Approximating Integrals via Monte Carlo and Deterministic Methods
This book is designed to introduce graduate students and researchers to the primary methods useful for approximating integrals. The emphasis is on those methods that have been found to be of practical use, and although the focus is on approximating higher- dimensional integrals the lower-dimensional case is also covered. Included in the book are asymptotic techniques, multiple quadrature and quasi-random techniques as well as a complete development of Monte Carlo algorithms. For the Monte Carlo section importance sampling methods, variance reduction techniques and the primary Markov Chain Monte Carlo algorithms are covered. This book brings these various techniques together for the first time, and hence provides an accessible textbook and reference for researchers in a wide variety of disciplines.
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adaptive importance sampling applications approach approximating integrals asymptotic expansion Bayesian Chapter choice choose compute consider construct context control variate convergence coordinate defined denote density proportional dimension dimensional envelopes Establish Evans and Swartz exact example Exercise exists finite following result function evaluations Gauss rules Gibbs sampling given Halton sequence Hessian matrix implies importance sampling independent integrand integration problems interpolation interval inverse iterations kernel Laplace Laplace approximation lattice point rule Lemma linear Markov chain matrix MCMC monomial Monte Carlo methods multivariate Student normalizing constant Note obtain orthogonal polynomials of degree probability Proof Prove quadrature rule random variable randomized quadrature rule rectangle rejection algorithm relative error respect Riemann integral sampler satisfies simulation stationary distribution statistical subgroup Suppose symmetric systematic sampling Theorem transformation trapezoid rule typically variance reduction vector weight function