Statistical algorithms | Approximations
In mathematics, Laplace's approximation fits an un-normalised Gaussian approximation to a (twice differentiable) un-normalised target density. In Bayesian statistical inference this is useful to simultaneously approximate the posterior and the marginal likelihood, see also Approximate inference. The method works by matching the log density and curvature at a mode of the target density. For example, a (possibly non-linear) regression or classification model with data set comprising inputs and outputs has (unknown) parameter vector of length . The likelihood is denoted and the parameter prior . The joint density of outputs and parameters is the object of inferential desire The joint is equal to the product of the likelihood and the prior and by Bayes' rule, equal to the product of the marginal likelihood and posterior . Seen as a function of the joint is an un-normalised density. In Laplace's approximation we approximate the joint by an un-normalised Gaussian , where we use to denote approximate density, for un-normalised density and is a constant (independent of ). Since the marginal likelihood doesn't depend on the parameter and the posterior normalises over we can immediately identify them with and of our approximation, respectively. Laplace's approximation is where we have defined where is the location of a mode of the joint target density, also known as the maximum a posteriori or MAP point and is the positive definite matrix of second derivatives of the negative log joint target density at the mode . Thus, the Gaussian approximation matches the value and the curvature of the un-normalised target density at the mode. The value of is usually found using a gradient based method, e.g. Newton's method. In summary, we have for the approximate posterior over and the approximate log marginal likelihood respectively. In the special case of Bayesian linear regression with a Gaussian prior, the approximation is exact. The main weaknesses of Laplace's approximation are that it is symmetric around the mode and that it is very local: the entire approximation is derived from properties at a single point of the target density. Laplace's method is widely used and was pioneered in the context of neural networks by David MacKay and for Gaussian processes by Williams and Barber, see references. (Wikipedia).
Differential Equations | The Laplace Transform of a Derivative
We establish a formula involving the Laplace transform of the derivative of a function. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Laplace Transform
Discrete Laplace Equation | Lecture 62 | Numerical Methods for Engineers
Derivation of the discrete Laplace equation using the central difference approximations for the partial derivatives. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscr
From playlist Numerical Methods for Engineers
Introduction to Laplace Transforms
Introduction to Laplace Transforms A full introduction. The definition is given, remarks are made, and an example of finding the laplace transform of a function with the definition is done.
From playlist Differential Equations
C75 Introduction to the Laplace Transform
Another method of solving differential equations is by firs transforming the equation using the Laplace transform. It is a set of instructions, just like differential and integration. In fact, a function is multiplied by e to the power negative s times t and the improper integral from ze
From playlist Differential Equations
Differential Equations | Laplace Transform of a Piecewise Function
We find the Laplace transform of a piecewise function using the unit step function. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Laplace Transform
A quick run over different approximation methods an why we would use them. The reference to the Skogestad method is to this video: https://youtu.be/pSG1FBxCvkE
From playlist Laplace
We see some examples of how we can use the properties of solutions to Laplace's Equation to "guess" solutions for the electric potential in some simple cases.
From playlist Phys 331 Uploads
Introduction to Laplace transforms
Free ebook https://bookboon.com/en/partial-differential-equations-ebook A basic introduction to the Laplace transform. We define it and show how to calculate Laplace transforms from the definition. We also discuss inverse transforms and how to use a table of transforms. Such ideas have
From playlist Partial differential equations
Derivation of Laplace's Equation In this video, I derive Laplace's equation from first principles, staring with a fluid in equilibrium and using the divergence theorem. This derivation is particularly elegant and coordinate-free. Enjoy! Subscribe to my channel: https://www.youtube.com/c/
From playlist Partial Differential Equations
Introduction to linear analysis
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From playlist Laplace
8ECM Invited Lecture: Nick Trefethen
From playlist 8ECM Invited Lectures
Lecture 23: Physically Based Animation and PDEs (CMU 15-462/662)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz2emSh0UQ5iOdT2xRHFHL7E Course information: http://15462.courses.cs.cmu.edu/
From playlist Computer Graphics (CMU 15-462/662)
Control of fluid motion by Mythily Ramaswamy
Program : Integrable systems in Mathematics, Condensed Matter and Statistical Physics ORGANIZERS : Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE & TIME : 16 July 2018 to 10 August 2018 VENUE : Ramanujan L
From playlist Integrable systems in Mathematics, Condensed Matter and Statistical Physics
Lecture 18: The Laplace Operator (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Stirling's Incredible Approximation // Gamma Functions, Gaussians, and Laplace's Method
We prove Stirling's Formula that approximates n! using Laplace's Method. ►Get my favorite, free calculator app for your phone or tablet: MAPLE CALCULATOR: https://www.maplesoft.com/products/maplecalculator/download.aspx?p=TC-9857 ►Check out MAPLE LEARN for your browser to make beautiful gr
From playlist Cool Math Series
Inverting the Z transform and Z transform of systems
I move from signals to systems in describing discrete systems in the z domain
From playlist Discrete
Benjamin Stamm: A perturbation-method-based post-processing of planewave approximations for
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From playlist HIM Lectures: Trimester Program "Multiscale Problems"
Differential Equations: Lecture 7.1 Definition of the Laplace Transform
This is a real classroom lecture on Differential Equations. I covered section 7.1 which is on the Definition of the Laplace Transform. I hope this video is helpful to someone. If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel
From playlist Differential Equations Full Lectures