Ancestral Sampling PDF
Preview Ancestral Sampling
Robert Collins CSE586, PSU Intro to Sampling Methods CSE586 Computer Vision II Penn State Univ Robert Collins Topics to be Covered CSE586, PSU Monte Carlo Integration Sampling and Expected Values Inverse Transform Sampling (CDF) Ancestral Sampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo Robert Collins Integration and Area CSE586, PSU http://videolectures.net/mlss08au_freitas_asm/ Robert Collins Integration and Area CSE586, PSU http://videolectures.net/mlss08au_freitas_asm/ Robert Collins Integration and Area CSE586, PSU total # boxes http://videolectures.net/mlss08au_freitas_asm/ Robert Collins Integration and Area CSE586, PSU http://videolectures.net/mlss08au_freitas_asm/ Robert Collins Integration and Area CSE586, PSU • As we use more samples, our answer should get more and more accurate • Doesn’t matter what the shape looks like (1,1) (1,1) (0,0) (0,0) arbitrary region area under curve (even disconnected) aka integration! Robert Collins Monte Carlo Integration CSE586, PSU Goal: compute definite integral of function f(x) from a to b Generate N uniform random upper bound samples in upper bound volume c N f(x) a b K Answer= * Area of upper bound volume N K K = * (b-a)(c-0) N count the K samples that fall below the f(x) curve Robert Collins Motivation: Normalizing Constant CSE586, PSU Sampling-based integration is useful for computing the normalizing constant that turns an arbitrary non-negative function f(x) into a probability density function p(x). Z Compute this via sampling (Monte Carlo Integration). Then: 1 Z Note: for complicated, multidimensional functions, this is the ONLY way we can compute this normalizing constant. Robert Collins Motivation : Expected Values CSE586, PSU If we can generate random samples x from a given i distribution P(x), then we can estimate expected values of functions under this distribution by summation, rather than integration. That is, we can approximate: by first generating N i.i.d. samples from P(x) and then forming the empirical estimate:
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