{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/krasserm/bayesian-machine-learning/blob/master/latent_variable_models_part_1.ipynb)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "try:\n", " # Check if notebook is running in Google Colab\n", " import google.colab\n", " # Get additional files from Github\n", " !wget https://raw.githubusercontent.com/krasserm/bayesian-machine-learning/master/latent_variable_models_util.py\n", " # Install additional dependencies\n", " !pip install daft==0.1.0\n", "except:\n", " pass" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Latent variable models - part 1: Gaussian mixture models and the EM algorithm\n", "\n", "This is part 1 of a two-part series of articles about latent variable models. Part 1 covers the expectation maximization (EM) algorithm and its application to Gaussian mixture models. [Part 2](latent_variable_models_part_2.ipynb) covers approximate inference and variational autoencoders.\n", "\n", "## Introduction\n", "\n", "Given a probabilistic model $p(\\mathbf{x} \\lvert \\boldsymbol{\\theta})$ and $N$ observations $\\mathbf{X} = \\left\\{ \\mathbf{x}_1, \\ldots, \\mathbf{x}_N \\right\\}$ we often want to find a value for parameter $\\boldsymbol{\\theta}$ that maximizes the likelihood function $p(\\mathbf{X} \\lvert \\boldsymbol{\\theta})$, a function of parameter $\\boldsymbol{\\theta}$. This is known as [maximimum likelihood estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE). \n", "\n", "$$\n", "\\boldsymbol{\\theta}_{MLE} = \\underset{\\boldsymbol{\\theta}}{\\mathrm{argmax}} p(\\mathbf{X} \\lvert \\boldsymbol{\\theta})\n", "\\tag{1}\n", "$$\n", "\n", "If the model is a simple probability distribution, like a single Gaussian, for example, then $\\boldsymbol{\\theta}_{MLE} = \\left\\{ \\boldsymbol{\\mu}_{MLE}, \\boldsymbol{\\Sigma}_{MLE} \\right\\}$ has an analytical solution. A common approach for more complex models is *gradient descent* using the *negative log likelihood*, $-\\log p(\\mathbf{X} \\lvert \\boldsymbol{\\theta})$, as loss function. This can easily be implemented with frameworks like PyTorch or Tensorflow provided that $p(\\mathbf{X} \\lvert \\boldsymbol{\\theta})$ is differentiable w.r.t. $\\boldsymbol{\\theta}$. But this is not necessarily the most efficient approach.\n", "\n", "## Gaussian mixture model\n", "\n", "MLE can often be simplified by introducing *latent variables*. A latent variable model makes the assumption that an observation $\\mathbf{x}_i$ is caused by some underlying latent variable, a variable that cannot be observed directly but can be inferred from observed variables and parameters. For example, the following plot shows observations in 2-dimensional space and one can see that their overall distribution doesn't seem follow a simple distribution like a single Gaussian." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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