The GHK algorithm (Geweke, Hajivassiliou and Keane) is an importance sampling method for simulating choice probabilities in the multivariate probit model. These simulated probabilities can be used to recover parameter estimates from the maximized likelihood equation using any one of the usual well known maximization methods (Newton's method, BFGS, etc.). Train has well documented steps for implementing this algorithm for a multinomial probit model. What follows here will applies to the binary multivariate probit model. Consider the case where one is attempting to evaluate the choice probability of where and where we can take as choices and as individuals or observations, is the mean and is the covariance matrix of the model. The probability of observing choice is Where and, Unless is small (less than or equal to 2) there is no closed form solution for the integrals defined above (some work has been done with ). The alternative to evaluating these integrals closed form or by quadrature methods is to use simulation. GHK is a simulation method to simulate the probability above using importance sampling methods. Evaluating is simplified by recognizing that the latent data model can be rewritten using a Cholesky factorization, . This gives where the terms are distributed . Using this factorization and the fact that the are distributed independently one can simulate draws from a truncated multivariate normal distribution using draws from a univariate random normal. For example, if the region of truncation has lower and upper limits equal to (including a,b = ) then the task becomes Note: , substituting: Rearranging above, Now all one needs to do is iteratively draw from the truncated univariate normal distribution with the given bounds above. This can be done by the inverse CDF method and noting the truncated normal distribution is given by, Where will be a number between 0 and 1 because the above is a CDF. This suggests to generate random draws from the truncated distribution one has to solve for giving, where and and is the standard normal CDF. With such draws one can reconstruct the by its simplified equation using the Cholesky factorization. These draws will be conditional on the draws coming before and using properties of normals the product of the conditional PDFs will be the joint distribution of the , Where is the multivariate normal distribution. Because conditional on is restricted to the set by the setup using the Cholesky factorization then we know that is a truncated multivariate normal. The distribution function of a truncated normal is, Therefore, has distribution, where is the standard normal pdf for choice . Because the above standardization makes each term mean 0 variance 1. Let the denominator and the numerator where is the multivariate normal PDF. Going back to the original goal, to evaluate the Using importance sampling we can evaluate this integral, This is well approximated by . (Wikipedia).
The Euclidean Algorithm: How and Why, Visually
We explain the Euclidean algorithm to compute the gcd, using visual intuition. You'll never forget it once you see the how and why. Then we write it out formally and do an example. This is part of a playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm
From playlist GCDs and Euclidean algorithm
Jana Cslovjecsek: Efficient algorithms for multistage stochastic integer programming using proximity
We consider the problem of solving integer programs of the form min {c^T x : Ax = b; x geq 0}, where A is a multistage stochastic matrix. We give an algorithm that solves this problem in fixed-parameter time f(d; ||A||_infty) n log^O(2d) n, where f is a computable function, d is the treed
From playlist Workshop: Parametrized complexity and discrete optimization
Introduction to Number Theory (Part 4)
The Euclidean algorithm is established and Bezout's theorem is proved.
From playlist Introduction to Number Theory
Euclidean Algorithm and GCDs (Ex. 1)
This video gives an example of how and why the Euclidean algorithm is used to find the gcd of two numbers. Like so: gcd(x,y) = ?. -Here's a second example:http://youtu.be/CtUsUnHz9ek -Here's an example of using the Euclidean algorithm to find a multiplicative inverse: http://youtu.be/K5nb
From playlist Cryptography and Coding Theory
Tony Yue Yu - 4/4 The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
Notes: https://nextcloud.ihes.fr/index.php/s/T6zEGCcJPS5JL4d 4/4 - Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs. --- We show that the naive counts of rational curves in an affine log
From playlist Tony Yue Yu - The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
Euclidean Algorithm and GCDs (Ex. 2)
This video gives an example of how the Euclidean algorithm is used to find the gcd of two numbers. Like so: gcd(x,y) = ?. -Here's the first example: http://youtu.be/WA4nP-iPYKE -Here's a video for using the Euclidean algorithm to find a multiplicative inverse: http://youtu.be/K5nbGbN5Trs
From playlist Cryptography and Coding Theory
The extended Euclidean algorithm in one simple idea
An intuitive explanation of the extended Euclidean algorithm as a simple modification of the Euclidean algorithm. This video is part of playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm9Y---qlNxXccpwYQfllCrHRJWwMky-
From playlist GCDs and Euclidean algorithm
GCD, Euclidean Algorithm and Bezout Coefficients
GCD, Euclidean Algorithm and Bezout Coefficients check out an earlier video on gcd. https://youtu.be/mQMksLNscY4
From playlist Elementary Number Theory
Membrane Excitability and Synaptic Plasticity (Lecture 1) by Suhita Nadkarni
PROGRAM ICTP-ICTS WINTER SCHOOL ON QUANTITATIVE SYSTEMS BIOLOGY (ONLINE) ORGANIZERS Vijaykumar Krishnamurthy (ICTS-TIFR, India), Venkatesh N. Murthy (Harvard University, USA), Sharad Ramanathan (Harvard University, USA), Sanjay Sane (NCBS-TIFR, India) and Vatsala Thirumalai (NCBS-TIFR, I
From playlist ICTP-ICTS Winter School on Quantitative Systems Biology (ONLINE)
Tony Yue Yu - 1/4 The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
Notes: https://nextcloud.ihes.fr/index.php/s/GwJbsQ8xMW2ifb8 1/4 - Motivation and ideas from mirror symmetry, main results. --- We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple wa
From playlist Tony Yue Yu - The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
Béa de Laporte - Landau-Ginzburg potentials via projective representations
Many interesting spaces arise as partial compactifications of Fock-Goncharov's cluster varieties, among them (affine cones over) flag varieties which are important objects in representation theory of algebraic groups. Due to a construction of Gross-Hacking-Keel-Kontsevich those partial com
From playlist Combinatorics and Arithmetic for Physics: Special Days 2022
Extended Euclidean Algorithm to Solve Linear Diophantine Equation
#shorts #mathonshorts Check out the videos on the GCD, Euclidean algorithms here Two Basic Theorems on gcd (Greatest Common Divisors) of Two Integers (Bezout's Identity) https://youtu.be/mQMksLNscY4 An Example of GCD, and Extended Euclidean Algorithm In Finding the Bezout Coefficients h
From playlist Elementary Number Theory
If you are interested in learning more about this topic, please visit http://www.gcflearnfree.org/ to view the entire tutorial on our website. It includes instructional text, informational graphics, examples, and even interactives for you to practice and apply what you've learned.
From playlist Machine Learning
Winter School JTP: Homological mirror symmetry for log Calabi-Yau surfaces, Ailsa Keating
Given a log Calabi-Yau surface Y with maximal boundary D, I’ll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I’ll exp
From playlist Winter School on “Connections between representation Winter School on “Connections between representation theory and geometry"
Some results on affine Deligne–Lusztig varieties – Xuhua He – ICM2018
Lie Theory and Generalizations Invited Lecture 7.8 Some results on affine Deligne–Lusztig varieties Xuhua He Abstract: The study of affine Deligne–Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne–Lusztig varieties are purely Lie-theoretic i
From playlist Lie Theory and Generalizations
An inverse theorem for the Gowers norms over finite fields - Ziegler
Tamar Ziegler Technion - Israel Institute of Technology June 18, 2010 For more videos, visit http://video.ias.edu
From playlist Mathematics
Michael Kemeny: The moduli of singular curves on K3 surfaces
In this talk we will study the moduli space Zg of smooth genus g curves admitting a singular model on a K3 surface. Using the Mori-Mukai approach of rank two, non-Abelian Brill-Noether loci we will work out the dimension of Zg, and further we will work out the Brill-Noether theory of curve
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Mod-05 Lec-37 Backward Induction
Game Theory and Economics by Dr. Debarshi Das, Department of Humanities and Social Sciences, IIT Guwahati. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist IIT Guwahati: Game Theory and Economics | CosmoLearning.org Economics
Chapter 4 - Solving Linear Equations with Technology - IB Math Studies (Math SL)
Hello and welcome to What The Math. This is a Chapter 4 video about linear equations and using GDC to solve various linear functions. This is a part of Chapter 4 from Harris Publication version of IB math book by Haese.
From playlist IB Math Studies Chapter 4
Tropical Lagrangian sections and Looijenga pairs - Andrew Hanlon
Joint IAS/Princeton University Symplectic Geometry Seminar Topic: Tropical Lagrangian sections and Looijenga pairs Speaker: Andrew Hanlon Affiliation: Stony Brook University Date: October 11, 2021 We will discuss the first steps in an approach to proving homological mirror symmetry for
From playlist Mathematics