2016_07_07_notes_on_advi

2016_07_07_notes_on_advi.ipynb

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   "source": [
    "# Notes on Automatic Differentiation Variational Inference\n",
    "\n",
    "Alp Kucukelbir, Dustin Tran, Rajesh Ranganath, Andrew Gelman, David M. Blei\n",
    "\n",
    "http://arxiv.org/abs/1603.00788"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pylab as plt\n",
    "import seaborn as sns\n",
    "import scipy.stats\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variational Inference\n",
    "\n",
    "__What we want:__\n",
    "\n",
    "* the posterior distribution, or an estimate thereof, for some model over some dataset\n",
    "\n",
    "__What we have:__\n",
    "\n",
    "* a dataset $x_{1:N}$ of $N$ observations\n",
    "* a model, generally containing latent variables $\\theta$\n",
    "* (the model is often expressed as a probabilistic graphical model with likelihood $p(x \\mid \\theta)$ and prior $p(\\theta)$)\n",
    "* so we then have the join probability distribution $p(x, \\theta) = p\\left(x\\mid\\theta\\right)p(\\theta)$\n",
    "\n",
    "__Challenges:__\n",
    "\n",
    "* non-conjugate models (can be complex or no closed-form solutions)\n",
    "* MCMC can be slow, or some samplers may not converge for some models\n",
    "\n",
    "__A way to proceed:__\n",
    "\n",
    "* find a distribution that is approximates the posterior\n",
    "* find a cost function that represents how close the approximate distribution is from the posterior\n",
    "* minimize that cost function\n",
    "\n",
    "__Some constraints:__\n",
    "\n",
    "* only consider differentiable models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### K-L divergence (the 'cost function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence\n",
    "\n",
    "Continuous divergence\n",
    "\n",
    "$$D_{\\mathrm KL}(q \\mid\\mid p)  = \\int_{\\theta} q(\\theta) \\log\\frac{q(\\theta)}{p(\\theta\\mid x)}\n",
    "            = \\mathbb{E}_q \\left[ \\log \\frac{q(\\theta)}{p(\\theta \\mid x)} \\right] \\geq 0,\n",
    "$$\n",
    "\n",
    "between an approximate distribution $q$ and the posterior $p(\\theta \\mid x)$.\n",
    "* not symmetric in exchange between $q$ and $p$\n",
    "* kindof like a distance function between distributions, but not a true metric\n",
    "* Bishop, 2006 gives a discussion by VI approaches generally use $D_{\\mathrm KL}(q \\mid\\mid p)$ instead of $D_{\\mathrm KL}(p \\mid\\mid q)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "outputs": [
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kfOEKP0bpgPWNwukwzg95DL5jpaQjCV+MNkn4whWFapFIKMJYeGyorzsRy1Co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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11307fb90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t1 = scipy.stats.norm(-2.5, 0.1)\n",
    "t2 = scipy.stats.norm(-2.8, 0.1)\n",
    "\n",
    "x = np.linspace(-4,-1,100)\n",
    "\n",
    "plt.plot(x,t1.pdf(x))\n",
    "plt.plot(x,t2.pdf(x))\n",
    "plt.title(\"KL-divergence:\" + str(scipy.stats.entropy(t1.pdf(x), t2.pdf(x))));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gZOCvZUuEYuQqeUxz8N9ghOiHBPwJYwfdVHw4F21hNeBnh5DHt5ZViOEzhvfW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\nCAn4QggxIf5/LZUbwUWklC8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1131b0f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t3 = scipy.stats.norm(-3, 0.1)\n",
    "t4 = scipy.stats.norm(-2.5, 0.1)\n",
    "\n",
    "x = np.linspace(-4,-1,100)\n",
    "\n",
    "plt.plot(x,t3.pdf(x))\n",
    "plt.plot(x,t4.pdf(x))\n",
    "plt.title(\"KL-divergence:\" + str(scipy.stats.entropy(t3.pdf(x), t4.pdf(x))));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ELBO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* We want to minimize the KL divergence, but since it explicitly includes the posterior, we need a proxy measure that approximates the divergence. For this, we'll use the Evidence Lower BOund (ELBO):\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\log p(x) &= \\log\\int_{\\theta} p(x,\\theta) \\\\\n",
    "              &= \\log\\int_{\\theta} p(x,\\theta) \\frac{q(\\theta)}{q(\\theta)} \\\\\n",
    "              &= \\log\\left(\\mathbb{E}_q\\left[ \\frac{p(x,\\theta)}{q(\\theta)} \\right]\\right) \\\\\n",
    "              & \\left. \\geq \\mathbb{E}_q[\\log p(x,\\theta)] - \\mathbb{E}_q[\\log q(\\theta)] = \\mathrm{ELBO} = \\mathcal{L} \\right.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "which can be related to the KL divergence:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    D_{\\mathrm KL}(q \\mid\\mid p) &= \\mathbb{E}_q \\left[ \n",
    "        \\log \\frac{q(\\theta)}{p(\\theta \\mid x)} \n",
    "    \\right] \\\\\n",
    "        &= \\mathbb{E}_q \\left[ \\log \\frac{q(\\theta)}{p(\\theta,x)/p(x)} \\right] \\\\\n",
    "        &= \\mathbb{E}_q \\left[\\log q(\\theta)\\right] \n",
    "          -\\mathbb{E}_q \\left[\\log p(\\theta,x)\\right]\n",
    "          + \\log p(x) \\\\\n",
    "        &= -(\\mathrm{ELBO}) + \\log p(x) \\\\\n",
    "        &= -\\mathcal{L} + \\log p(x)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Since $D_{\\mathrm KL}(q \\mid\\mid p) \\geq 0$ and $\\log p(x) = \\mathrm{constant}$, maximizing the ELBO minimizes the KL divergence. As such, finding a good approximate posterior comes down to finding a good $q$, as well as a optimization problem to maximize the corresponding ELBO."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The importance of support-matching"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The gradient $\\nabla_{\\theta} \\log p(x,\\theta)$ is valid within the support of the prior:\n",
    "\n",
    "$$\n",
    "    \\mathrm{supp}(p(\\theta)) = \\left\\{\n",
    "        \\theta \\mid \\theta \\in \\mathbb{R}^{K} \\,\\mathrm{and}\\, p(\\theta)>0 \n",
    "    \\right\\}\n",
    "    \\subseteq \\mathbb{R}^{K},\n",
    "$$\n",
    "\n",
    "where $K$ is the dimension of the latent variable space.\n",
    "\n",
    "* This is due to the fact that if \n",
    "\n",
    "$$\n",
    "    \\mathrm{supp}(q) \\subsetneq \\mathrm{supp}(p) \n",
    "        \\implies \n",
    "    D_{\\mathrm KL}(q \\mid\\mid p) = \\mathbb{E}_{q}[\\log q] - \\mathbb{E}_{p}[\\log p] = -\\infty\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ADVI\n",
    "\n",
    "### Transformation to unbounded real coordinate space\n",
    "\n",
    "* Need to ensure that the support of the prior is unbounded, so we don't have to worry about mis-matched support\n",
    "* introduce a coordinate transformation, $T : \\mathrm{supp}(p(\\theta)) \\rightarrow \\mathbb{R}^{K}$, that maps the latent coordinates to the unbounded real space\n",
    "* variables in this mapped space are $\\zeta = T(\\theta)$\n",
    "\n",
    "$$\n",
    "    p(x, \\zeta) = p\\left( x, T^{-1}(\\theta) \\right)\n",
    "    \\left| \n",
    "        \\det J_{T^{-1}} (\\zeta)\n",
    "    \\right|\n",
    "$$\n",
    "\n",
    "* the Jacobian ensures that the transformed density integrates to one\n",
    "\n",
    "### Mean-field Gaussian\n",
    "\n",
    "* one approach is to try a factorized Gaussian solution in the real-valued coordinate space\n",
    "\n",
    "$$\n",
    "    q(\\zeta, \\phi) = \\mathcal{N}(\\zeta; \\mu, \\mathrm{diag}(\\sigma^2))\n",
    "    = \\prod_{k=1}^{K} \\mathcal{N}(\\zeta_k; \\mu_k, \\sigma_k^2)\n",
    "$$\n",
    "\n",
    "where $\\phi = (\\mu_1, \\dots, \\mu_K, \\sigma_1^2, \\dots, \\sigma_K^2)$\n",
    "\n",
    "* however, these parameters live in the set of $\\Phi = \\left\\{ \\mathbb{R}^{K}, \\mathbb{R}^{K}_{\\gt 0} \\right\\}$\n",
    "* reparameterize as $\\omega = \\log(\\sigma)$, then\n",
    "\n",
    "$$\n",
    "    q(\\zeta, \\phi) = \\mathcal{N}\\left(\\zeta; \\mu, \\mathrm{diag}(\\exp(\\omega)^2)\\right)\n",
    "$$\n",
    "\n",
    "* where the parameters $\\phi = (\\mu_1, \\dots, \\mu_K, \\omega_1, \\dots, \\omega_K)$ are unconstrained in $\\mathbb{R}^{2K}$\n",
    "\n",
    "### Full-rank Gaussian\n",
    "\n",
    "* another approach is to use a full-rank Gaussian,\n",
    "\n",
    "$$\n",
    "    q(\\zeta; \\phi) = \\mathcal{N}(\\zeta; \\mu, \\Sigma)\n",
    "$$\n",
    "\n",
    "where $\\phi = (\\mu, \\Sigma)$ are the mean values, $\\mu$, and the covariance matrix $\\Sigma$\n",
    "\n",
    "* to ensure that $\\Sigma$ is positive semidefinite, it can be reparameterize as $\\Sigma = LL^{T}$, using the non-unique definition of the Cholesky factorization, such that $L$ is in the unconstrained space of lower-triangle matrices with $K(K + 1)/2$ real-valued entries. So then\n",
    "\n",
    "$$\n",
    "    q(\\zeta; \\phi) = \\mathcal{N}(\\zeta; \\mu, L L^{T})\n",
    "$$\n",
    "\n",
    "where $\\phi = (\\mu, L)$ are unconstrained in $\\mathbb{R}^{K + K(K+1)/2}$\n",
    "\n",
    "* the full-rank Gaussian is a generalization of the mean-field approximation, and the off-diagonal terms in the covariance matrix capture correlations in the posterior distribution\n",
    "* however, it is computationally more costly to compute"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Variation inference in the real-coodinate space\n",
    "\n",
    "* after employing the transformation function, the ELBO becomes\n",
    "\n",
    "$$\n",
    "    \\mathcal{L}(\\phi) = \n",
    "    \\mathbb{E}_{q(\\zeta;\\phi)} \\left[\n",
    "        \\log p \\left(\n",
    "            x, T^{-1}(\\zeta)\n",
    "        \\right)\n",
    "        +\n",
    "        \\log \\left| \n",
    "            \\det J_{T^{-1}} (\\zeta)\n",
    "        \\right|\n",
    "    \\right]\n",
    "    +\n",
    "    \\mathbb{H} \\left[\n",
    "        q(\\zeta; \\phi)\n",
    "    \\right]\n",
    "$$\n",
    "\n",
    "where $\\mathbb{H}$ is the entropy\n",
    "\n",
    "* the ELBO can then be optimized with respect to the variational parameters $\\phi$ without worrying about the constraining the support:\n",
    "\n",
    "$$\n",
    "    \\phi^{*} = {\\arg\\max}_{\\phi} \\mathcal{L}(\\phi)\n",
    "$$\n",
    "\n",
    "* which can be tackled via gradient ascent:\n",
    "\n",
    "$$\n",
    "    \\begin{align}\n",
    "        \\mu^{(i+1)} &\\leftarrow \\mu^{(i)} + \\alpha \\nabla_{\\mu} \\mathcal{L}, \\\\\n",
    "        \\omega^{(i+1)} &\\leftarrow \\omega^{(i)} + \\alpha \\nabla_{\\omega} \\mathcal{L},\n",
    "            & \\mathrm{(mean \\, field)} \\\\\n",
    "        L^{(i+1)} &\\leftarrow L^{(i)} + \\alpha \\nabla_{L} \\mathcal{L},\n",
    "            & \\mathrm{(full \\, rank)}\n",
    "    \\end{align}\n",
    "$$\n",
    "\n",
    "where $\\nabla \\mathcal{L}$ is the gradient of the ELBO."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### EBLO gradients\n",
    "\n",
    "* we would like to compute the gradients of the ELBO using automatic differentiation methods, but the ELBO has the unknown expectation $\\mathbb{E}_{q(\\zeta;\\phi)}$\n",
    "* to proceed, we'll apply another transformation that will ultimately allow us to take the gradients inside the expectation\n",
    "* specifically, the transformation $S_{\\phi}$ absorbs the variational parameters $\\phi$, and converts the variational approximation Gaussian into a standard Gaussian\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\eta = S_{\\phi}(\\zeta) &= \\mathrm{diag}(\\exp(\\omega)^2)^{-1} (\\zeta - \\mu)\n",
    "        & \\mathrm{(mean \\, field)} \\\\\n",
    "    \\eta = S_{\\phi}(\\zeta) &= L^{-1} (\\zeta - \\mu) \n",
    "        & \\mathrm{(full \\, rank)}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The variational density then becomes:\n",
    "$$\n",
    "    q(\\eta) = \\mathcal{N}(\\eta; 0, I) = \\prod_{k=1}^{K} \\mathcal{N}(\\eta_k; 0, I),\n",
    "$$\n",
    "\n",
    "and the optimization problem becomes:\n",
    "\n",
    "$$\n",
    "    \\phi^{*} = \\arg\\max_{\\phi}\n",
    "    \\mathbb{E}_{\\mathcal{N}(\\eta; 0, I)} \\left[\n",
    "        \\log p \\left(\n",
    "            x, T^{-1}\\left(S_{\\phi}^{-1}(\\eta)\\right)\n",
    "        \\right)\n",
    "        +\n",
    "        \\log \\left| \n",
    "            \\det J_{T^{-1}} \\left(S_{\\phi}^{-1}(\\eta)\\right)\n",
    "        \\right|\n",
    "    \\right]\n",
    "    +\n",
    "    \\mathbb{H} \\left[\n",
    "        q(\\zeta; \\phi)\n",
    "    \\right]\n",
    "$$\n",
    "\n",
    "* since the expectation no longer depends on $\\phi$, we can push the gradient inside of the expectation:\n",
    "\n",
    "$$\n",
    "    \\nabla_{\\mu} \\mathcal{L} = \n",
    "    \\mathbb{E}_{\\mathcal{N}(\\eta)} \\left[\n",
    "        \\nabla_{\\theta} \\log p(x, \\theta)\n",
    "        \\nabla_{\\zeta} T^{-1}(\\zeta)\n",
    "        +\n",
    "        \\nabla_{\\zeta} \\log \\left|\n",
    "            \\det J_{T^{-1}}(\\zeta)\n",
    "        \\right|\n",
    "    \\right]\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\nabla_{\\omega} \\mathcal{L} &= \n",
    "    \\mathbb{E}_{\\mathcal{N}(\\eta)} \\left[\n",
    "        \\left(        \n",
    "            \\nabla_{\\theta} \\log p(x, \\theta)\n",
    "            \\nabla_{\\zeta} T^{-1}(\\zeta)\n",
    "            +\n",
    "            \\nabla_{\\zeta} \\log \\left|\n",
    "                \\det J_{T^{-1}}(\\zeta)\n",
    "            \\right|\n",
    "        \\right)\n",
    "        \\eta^{T} \\mathrm{diag}(\\exp(\\omega))\n",
    "    \\right]\n",
    "    + 1         & \\mathrm{(mean \\, field)} \\\\\n",
    "    \\nabla_{L} \\mathcal{L} &= \n",
    "    \\mathbb{E}_{\\mathcal{N}(\\eta)} \\left[\n",
    "        \\left(        \n",
    "            \\nabla_{\\theta} \\log p(x, \\theta)\n",
    "            \\nabla_{\\zeta} T^{-1}(\\zeta)\n",
    "            +\n",
    "            \\nabla_{\\zeta} \\log \\left|\n",
    "                \\det J_{T^{-1}}(\\zeta)\n",
    "            \\right|\n",
    "        \\right)\n",
    "        \\eta^{T}\n",
    "    \\right]\n",
    "    +\n",
    "    (L^{-1})^{T} & \\mathrm{(full \\, rank)} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "* the gradients inside the expectation can now be computed using automatic differentiation\n",
    "* the expectation itself can be estimated using MC integration\n",
    "* that is, we can sample from the standard Gaussian and evaluate the emperical mean of the gradients in within the expectation\n",
    "* this procedure gives noisy unbiased gradients of the ELBO of any differentiable probability model, and we can further use those gradients for a stochastic optimization for the variational inference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stochastic gradient ascent\n",
    "\n",
    "* armed with the noisy gradients, the optimization of the ELBO can proceed via the gradient ascent approach discussed above\n",
    "* the stepsize, $\\alpha$ can be made to be made to vary, and a procedure outlined in the paper discusses the calculation of the stepsize at each iteration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complexity\n",
    "\n",
    "* ADVI has complexity $\\vartheta(NMK)$ per iteration, where $N$ is the number of datapoints, $M$ is the number of MC samples, and $K$ is the number of latent variables\n",
    "* scaling to large datasets can be done by subsampling. For a minibatch of size $B \\ll N$, the complexity per iteration is $\\vartheta(BMK)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate a small dataset dataset with fixed probability `p=0.2` of an entry being a 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0]\n"
     ]
    }
   ],
   "source": [
    "data = np.random.binomial(1, 0.2, size=20)\n",
    "print data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll make a simple, but not trivial, model to try out the ADVI methods in `pymc3` and `Stan`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## pymc3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I think `pymc3` currently only supports the Mean-Field version of ADVI."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Couldn't import dot_parser, loading of dot files will not be possible.\n"
     ]
    }
   ],
   "source": [
    "import pymc3 as pm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Applied log-transform to alpha and added transformed alpha_log to model.\n",
      "Applied log-transform to beta and added transformed beta_log to model.\n",
      "Applied logodds-transform to p and added transformed p_logodds to model.\n"
     ]
    }
   ],
   "source": [
    "with pm.Model() as model:\n",
    "    alpha = pm.HalfCauchy('alpha', 20)\n",
    "    beta  = pm.HalfCauchy('beta', 50)\n",
    "    \n",
    "    prior = pm.Beta('p', alpha=alpha, beta=beta)\n",
    "    likelihood = pm.Binomial('binomial', n=len(data), p=prior, observed=sum(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For `pymc3`, `accurate_elbo=False` means that the number of MC samples per iteration is set to $M=1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0 [0%]: ELBO = -3.9\n",
      "Iteration 400 [10%]: Average ELBO = -28.8\n",
      "Iteration 800 [20%]: Average ELBO = -24.09\n",
      "Iteration 1200 [30%]: Average ELBO = -20.25\n",
      "Iteration 1600 [40%]: Average ELBO = -18.58\n",
      "Iteration 2000 [50%]: Average ELBO = -17.92\n",
      "Iteration 2400 [60%]: Average ELBO = -15.24\n",
      "Iteration 2800 [70%]: Average ELBO = -12.94\n",
      "Iteration 3200 [80%]: Average ELBO = -11.75\n",
      "Iteration 3600 [90%]: Average ELBO = -10.62\n",
      "Finished [100%]: Average ELBO = -10.56\n",
      "CPU times: user 3.51 s, sys: 369 ms, total: 3.88 s\n",
      "Wall time: 10.8 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "advi_fit = pm.variational.advi(model=model, n=4000, accurate_elbo=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wGW9/BcL/3+Cce9bMmoF/BN6DVxUW3QW8HZgKdAJ7o8o7gLqYspSMD1Wzq/UQ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\nvlFSERER3/x/L1vCyL3kpioAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1188befd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(advi_fit.elbo_vals)\n",
    "plt.xlabel('Iteration')\n",
    "plt.ylabel(\"ELBO\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ADVIFit(means={'alpha_log_': array(2.7251985780730963), 'beta_log_': array(3.750899891216531), 'p_logodds_': array(-1.0869779295409336)}, stds={'alpha_log_': 0.61737582209059227, 'beta_log_': 0.64250377477510057, 'p_logodds_': 0.57563120930079181}, elbo_vals=array([ -3.90476844,  -2.82537291, -42.19240793, ..., -11.15347832,\n",
      "        -3.4346194 ,  -4.77228724]))\n"
     ]
    }
   ],
   "source": [
    "print advi_fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the other hand, with `accurate_elbo=True`, $M=100$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0 [0%]: ELBO = -35.23\n",
      "Iteration 400 [10%]: Average ELBO = -26.89\n",
      "Iteration 800 [20%]: Average ELBO = -18.48\n",
      "Iteration 1200 [30%]: Average ELBO = -12.9\n",
      "Iteration 1600 [40%]: Average ELBO = -9.64\n",
      "Iteration 2000 [50%]: Average ELBO = -7.33\n",
      "Iteration 2400 [60%]: Average ELBO = -5.98\n",
      "Iteration 2800 [70%]: Average ELBO = -5.18\n",
      "Iteration 3200 [80%]: Average ELBO = -4.76\n",
      "Iteration 3600 [90%]: Average ELBO = -4.58\n",
      "Finished [100%]: Average ELBO = -4.53\n",
      "CPU times: user 25.3 s, sys: 576 ms, total: 25.9 s\n",
      "Wall time: 33.1 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "advi_fit = pm.variational.advi(model=model, n=4000, accurate_elbo=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BzfEJKXlKCns2EK+n07WLiKSrcG0eNwJ/N8b8CnjTd/484CKcgXwZo19J9OMx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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11adb8990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(advi_fit.elbo_vals)\n",
    "plt.xlabel('Iteration')\n",
    "plt.ylabel(\"ELBO\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ADVIFit(means={'alpha_log_': array(2.0496932971415927), 'beta_log_': array(3.6787438305091418), 'p_logodds_': array(-1.6862302056890575)}, stds={'alpha_log_': 0.35386626709018154, 'beta_log_': 0.3674171620327436, 'p_logodds_': 0.33450676412542907}, elbo_vals=array([-35.22716538, -23.25196236, -37.7726848 , ...,  -4.28355296,\n",
      "        -4.70956574,  -4.35644914]))\n"
     ]
    }
   ],
   "source": [
    "print advi_fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get a good look at the approximate posterior, we can use the `sample_vp` function, which draws Gaussian samples in the real-space, then transforms the samples back to latent variable space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "advi_trace = pm.variational.sample_vp(advi_fit, draws=5000, model=model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare with the NUTS sampler:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 5000 of 5000 complete in 14.9 sec"
     ]
    }
   ],
   "source": [
    "with model:\n",
    "    step = pm.NUTS()\n",
    "    nuts_trace = pm.sample(5000, step)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lEAeAi4E9ZnYOcDCv7xBwmpl1ABniU9o3ljL9KzA7ferY6CST09nVm961YLrW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\nSGSUnEVERCKj5CwiIhKZ/wPT2S6dJSrMuQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11dd6f990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8,8))\n",
    "plt.subplot(2,2,1)\n",
    "\n",
    "plt.hist(advi_trace['p'], range=[0,1], bins=50, normed=True, label='ADVI', alpha=0.5)\n",
    "plt.hist(nuts_trace['p'], range=[0,1], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.xlabel(\"p\")\n",
    "\n",
    "plt.subplot(2,2,2)\n",
    "plt.hist(advi_trace['alpha'], range=[0,100], bins=50, normed=True, label='ADVI', alpha=0.5)\n",
    "plt.hist(nuts_trace['alpha'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.xlabel(\"alpha\")\n",
    "plt.legend()\n",
    "\n",
    "plt.subplot(2,2,4)\n",
    "plt.hist(advi_trace['beta'], range=[0,100], bins=50, normed=True, label='ADVI', alpha=0.5)\n",
    "plt.hist(nuts_trace['beta'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.xlabel(\"beta\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pystan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from pystan import StanModel, stan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "model_code = \"\"\"\n",
    "    data {\n",
    "        int<lower=0> N;\n",
    "        int<lower=0,upper=1> y[N];\n",
    "    }\n",
    "    parameters {\n",
    "        real<lower=0> alpha;\n",
    "        real<lower=0> beta;\n",
    "        real<lower=0, upper=1> p;\n",
    "    }\n",
    "    model {\n",
    "        alpha ~ cauchy(0, 20);\n",
    "        beta ~ cauchy(0, 50);\n",
    "        p ~ beta(alpha, beta);\n",
    "        for (n in 1:N)\n",
    "            y[n] ~ bernoulli(p);\n",
    "    }\n",
    "\"\"\"\n",
    "\n",
    "stan_data = {\n",
    "    'N':len(data),\n",
    "    'y':data\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pystan NUTS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, for comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "stan_nuts = stan(model_code=model_code, data=stan_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Inference for Stan model: anon_model_2df7ffd52b91f10700e84d4a94d52b93.\n",
      "4 chains, each with iter=2000; warmup=1000; thin=1; \n",
      "post-warmup draws per chain=1000, total post-warmup draws=4000.\n",
      "\n",
      "        mean se_mean     sd   2.5%    25%    50%    75%  97.5%  n_eff   Rhat\n",
      "alpha  33.27   11.63  47.95   1.84   7.87  15.29  30.69 190.23   17.0   1.22\n",
      "beta  176.84   36.37 241.25  12.47  43.91  84.18 182.95  895.4   44.0   1.09\n",
      "p       0.16  3.9e-3   0.07   0.04   0.11   0.15   0.21   0.33  370.0   1.02\n",
      "lp__   -4.51    0.08   1.37  -7.95  -5.29  -4.23  -3.45  -2.82  279.0   1.01\n",
      "\n",
      "Samples were drawn using NUTS at Wed Jul 13 10:46:45 2016.\n",
      "For each parameter, n_eff is a crude measure of effective sample size,\n",
      "and Rhat is the potential scale reduction factor on split chains (at \n",
      "convergence, Rhat=1).\n"
     ]
    }
   ],
   "source": [
    "print stan_nuts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pystan Mean-field\n",
    "\n",
    "Note: although Stan's ADVI feature has been around for a couple months, the PyStan integration for ADVI was pushed literally yesterday, so no guarantees on the bleeding edge."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "stan_model = StanModel(model_code=model_code)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:pystan:Automatic Differentiation Variational Inference (ADVI) is an EXPERIMENTAL ALGORITHM.\n",
      "WARNING:pystan:ADVI samples may be found on the filesystem in the file `/var/folders/ks/ny8mvlt11hbfnk6vcdd8ghx4b6qs2l/T/tmpYzZ6_3/output.csv`\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "OrderedDict([('args', {'algorithm': 'MEANFIELD', 'random_seed': '89366828', 'tol_rel_obj': 0.01, 'eval_elbo': 100, 'sample_file': '/var/folders/ks/ny8mvlt11hbfnk6vcdd8ghx4b6qs2l/T/tmpYzZ6_3/output.csv', 'output_samples': 10000, 'init_radius': 2.0, 'chain_id': 1, 'iter': 10000, 'adapt_engaged': True, 'init': 'random', 'eta': 1.0, 'grad_samples': 1, 'elbo_samples': 100, 'append_samples': False, 'enable_random_init': False, 'method': 'VARIATIONAL', 'adapt_iter': 50}), ('mean_pars', [])])\n"
     ]
    }
   ],
   "source": [
    "results = stan_model.vb(data=stan_data, iter=10000, output_samples=10000, algorithm='meanfield')\n",
    "print(results)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The model samples are currently being written to disk, so we'll import them back in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "meanfield_samples = pd.read_csv(\n",
    "    results['args']['sample_file'],\n",
    "    skiprows=7,\n",
    "    names=['alpha','beta','p']\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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w193vplDT7YjItHTPWaR8dBOW1B0Tc/dgso3TzgMCM1sNHA3cZGbnuPveqXaarJJZvkrp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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11dd6f3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8,8))\n",
    "plt.subplot(2,2,1)\n",
    "\n",
    "plt.hist(stan_nuts['p'], range=[0,1], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(meanfield_samples['p'].values, range=[0,1], bins=50,\n",
    "         normed=True, label='Mean-field ADVI', alpha=0.5)\n",
    "plt.xlabel(\"p\")\n",
    "\n",
    "plt.subplot(2,2,2)\n",
    "plt.hist(stan_nuts['alpha'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(meanfield_samples['alpha'], range=[0,100], bins=50,\n",
    "         normed=True, label='Mean-field ADVI', alpha=0.5)\n",
    "plt.xlabel(\"alpha\")\n",
    "plt.legend()\n",
    "\n",
    "plt.subplot(2,2,4)\n",
    "plt.hist(stan_nuts['beta'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(meanfield_samples['beta'].values, range=[0,100], bins=50,\n",
    "         normed=True, label='Mean-field ADVI', alpha=0.5)\n",
    "plt.xlabel(\"beta\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pystan Full-rank\n",
    "\n",
    "We can also look at the output of the Full-Rank flavour of ADVI:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:pystan:Automatic Differentiation Variational Inference (ADVI) is an EXPERIMENTAL ALGORITHM.\n",
      "WARNING:pystan:ADVI samples may be found on the filesystem in the file `/var/folders/ks/ny8mvlt11hbfnk6vcdd8ghx4b6qs2l/T/tmpIbhBl1/output.csv`\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "OrderedDict([('args', {'algorithm': 'FULLRANK', 'random_seed': '1131162010', 'tol_rel_obj': 0.01, 'eval_elbo': 100, 'sample_file': '/var/folders/ks/ny8mvlt11hbfnk6vcdd8ghx4b6qs2l/T/tmpIbhBl1/output.csv', 'output_samples': 10000, 'init_radius': 2.0, 'chain_id': 1, 'iter': 10000, 'adapt_engaged': True, 'init': 'random', 'eta': 1.0, 'grad_samples': 1, 'elbo_samples': 100, 'append_samples': False, 'enable_random_init': False, 'method': 'VARIATIONAL', 'adapt_iter': 50}), ('mean_pars', [])])\n"
     ]
    }
   ],
   "source": [
    "fullrank_results = stan_model.vb(data=stan_data, iter=10000,\n",
    "                                 output_samples=10000, algorithm='fullrank')\n",
    "print fullrank_results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "fullrank_samples = pd.read_csv(\n",
    "    fullrank_results['args']['sample_file'],\n",
    "    skiprows=7,\n",
    "    names=['alpha','beta','p']\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jc6Kmy8C87XOta25K0NmZmfcz1Uq932854h5j3ONbgZ3AxcB2Mzsb2FXR9jBw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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11dd6f550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8,8))\n",
    "plt.subplot(2,2,1)\n",
    "\n",
    "plt.hist(stan_nuts['p'], range=[0,1], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(fullrank_samples['p'].values, range=[0,1], bins=50,\n",
    "         normed=True, label='Full-rank ADVI', alpha=0.5)\n",
    "plt.xlabel(\"p\")\n",
    "\n",
    "plt.subplot(2,2,2)\n",
    "plt.hist(stan_nuts['alpha'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(fullrank_samples['alpha'], range=[0,100], bins=50,\n",
    "         normed=True, label='Full-rank ADVI', alpha=0.5)\n",
    "plt.xlabel(\"alpha\")\n",
    "plt.legend()\n",
    "\n",
    "plt.subplot(2,2,4)\n",
    "plt.hist(stan_nuts['beta'], range=[0,100], bins=50, normed=True, label='NUTS', alpha=0.5)\n",
    "plt.hist(fullrank_samples['beta'].values, range=[0,100],\n",
    "         bins=50, normed=True, label='Full-rank ADVI', alpha=0.5)\n",
    "plt.xlabel(\"beta\");"
   ]
  }
 ],
 "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": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}