10000 # number of data points\n", "D = 2 # data dimension\n", "K = 5 # number of clusters\n", "\n", "beta = Normal(loc=tf.zeros([K, D]), scale=tf.ones([K, D]))\n", "z = Categorical(logits=tf.zeros([N, K]))\n", "x = Normal(loc=tf.gather(beta, z), scale=tf.ones([N, D]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is used below in Figure 6 (left/right) and Figure * (variational EM).\n", "\n", "__Figure 6__ __(left)__. Variational inference\n", "(Jordan, Ghahramani, Jaakkola, & Saul, 1999).\n", "It performs inference on the model defined in Figure 5." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import numpy as np\n", "import tensorflow as tf\n", "from edward.models import Categorical, Normal\n", "\n", "x_train = np.zeros([N, D])\n", "\n", "qbeta = Normal(loc=tf.Variable(tf.zeros([K, D])),\n", " scale=tf.exp(tf.Variable(tf.zeros([K, D]))))\n", "qz = Categorical(logits=tf.Variable(tf.zeros([N, K])))\n", "\n", "inference = ed.VariationalInference({beta: qbeta, z: qz}, data={x: x_train})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Figure 6__ __(right)__. Monte Carlo (Robert & Casella, 1999).\n", "It performs inference on the model defined in Figure 5." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import numpy as np\n", "import tensorflow as tf\n", "from edward.models import Empirical\n", "\n", "x_train = np.zeros([N, D])\n", "\n", "T = 10000 # number of samples\n", "qbeta = Empirical(params=tf.Variable(tf.zeros([T, K, D])))\n", "qz = Empirical(params=tf.Variable(tf.zeros([T, N])))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Figure 7__. Generative adversarial network\n", "(Goodfellow et al., 2014)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import numpy as np\n", "import tensorflow as tf\n", "from edward.models import Normal\n", "from keras.layers import Dense\n", "\n", "N = 55000 # number of data points\n", "d = 50 # latent dimension\n", "\n", "def generative_network(eps):\n", " h = Dense(256, activation='relu')(eps)\n", " return Dense(28 * 28, activation=None)(h)\n", "\n", "def discriminative_network(x):\n", " h = Dense(28 * 28, activation='relu')(x)\n", " return Dense(1, activation=None)(h)\n", "\n", "# DATA\n", "x_train = np.zeros([N, 28 * 28])\n", "\n", "# MODEL\n", "eps = Normal(loc=tf.zeros([N, d]), scale=tf.ones([N, d]))\n", "x = generative_network(eps)\n", "\n", "# INFERENCE\n", "inference = ed.GANInference(data={x: x_train},\n", " discriminator=discriminative_network)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For an example of it in use, see the\n", "[generative adversarial networks](http://edwardlib.org/tutorials/gan) tutorial.\n", "\n", "__Figure *__. Variational EM (Radford M. Neal & Hinton, 1993).\n", "It performs inference on the model defined in Figure 5." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import numpy as np\n", "import tensorflow as tf\n", "from edward.models import Categorical, PointMass\n", "\n", "# DATA\n", "x_train = np.zeros([N, D])\n", "\n", "# INFERENCE\n", "qbeta = PointMass(params=tf.Variable(tf.zeros([K, D])))\n", "qz = Categorical(logits=tf.Variable(tf.zeros([N, K])))\n", "\n", "inference_e = ed.VariationalInference({z: qz}, data={x: x_train, beta: qbeta})\n", "inference_m = ed.MAP({beta: qbeta}, data={x: x_train, z: qz})\n", "\n", "inference_e.initialize()\n", "inference_m.initialize()\n", "\n", "tf.initialize_all_variables().run()\n", "\n", "for _ in range(10000):\n", " inference_e.update()\n", " inference_m.update()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For more details, see the\n", "[inference compositionality](http://edwardlib.org/api/inference-compositionality) webpage.\n", "See\n", "[examples/factor_analysis.py](https://github.com/blei-lab/edward/blob/master/examples/factor_analysis.py) for\n", "a version performing Monte Carlo EM for logistic factor analysis\n", "in the Github repository.\n", "It leverages Hamiltonian Monte Carlo for the E-step to perform maximum\n", "marginal a posteriori.\n", "\n", "__Figure *__. Data subsampling." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import tensorflow as tf\n", "from edward.models import Categorical, Normal\n", "\n", "N = 10000 # number of data points\n", "M = 128 # batch size during training\n", "D = 2 # data dimension\n", "K = 5 # number of clusters\n", "\n", "# DATA\n", "x_batch = tf.placeholder(tf.float32, [M, D])\n", "\n", "# MODEL\n", "beta = Normal(loc=tf.zeros([K, D]), scale=tf.ones([K, D]))\n", "z = Categorical(logits=tf.zeros([M, K]))\n", "x = Normal(loc=tf.gather(beta, z), scale=tf.ones([M, D]))\n", "\n", "# INFERENCE\n", "qbeta = Normal(loc=tf.Variable(tf.zeros([K, D])),\n", " scale=tf.nn.softplus(tf.Variable(tf.zeros([K, D]))))\n", "qz = Categorical(logits=tf.Variable(tf.zeros([M, D])))\n", "\n", "inference = ed.VariationalInference({beta: qbeta, z: qz}, data={x: x_batch})\n", "inference.initialize(scale={x: float(N) / M, z: float(N) / M})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For more details, see the\n", "[data subsampling](http://edwardlib.org/api/inference-data-subsampling) webpage.\n", "\n", "## Section 5. Experiments\n", "\n", "__Figure 9__. Bayesian logistic regression with Hamiltonian Monte Carlo." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import edward as ed\n", "import numpy as np\n", "import tensorflow as tf\n", "from edward.models import Bernoulli, Empirical, Normal\n", "\n", "N = 581012 # number of data points\n", "D = 54 # number of features\n", "T = 100 # number of empirical samples\n", "\n", "# DATA\n", "x_data = np.zeros([N, D])\n", "y_data = np.zeros([N])\n", "\n", "# MODEL\n", "x = tf.Variable(x_data, trainable=False)\n", "beta = Normal(loc=tf.zeros(D), scale=tf.ones(D))\n", "y = Bernoulli(logits=ed.dot(x, beta))\n", "\n", "# INFERENCE\n", "qbeta = Empirical(params=tf.Variable(tf.zeros([T, D])))\n", "inference = ed.HMC({beta: qbeta}, data={y: y_data})\n", "inference.run(step_size=0.5 / N, n_steps=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For an example of it in use, see\n", "[examples/bayesian_logistic_regression.py](https://github.com/blei-lab/edward/blob/master/examples/bayesian_logistic_regression.py)\n", "in the Github repository.\n", "\n", "## Appendix A. Model Examples\n", "\n", "__Figure 10__. Bayesian neural network for classification (Denker, Schwartz, Wittner, & Solla, 1987)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Bernoulli, Normal\n", "\n", "N = 1000 # number of data points\n", "D = 528 # number of features\n", "H = 256 # hidden layer size\n", "\n", "W_0 = Normal(loc=tf.zeros([D, H]), scale=tf.ones([D, H]))\n", "W_1 = Normal(loc=tf.zeros([H, 1]), scale=tf.ones([H, 1]))\n", "b_0 = Normal(loc=tf.zeros(H), scale=tf.ones(H))\n", "b_1 = Normal(loc=tf.zeros(1), scale=tf.ones(1))\n", "\n", "x = tf.placeholder(tf.float32, [N, D])\n", "y = Bernoulli(logits=tf.matmul(tf.nn.tanh(tf.matmul(x, W_0) + b_0), W_1) + b_1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For an example of it in use, see\n", "[examples/getting_started_example.py](https://github.com/blei-lab/edward/blob/master/examples/getting_started_example.py)\n", "in the Github repository.\n", "\n", "__Figure 11__. Latent Dirichlet allocation (D. M. Blei, Ng, & Jordan, 2003)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Categorical, Dirichlet\n", "\n", "D = 4 # number of documents\n", "N = [11502, 213, 1523, 1351] # words per doc\n", "K = 10 # number of topics\n", "V = 100000 # vocabulary size\n", "\n", "theta = Dirichlet(tf.zeros([D, K]) + 0.1)\n", "phi = Dirichlet(tf.zeros([K, V]) + 0.05)\n", "z = [[0] * N] * D\n", "w = [[0] * N] * D\n", "for d in range(D):\n", " for n in range(N[d]):\n", " z[d][n] = Categorical(theta[d, :])\n", " w[d][n] = Categorical(phi[z[d][n], :])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Figure 12__. Gaussian matrix factorization\n", "(Salakhutdinov & Mnih, 2011)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import Normal\n", "\n", "N = 10\n", "M = 10\n", "K = 5 # latent dimension\n", "\n", "U = Normal(loc=tf.zeros([M, K]), scale=tf.ones([M, K]))\n", "V = Normal(loc=tf.zeros([N, K]), scale=tf.ones([N, K]))\n", "Y = Normal(loc=tf.matmul(U, V, transpose_b=True), scale=tf.ones([N, M]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Figure 13__. Dirichlet process mixture model (Antoniak, 1974)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import tensorflow as tf\n", "from edward.models import DirichletProcess, Normal\n", "\n", "N = 1000 # number of data points\n", "D = 5 # data dimensionality\n", "\n", "dp = DirichletProcess(alpha=1.0, base=Normal(loc=tf.zeros(D), scale=tf.ones(D)))\n", "mu = dp.sample(N)\n", "x = Normal(loc=loc, scale=tf.ones([N, D]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To see the essential component defining the DirichletProcess, see\n", "[examples/pp_dirichlet_process.py](https://github.com/blei-lab/edward/blob/master/examples/pp_dirichlet_process.py)\n", "in the Github repository. Its source implementation can be found at\n", "[edward/models/dirichlet_process.py](https://github.com/blei-lab/edward/blob/master/edward/models/dirichlet_process.py).\n", "\n", "## Appendix B. Inference Examples\n", "\n", "__Figure *__. Stochastic variational inference (M. D. Hoffman, Blei, Wang, & Paisley, 2013). \n", "For more details, see the\n", "[data subsampling](http://edwardlib.org/api/inference-data-subsampling) webpage.\n", "\n", "## Appendix C. Complete Examples\n", "\n", "__Figure 15__. Variational auto-encoder\n", "(Kingma & Welling, 2014; Rezende et al., 2014).\n", "See the script\n", "\\href{https://github.com/blei-lab/edward/blob/master/examples/vae.py}{\\texttt{examples/vae.py}}\n", "in the Github repository.\n", "\n", "__Figure 16__. Exponential family embedding (Rudolph, Ruiz, Mandt, & Blei, 2016).\n", "A Github repository with comprehensive features is available at\n", "[mariru/exponential_family_embeddings](https://github.com/mariru/exponential_family_embeddings)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.12" } }, "nbformat": 4, "nbformat_minor": 1 }