1Blurb:: 2Importance sampling 3Description:: 4The \c importance_sampling method is based on ideas in reliability modeling. 5 6An initial Latin Hypercube sampling is performed to generate an initial set of samples. 7These initial samples are augmented with samples from an importance density as follows: 8\li The variables are transformed to standard normal space. 9\li In the transformed space, 10 the importance density is a set of normal densities centered around points which 11 are in the failure region. 12\li Note that this is similar in spirit to the reliability 13 methods, in which importance sampling is centered around a Most Probable Point (MPP). 14\li In the case of the LHS samples, the importance sampling density will simply by 15 a mixture of normal distributions centered around points in the failure region. 16 17<b> Options </b> 18 19Choose one of the importance sampling options: 20\li \c import 21\li \c adapt_import 22\li \c mm_adapt_import 23 24The options for importance sampling are as follows: \c import centers a sampling 25density at one of the initial LHS samples identified in the failure region. 26It then generates the importance samples, weights them by their probability of occurence 27given the original density, and calculates the required probability (CDF or CCDF level). 28\c adapt_import is the same as \c import but is performed iteratively until the 29failure probability estimate converges. 30\c mm_adapt_import starts with all of the samples located in the failure region 31to build a multimodal sampling density. First, it uses a small number of samples around 32each of the initial samples in the failure region. Note that these samples 33are allocated to the different points based on their relative probabilities of occurrence: 34more probable points get more samples. This early part of the approach is done 35to search for "representative" points. Once these are located, the multimodal sampling 36density is set and then \c mm_adapt_import proceeds similarly to \c adapt_import (sample 37until convergence). 38 39Topics:: uncertainty_quantification, aleatory_uncertainty_quantification_methods, sampling 40Examples:: 41Theory:: 42Importance sampling is a method that allows one to estimate statistical 43quantities such as failure probabilities (e.g. the probability that 44a response quantity will exceed a threshold or fall below a threshold value) 45in a way that is more efficient than Monte Carlo sampling. The core idea 46in importance sampling is that one generates samples that preferentially 47samples important regions in the space (e.g. in or near the failure region 48or user-defined region of interest), and then appropriately weights 49the samples to obtain an unbiased estimate of the failure probability 50\cite Srinivasan2002. 51In importance sampling, the samples are generated from a density which is 52called the importance density: it is not the original probability density 53of the input distributions. The importance density should be centered near the 54failure region of interest. For black-box simulations such as those commonly 55interfaced with %Dakota, it is difficult to specify the importance density a priori: 56the user often does not know where the failure region lies, especially in a high-dimensional 57space. \cite Swiler2010. 58We have developed two importance sampling approaches which do not 59rely on the user explicitly specifying an importance density. 60 61Faq:: 62See_Also:: method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-polynomial_chaos, method-stoch_collocation 63