{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Variational Inference\n", "\n", "Inference and learning often involve intractable integrals. \n", "\n", "Prime examples include __bayesian inference__\n", "\n", "$$p(y | \\mathbf{x}, \\mathcal{D})=\\int p(y, \\mathbf{w} | \\mathbf{x}, \\mathcal{D}) \\mathrm{d} \\mathbf{w}=\\int p(y | \\mathbf{x}, \\mathbf{w}) p(\\mathbf{w} | \\mathcal{D}) \\mathrm{d} \\mathbf{w}$$\n", "\n", "\n", "or marginalization of unseen variables\n", "\n", "$$L(\\boldsymbol{\\theta})=p(\\mathcal{D} ; \\boldsymbol{\\theta})=\\int_{\\mathbf{u}} p(\\mathbf{u}, \\mathcal{D} ; \\boldsymbol{\\theta}) \\mathrm{d} \\mathbf{u}$$\n", "\n", "\n", "There are two possible methods to approximate such integrals:\n", "- Monte Carlo estimate through sampling\n", "- Variational approach\n", "\n", "In this notebook I will focus on this second approach.\n", "\n", "## Kullback-Leibler divergence\n", "\n", "The KL divergence is a fundamental concept in variational inference and consequently for variational autoencoders.\n", "The KL divergence between two distributions $p$ and $q$ is:\n", "\n", "$$\\mathrm{KL}(p \\| q)=\\int p(\\mathbf{x}) \\,\\log \\left(\\frac{p(\\mathbf{x})}{q(\\mathbf{x})} \\right)\\mathrm{d} \\mathbf{x}=\\mathbb{E}_{p(\\mathbf{x})}\\left[\\log \\frac{p(\\mathbf{x})}{q(\\mathbf{x})}\\right]$$\n", "\n", "Properties of KL divergence:\n", "- $\\mathrm{KL}(p \\| q)=0$ iff $p = q$ \n", "- $\\mathrm{KL}(p \\| q)\\neq \\mathrm{KL}(q \\| p)$ (non-commutative)\n", "- $\\mathrm{KL}(p \\| q)\\geq0$ (always non-negative)\n", "\n", "## Variational principle\n", "\n", "Given a joint distribution $p(\\mathbf{x}, \\mathbf{y})$, the Variational principle states that we can __formulate inference tasks__ such as marginalization $p(\\mathbf{x})=\\int p(\\mathbf{x}, \\mathbf{y}) \\mathrm{d} \\mathbf{y}$, and conditioning $p(\\mathbf{y} | \\mathbf{x})$, __as optimization problems__.\n", "\n", "Specifically, the maximisation of variational free energy\n", "\n", "$$\\mathcal{F}(\\mathbf{x}, q)=\\mathbb{E}_{q(\\mathbf{y})}\\left[\\log \\frac{p(\\mathbf{x}, \\mathbf{y})}{q(\\mathbf{y})}\\right]$$\n", "\n", "leads to \n", "- $\\log p(\\mathbf{x})=\\max _{q(\\mathbf{y})} \\mathcal{F}(\\mathbf{x}, q)$\n", "- $p(\\mathbf{y} | \\mathbf{x})=\\operatorname{argmax}_{q(\\mathbf{y})} \\mathcal{F}(\\mathbf{x}, q)$\n", "\n", "\n", "By separating the joint distribution $p$ in the formulation of the free energy, we find that\n", "$$\\log p\\left(\\mathbf{x}\\right)=\\mathrm{KL}\\left(q(\\mathbf{y}) \\| p\\left(\\mathbf{y} | \\mathbf{x}\\right)\\right)+\\mathcal{F}\\left(\\mathbf{x}, q\\right)=\\mathrm{const}$$\n", "\n", "Meaning that maximising the variational free energy is equivalent to minimising the KL divergence $\\mathrm{KL}(q \\| p)$\n", "\n", "Since the KL-divergence is always non-negative, $\\mathcal{F}$ is also referred to as the Evidence Lower Bound (ELBO), since it provides a lower bound for the marginal likelihood.\n", "$$\\log p\\left(\\mathbf{x}\\right)\\geq\\mathcal{F}\\left(\\mathbf{x}, q\\right)$$\n", "\n", "In variational inference the $q$ distribution involved in the ELBO is parametrised as $q(\\mathbf{y}; \\mathbf{\\theta})$, and the parameters are optimised to push the ELBO as high as possible." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example implementation: variational inference in 1D" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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