Variational Inference: Foundations and Modern Methods (NIPS 2016 tutorial)

David Blei, Rajesh Ranganath, Shakir Mohamed.
One of the core problems of modern statistics and machine learning is to approximate difficult-to-compute probability distributions. This problem is especially important in probabilistic modeling, which frames all inference about unknown quantities as a calculation about a conditional distribution. In this tutorial we review and discuss variational inference (VI), a method a that approximates probability distributions through optimization. VI has been used in myriad applications in machine learning and tends to be faster than more traditional methods, such as Markov chain Monte Carlo sampling. Brought into machine learning in the 1990s, recent advances and easier implementation have renewed interest and application of this class of methods. This tutorial aims to provide both an introduction to VI with a modern view of the field, and an overview of the role that probabilistic inference plays in many of the central areas of machine learning.
The tutorial has three parts. First, we provide a broad review of variational inference from several perspectives. This part serves as an introduction (or review) of its central concepts. Second, we develop and connect some of the pivotal tools for VI that have been developed in the last few years, tools like Monte Carlo gradient estimation, black box variational inference, stochastic approximation, and variational auto-encoders. These methods have lead to a resurgence of research and applications of VI. Finally, we discuss some of the unsolved problems in VI and point to promising research directions.

[@yididiyayilma900]

30:00 The most intuitive explaination for stochastic optimization I have ever heard so far.

[@Filaaaix]

At 1:27:30 => I didn't really get how you derive the Auxiliary variational bound. Is there a good source where it's explained more thoroughly?

[@nicodetullio]

lo

[@alexeygritsenko9955]

At 44:43 - why does the score function have expectation of zero?

[@mataneyal]

\begin{equation} \begin{aligned} \mathbb{E}_q [ abla_ u g(z; u)] &= \mathbb{E}_q [ abla_ u \log p(x,z) - abla_ u \log q(z; u)]\\ &= - \mathbb{E}_q [ abla_ u \log q(z; u)] ~ \textrm{($\log p(x,z)$ is not a function of $ u$)}\\ &= - \int q(z; u) abla_ u \log q(z; u) \\ &= - \int abla_ u q(z; u)~\textrm{(Log Derivative Trick)}\\ &= 0 ~\textrm{($q(z; u)$ is a continuous probability distribution)} \end{aligned} \end{equation}

[@maxturgeon89]

Chain rule and dominated convergence theorem