Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes: 1. * To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables. 2. * To derive a lower bound for the marginal likelihood (sometimes called the evidence) of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.) In the former purpose (that of approximating a posterior probability), variational Bayes is an alternative to Monte Carlo sampling methods—particularly, Markov chain Monte Carlo methods such as Gibbs sampling—for taking a fully Bayesian approach to statistical inference over complex distributions that are difficult to evaluate directly or sample. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. Variational Bayes can be seen as an extension of the expectation-maximization (EM) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be solved analytically. For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed. However, deriving the set of equations used to update the parameters iteratively often requires a large amount of work compared with deriving the comparable Gibbs sampling equations. This is the case even for many models that are conceptually quite simple, as is demonstrated below in the case of a basic non-hierarchical model with only two parameters and no latent variables. (Wikipedia).
Christine Keribin: Variational Bayes methods and algorithms - Part 1
Abstract: Bayesian posterior distributions can be numerically intractable, even by the means of Markov Chain Monte Carlo methods. Bayesian variational methods can then be used to compute directly (and fast) a deterministic approximation of these posterior distributions. In this course, I d
From playlist Probability and Statistics
Free ebook http://tinyurl.com/EngMathYT I show how to solve differential equations by applying the method of variation of parameters for those wanting to review their understanding.
From playlist Differential equations
Bayesian vs frequentist statistics probability - part 1
This video provides an intuitive explanation of the difference between Bayesian and classical frequentist statistics. If you are interested in seeing more of the material, arranged into a playlist, please visit: https://www.youtube.com/playlist?list=PLFDbGp5YzjqXQ4oE4w9GVWdiokWB9gEpm Unfo
From playlist Bayesian statistics: a comprehensive course
Bayesian vs frequentist statistics
This video provides an intuitive explanation of the difference between Bayesian and classical frequentist statistics. If you are interested in seeing more of the material, arranged into a playlist, please visit: https://www.youtube.com/playlist?list=PLFDbGp5YzjqXQ4oE4w9GVWdiokWB9gEpm Un
From playlist Bayesian statistics: a comprehensive course
Variation of Constants / Parameters
Download the free PDF http://tinyurl.com/EngMathYT A basic illustration of how to apply the variation of constants / parameters method to solve second order differential equations.
From playlist Differential equations
Stanford CS330: Deep Multi-task and Meta Learning | 2020 | Lecture 8 - Bayesian Meta-Learning
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/ai To follow along with the course, visit: https://cs330.stanford.edu/ To view all online courses and programs offered by Stanford, visit: http://online.stanford.
From playlist Stanford CS330: Deep Multi-task and Meta Learning | Autumn 2020
Variation of Parameters for Systems of Differential Equations
This is the second part of the variation of parameters-extravaganza! In this video, I show you how to use the same method in the last video to solve inhomogeneous systems of differential equations. Witness how linear algebra makes this method so elegant!
From playlist Differential equations
Nineteenth Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series Talk
Date: Wednesday, March 24, 2021, 10:00am Eastern Time Zone (US & Canada) Speaker: Marcelo Pereyra, Heriot-Watt University Abstract: Play & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) es
From playlist Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series
Variational Bayesian NNs and Resolution of Singularities - Singular Learning Theory Seminar 35
Edmund Lau presents recent work jointly with Susan Wei, on variational inference, Bayesian neural networks and how this field can be improved using ideas from singular learning theory. You can join this seminar from anywhere, on any device, at https://www.metauni.org. All are welcome. Th
From playlist Singular Learning Theory
Stanford CS330: Multi-Task and Meta-Learning, 2019 | Lecture 5 - Bayesian Meta-Learning
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/ai Assistant Professor Chelsea Finn, Stanford University http://cs330.stanford.edu/
From playlist Stanford CS330: Deep Multi-Task and Meta Learning
Stanford CS330: Deep Multi-task & Meta Learning I 2021 I Lecture 7
For more information about Stanford's Artificial Intelligence professional and graduate programs visit: https://stanford.io/ai To follow along with the course, visit: http://cs330.stanford.edu/fall2021/index.html To view all online courses and programs offered by Stanford, visit: http:/
From playlist Stanford CS330: Deep Multi-Task & Meta Learning I Autumn 2021I Professor Chelsea Finn
Stanford CS330 Deep Multi-Task & Meta Learning - Bayesian Meta-Learning l 2022 I Lecture 12
For more information about Stanford's Artificial Intelligence programs visit: https://stanford.io/ai To follow along with the course, visit: https://cs330.stanford.edu/ To view all online courses and programs offered by Stanford, visit: http://online.stanford.edu Chelsea Finn Computer
From playlist Stanford CS330: Deep Multi-Task and Meta Learning I Autumn 2022
Statistical Rethinking - Lecture 01
The Golem of Prague / Small World and Large Worlds: Chapters 1 and 2 of 'Statistical Rethinking: A Bayesian Course with R Examples'.
From playlist Statistical Rethinking Winter 2015
Differential Equations | Variation of Parameters.
We derive the general form for a solution to a differential equation using variation of parameters. http://www.michael-penn.net
From playlist Differential Equations
Variational Bayes: An Overview and Risk-Sensitive Formulations by Harsha Honnappa
PROGRAM: ADVANCES IN APPLIED PROBABILITY ORGANIZERS: Vivek Borkar, Sandeep Juneja, Kavita Ramanan, Devavrat Shah, and Piyush Srivastava DATE & TIME: 05 August 2019 to 17 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Applied probability has seen a revolutionary growth in resear
From playlist Advances in Applied Probability 2019
C33 Example problem using variation of parameters
Another example problem using the method of variation of parameters on second-order, linear, ordinary DE's.
From playlist Differential Equations
A. Eberle: Couplings & converg. to equilibrium f. Langevin dyn. & Hamiltonian Monte Carlo methods
The lecture was held within the framework of the Hausdorff Trimester Program: Kinetic Theory Abstract: Coupling methods provide a powerful approach to quantify convergence to equilibrium of Markov processes in appropriately chosen Wasserstein distances. This talk will give an overview on
From playlist Workshop: Probabilistic and variational methods in kinetic theory
The virtue of Bayesian analysis in food risk assessment, Jukka Ranta - Bayes@Lund 2018
Find more info about Bayes@Lund, including slides, here: https://bayesat.github.io/lund2018/bayes_at_lund_2018.html
From playlist Bayes@Lund 2018
Derive the Variation of Parameters Formula to Solve Linear Second Order Nonhomogeneous DEs
This video derives or proves the variation of parameters formula used to find a particular solution and solve linear second order nonhomogeneous differential equations. Site: http://mathispower4u.com
From playlist Linear Second Order Nonhomogeneous Differential Equations: Variation of Parameters