Application Of Importance Sampling Research Articles

Extensions of the deep Galerkin method

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos(2018)[25] to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker–Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to nonlinear a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled by a novel application of importance sampling. Secondly, we tackle a number of Hamilton–Jacobi–Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique inspired by the policy improvement algorithms (PIA).

Error Floor Estimation of LDPC Decoders — A Code Independent Approach to Measuring the Harmfulness of Trapping Sets

The linear state-space model is a well-known code-independent method to estimate the contribution of a trapping set (TS) structure to the error floor of low-density parity-check (LDPC) codes. In this paper, we provide an in-depth analysis of this method by incorporating a more accurate model for the incoming messages to the TS structure that takes into account the randomness and the correlation among such messages. Based on this analysis, we demonstrate that both randomness and correlation result in the over-estimation of the failure probability of the TS. We then propose an alternate code-independent technique for the error floor estimation of iterative LDPC decoders that can accurately estimate the contribution of different TS structures in the error floor. Compared to the linear state-space model, the proposed method is not only more accurate, but also more general, in that, it is applicable to any saturating iterative message-passing decoder, symmetrically quantized or unquantized, over any memoryless binary-input output-symmetric channel. The proposed technique can be viewed as the local application of importance sampling (IS) to the message-passing algorithm over the subgraph induced by the TS in the code’s Tanner graph. In the message-passing process, to account for the effect of the rest of the Tanner graph, density evolution along with a simple correlation model is used to generate the messages coming into the TS from the rest of the Tanner graph. Extensive simulations demonstrate that the proposed technique can accurately estimate the error floor of LDPC codes over both additive white Gaussian noise (AWGN) channel and binary symmetric channel (BSC), for a variety of iterative decoding algorithms and quantization schemes.

The R Package MitISEM: Efficient and Robust Simulation Procedures for Bayesian Inference

This paper presents the R package MitISEM (mixture of t by importance sampling weighted expectation maximization) which provides an automatic and flexible two-stage method to approximate a non-elliptical target density kernel - typically a posterior density kernel - using an adaptive mixture of Student t densities as approximating density. In the first stage a mixture of Student t densities is fitted to the target using an expectation maximization algorithm where each step of the optimization procedure is weighted using importance sampling. In the second stage this mixture density is a candidate density for efficient and robust application of importance sampling or the Metropolis-Hastings (MH) method to estimate properties of the target distribution. The package enables Bayesian inference and prediction on model parameters and probabilities, in particular, for models where densities have multi-modal or other non-elliptical shapes like curved ridges. These shapes occur in research topics in several scientific fields. For instance, analysis of DNA data in bio-informatics, obtaining loans in the banking sector by heterogeneous groups in financial economics and analysis of education's effect on earned income in labor economics. The package MitISEM provides also an extended algorithm, 'sequential MitISEM', which substantially decreases computation time when the target density has to be approximated for increasing data samples. This occurs when the posterior or predictive density is updated with new observations and/or when one computes model probabilities using predictive likelihoods. We illustrate the MitISEM algorithm using three canonical statistical and econometric models that are characterized by several types of non-elliptical posterior shapes and that describe well-known data patterns in econometrics and finance. We show that MH using the candidate density obtained by MitISEM outperforms, in terms of numerical efficiency, MH using a simpler candidate, as well as the Gibbs sampler. The MitISEM approach is also used for Bayesian model comparison using predictive likelihoods.

The R Package Mitisem: Efficient and Robust Simulation Procedures for Bayesian Inference

This paper presents the R-package MitISEM (mixture of t by importance sampling weighted expectation maximization) which provides an automatic and flexible two-stage method to approximate a non-elliptical target density kernel -- typically a posterior density kernel -- using an adaptive mixture of Student-t densities as approximating density. In the first stage a mixture of Student-t densities is fitted to the target using an expectation maximization (EM) algorithm where each step of the optimization procedure is weighted using importance sampling. In the second stage this mixture density is a candidate density for efficient and robust application of importance sampling or the Metropolis-Hastings (MH) method to estimate properties of the target distribution. The package enables Bayesian inference and prediction on model parameters and probabilities, in particular, for models where densities have multi-modal or other non-elliptical shapes like curved ridges. These shapes occur in research topics in several scientific fields. For instance, analysis of DNA data in bio-informatics, obtaining loans in the banking sector by heterogeneous groups in financial economics and analysis of education's effect on earned income in labor economics. The package MitISEM provides also an extended algorithm, 'sequential MitISEM', which substantially decreases computation time when the target density has to be approximated for increasing data samples.

Incorporating Radiation in Noise-Induced Phase Evolution of Optical Solitons

This paper extends the application of importance sampling to include the leading order effect of dispersive radiation on the soliton's phase when simulating bit errors in optical communication systems that use optical solitons as bit carriers. A simple one-parameter model for the radiation is used to account for the most significant effect of radiation on phase, a mean shift that scales with noise bandwidth. This improved model is used to inform optimal biasing of paths used for importance-sampled Monte Carlo simulations, with the resulting numerics demonstrating improved targeting of phase values.

The R Package MitISEM: Efficient and Robust Simulation Procedures for Bayesian Inference

This paper presents the R-package MitISEM (mixture of t by importance sampling weighted expectation maximization) which provides an automatic and flexible two-stage method to approximate a non-elliptical target density kernel -- typically a posterior density kernel -- using an adaptive mixture of Student- t densities as approximating density. In the first stage a mixture of Student- t densities is fitted to the target using an expectation maximization (EM) algorithm where each step of the optimization procedure is weighted using importance sampling. In the second stage this mixture density is a candidate density for efficient and robust application of importance sampling or the Metropolis-Hastings (MH) method to estimate properties of the target distribution. The package enables Bayesian inference and prediction on model parameters and probabilities, in particular, for models where densities have multi-modal or other non-elliptical shapes like curved ridges. These shapes occur in research topics in several scientific fields. For instance, analysis of DNA data in bio-informatics, obtaining loans in the banking sector by heterogeneous groups in financial economics and analysis of education's effect on earned income in labor economics. The package MitISEM provides also an extended algorithm, 'sequential MitISEM', which substantially decreases computation time when the target density has to be approximated for increasing data samples.

Multi-Level Monte Carlo Simulations with Importance Sampling

We present an application of importance sampling to multi-asset options under the Heston and the Bates models as well as to the Heston-Hull-White and the Heston-Cox-Ingersoll-Ross models. Moreover, we provide an efficient importance sampling scheme in a Multi-Level Monte Carlo simulation. In all cases, we explain how the Greeks can be computed in the different simulation schemes using the Likelihood Ratio Method, and how combining it with importance sampling leads to a significant variance reduction for the Greeks.

Improving the efficiency of mean sound level calculations through importance sampling.

Outdoor sound levels vary in space and time due to randomness in acoustic source emissions and environmental effects on wave propagation. Sound levels may also appropriately be viewed as random due to uncertainties in the modeling process. To assess the impact of the randomness on sound level predictions, Monte Carlo sampling of the model input space can be used; that is, equally likely sets of model inputs are generated, to which a sound propagation model is applied repeatedly. To improve the efficiency of the calculations, it is highly desirable to sample primarily the acoustical frequency bands and environmental conditions (such as downwind propagation and hard ground surfaces) contributing most strongly to the sound level. Importance sampling techniques, which adaptively focus sampling effort on the most important parts of the input parameter space, can be used for this purpose. This paper explores application of importance sampling to prediction of sound level statistics. The technique is found to be...

Application of importance sampling to the computation of large deviations in nonequilibrium processes

We present an algorithm for finding the probabilities of rare events in nonequilibrium processes. The algorithm consists of evolving the system with a modified dynamics for which the required event occurs more frequently. By keeping track of the relative weight of phase-space trajectories generated by the modified and the original dynamics one can obtain the required probabilities. The algorithm is tested on two model systems of steady-state particle and heat transport where we find a huge improvement from direct simulation methods.

Low BER performance estimation of LDPC codes via application of importance sampling to trapping sets

We introduce an importance sampling (IS) method that successfully simulates the performance of Low density Parity Check (LDPC) Codes in an AWGN channel at very low bit error rates (BERs). By effectively finding and biasing bit node combinations that are the dominant sources of error events, called trapping sets, the developed technique provokes more frequent decoder failures. Consequently, fewer simulation runs and higher simulation gains are achieved.

A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons

We describe in detail the application of importance sampling to numerical simulations of large noise-induced perturbations in soliton-based optical transmission systems governed by the nonlinear Schrödinger equation. The method allows one to concentrate the samples in Monte Carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors, and it yields computational speedups of several orders of magnitude over standard Monte Carlo simulations. We demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude, frequency, timing, and phase fluctuations in a prototypical soliton-based communication system.

A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons

We demonstrate in detail the application of importance sampling to the numerical simulation of large noise-induced perturbations in soliton-based optical transmission systems governed by the nonlinear Schrödinger equation. The method allows one to concentrate numerical Monte Carlo simulations around the noise realizations that are most likely to produce the large pulse deformations connected with errors, and yields computational speedups of several orders of magnitude over standard Monte Carlo simulations. We demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude, frequency, and timing fluctuations in a prototypical soliton-based communication system.

Importance Sampling for Portfolio Credit Risk

Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rare-event simulation problem because default probabilities are low for highly rated obligors and because risk management is particularly concerned with rare but significant losses resulting from a large number of defaults. This makes importance sampling (IS) potentially attractive. But the application of IS is complicated by the mechanisms used to model dependence between obligors, and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used normal copula model of portfolio credit risk. The procedure has two parts: One applies IS conditional on a set of common factors affecting multiple obligors, the other applies IS to the factors themselves. The relative importance of the two parts of the procedure is determined by the strength of the dependence between obligors. We provide both theoretical and numerical support for the method.

Computing large signal distortions and bit-error ratios in DPSK transmission systems

We demonstrate the application of importance sampling (IS) to the simulation of a soliton-based differential phase-shift-keyed optical transmission system. We use an approximation to the pulse evolution given by perturbation theory and optimized with calculus of variations to guide full numerical Monte Carlo simulations employing IS. The method not only directly determines the bit-error ratio, but also predicts the manner in which large signal distortions occur.

Applications of importance sampling to polarization mode dispersion

Applications of importance sampling to polarization mode dispersion

Linkage analysis with sequential imputation.

Multilocus calculations, using all available information on all pedigree members, are important for linkage analysis. Exact calculation methods in linkage analysis are limited in either the number of loci or the number of pedigree members they can handle. In this article, we propose a Monte Carlo method for linkage analysis based on sequential imputation. Unlike exact methods, sequential imputation can handle large pedigrees with a moderate number of loci in its current implementation. This Monte Carlo method is an application of importance sampling, in which we sequentially impute ordered genotypes locus by locus, and then impute inheritance vectors conditioned on these genotypes. The resulting inheritance vectors, together with the importance sampling weights, are used to derive a consistent estimator of any linkage statistic of interest. The linkage statistic can be parametric or nonparametric; we focus on nonparametric linkage statistics. We demonstrate that accurate estimates can be achieved within a reasonable computing time. A simulation study illustrates the potential gain in power using our method for multilocus linkage analysis with large pedigrees. We simulated data at six markers under three models. We analyzed them using both sequential imputation and GENEHUNTER. GENEHUNTER had to drop between 38-54% of pedigree members, whereas our method was able to use all pedigree members. The power gains of using all pedigree members were substantial under 2 of the 3 models. We implemented sequential imputation for multilocus linkage analysis in a user-friendly software package called SIMPLE.

An importance sampling algorithm for the simulation of a GPS scheduler

AbstractA key requirement of modern telecommunications networks is the capability of providing a variety of Quality of Service (QoS) guarantees to different classes of users. In this scenario scheduling systems play a major role, thanks to their ability of offering different levels of service, while preventing some “greedy users” from degrading the QoS of other classes. A primary QoS parameter is the Cell Loss Probability (CLP), whose typical values are very small and therefore difficult to estimate through standard Monte Carlo (MC) simulation, since long run times are required to achieve accurate results. Under proper conditions, Importance Sampling (IS) techniques can speed up simulations involving rare events. In this paper we propose an application of IS to the simulation of a Generalized Processor Sharing (GPS) scheduler, fed by Markov Arrival Processes (MAP). The algorithms we propose are based on large deviations principles, which provide asymptotic results on the decay rate of per‐session queue length tail distributions in an idealized GPS discipline. In particular, we first present a scheme for the simulation of a two‐buffer system, then we extend the technique to the general case of a multi‐buffer scheduler. We subsequently employ our algorithms to simulate a packetized realistic system, namely the Packet‐by‐packet Generalized Processor Sharing.

Importance sampling for polarization-mode dispersion

We describe the application of importance sampling to Monte-Carlo simulations of polarization-mode dispersion (PMD) in optical fibers. The method allows rare differential group delay (DGD) events to be simulated much more efficiently than with standard Monte-Carlo methods and, thus, it can be used to assess PMD-induced system outage probabilities at realistic bit-error rates. We demonstrate the technique by accurately calculating the tails of the DGD probability distribution with a relatively small number of Monte-Carlo trials.

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.

Monte Carlo approaches to sampling forested tracts with lines or points

Several line- and point-based sampling methods can be employed to estimate the aggregate dimensions of trees standing on a forested tract or pieces of coarse woody debris lying on the forest floor. Line methods include line intersect sampling, horizontal line sampling, and transect relascope sampling; point methods include variable- and fixed-radius plot sampling, and point relascope sampling. We demonstrate that the line methods can be interpreted as applications of importance sampling and that point methods can be interpreted as two-stage applications of importance sampling and crude Monte Carlo. Interestingly, each of the line methods also can be implemented as a point method. Operationally, the two stages of a point method effectively reduce to a single stage. Estimators of target parameters are derived from the Monte Carlo approach for all six methods. Two new methods of estimating cross-sectional areas of slanted or tilted log-shaped objects are suggested for use in line intersect sampling. Boundary problems also are discussed.