QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, When to Stop Sampling: Answers and Further Questions - Fred Hickernell, Dec 12, 2017
- The document discusses methods for determining when to stop sampling in Monte Carlo integration to achieve a desired error tolerance.
- For independent and identically distributed (IID) sampling, the central limit theorem can be used to determine the necessary sample size based on the variance of the integrand.
- Quasi-Monte Carlo sampling can achieve faster convergence rates by using low-discrepancy point sets that more uniformly sample the domain. The error can be analyzed in the frequency domain based on the decay of the true Fourier coefficients.
- Bayesian cubature methods model the integrand as a Gaussian process, allowing inference of hyperparameters from sample points to improve integration accuracy.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk we consider the question of how to use QMC with an empirical dataset, such as a set of points generated by MCMC. Using ideas from partitioning for parallel computing, we apply recursive bisection to reorder the points, and then interleave the bits of the QMC coordinates to select the appropriate point from the dataset. Numerical tests show that in the case of known distributions this is almost as effective as applying QMC directly to the original distribution. The same recursive bisection can also be used to thin the dataset, by recursively bisecting down to many small subsets of points, and then randomly selecting one point from each subset. This makes it possible to reduce the size of the dataset greatly without significantly increasing the overall error. Co-author: Fei Xie
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk we consider the question of how to use QMC with an empirical dataset, such as a set of points generated by MCMC. Using ideas from partitioning for parallel computing, we apply recursive bisection to reorder the points, and then interleave the bits of the QMC coordinates to select the appropriate point from the dataset. Numerical tests show that in the case of known distributions this is almost as effective as applying QMC directly to the original distribution. The same recursive bisection can also be used to thin the dataset, by recursively bisecting down to many small subsets of points, and then randomly selecting one point from each subset. This makes it possible to reduce the size of the dataset greatly without significantly increasing the overall error. Co-author: Fei Xie
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Sequential quasi-Monte Carlo (SQMC) is a quasi-Monte Carlo (QMC) version of sequential Monte Carlo (or particle filtering), a popular class of Monte Carlo techniques used to carry out inference in state space models. In this talk I will first review the SQMC methodology as well as some theoretical results. Although SQMC converges faster than the usual Monte Carlo error rate its performance deteriorates quickly as the dimension of the hidden variable increases. However, I will show with an example that SQMC may perform well for some "high" dimensional problems. I will conclude this talk with some open problems and potential applications of SQMC in complicated settings.
In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Sequential quasi-Monte Carlo (SQMC) is a quasi-Monte Carlo (QMC) version of sequential Monte Carlo (or particle filtering), a popular class of Monte Carlo techniques used to carry out inference in state space models. In this talk I will first review the SQMC methodology as well as some theoretical results. Although SQMC converges faster than the usual Monte Carlo error rate its performance deteriorates quickly as the dimension of the hidden variable increases. However, I will show with an example that SQMC may perform well for some "high" dimensional problems. I will conclude this talk with some open problems and potential applications of SQMC in complicated settings.
In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...
Similar to QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, When to Stop Sampling: Answers and Further Questions - Fred Hickernell, Dec 12, 2017
My talk in the International Conference on Computational Finance 2019 (ICCF2019). The talk is about designing new efficient methods for option pricing under the rough Bergomi model.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Conference talk at the SIAM Conference on Financial Mathematics and Engineering, held in virtual format, June 1-4 2021, about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model".
- Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
We will describe and analyze accurate and efficient numerical algorithms to interpolate and approximate the integral of multivariate functions. The algorithms can be applied when we are given the function values at an arbitrary positioned, and usually small, existing sparse set of function values (samples), and additional samples are impossible, or difficult (e.g. expensive) to obtain. The methods are based on local, and global, tensor-product sparse quasi-interpolation methods that are exact for a class of sparse multivariate orthogonal polynomials.
Simplifying Gaussian Mixture Models Via Entropic Quantization (EUSIPCO 2009)Frank Nielsen
Slides for the paper presented at EUSIPCO 2009:
Simplifying Gaussian Mixture Models Via Entropic Quantization
http://www.eurasip.org/Proceedings/Eusipco/Eusipco2009/contents/papers/1569187249.pdf
Inria Tech Talk - La classification de données complexes avec MASSICCCStéphanie Roger
MASSICCC - Une plateforme SaaS pour le traitement de la classification de données complexes hétérogènes et incomplètes.
Dans ce Tech Talk venez découvrir, tester et apprendre à maîtriser MASSICCC (Massive clustering in cloud computing) une plateforme SaaS orientée utilisateurs, ainsi que ses trois familles d’algorithmes de #classification, fruits des dernières avancées des équipes de recherche Modal & Celeste de Inria, pour analyser et faire de l’apprentissage sur vos "Big Data" (ex : en immobilier, maintenance prédictive, santé, open data, etc. ).
MASSICCC c’est aussi :
- Un accès gratuit pour le test et la recherche sur https://massiccc.lille.inria.fr
- Un "one for all" de la classification
- Une forte interprétabilité des résultats (avec ses graphiques)
- Un mode SaaS qui vous permet un suivi des expériences (en cours ou terminées)
- Et des algorithmes open source qui sont réutilisables indépendamment.
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...Laurent Duval
This paper jointly addresses the problems of chromatogram baseline correction and noise reduction. The proposed approach is based on modeling the series of chromatogram peaks as sparse with sparse derivatives, and on modeling the baseline as a low-pass signal. A convex optimization problem is formulated so as to encapsulate these non-parametric models. To account for the positivity of chromatogram peaks, an asymmetric penalty functions is utilized. A robust, computationally efficient, iterative algorithm is developed that is guaranteed to converge to the unique optimal solution. The approach, termed Baseline Estimation And Denoising with Sparsity (BEADS), is evaluated and compared with two state-of-the-art methods using both simulated and real chromatogram data.
H2O World - Consensus Optimization and Machine Learning - Stephen BoydSri Ambati
H2O World 2015 - Stephen Boyd
Similar to QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, When to Stop Sampling: Answers and Further Questions - Fred Hickernell, Dec 12, 2017 (20)
Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
I will discuss paradigmatic statistical models of inference and learning from high dimensional data, such as sparse PCA and the perceptron neural network, in the sub-linear sparsity regime. In this limit the underlying hidden signal, i.e., the low-rank matrix in PCA or the neural network weights, has a number of non-zero components that scales sub-linearly with the total dimension of the vector. I will provide explicit low-dimensional variational formulas for the asymptotic mutual information between the signal and the data in suitable sparse limits. In the setting of support recovery these formulas imply sharp 0-1 phase transitions for the asymptotic minimum mean-square-error (or generalization error in the neural network setting). A similar phase transition was analyzed recently in the context of sparse high-dimensional linear regression by Reeves et al.
Many different measurement techniques are used to record neural activity in the brains of different organisms, including fMRI, EEG, MEG, lightsheet microscopy and direct recordings with electrodes. Each of these measurement modes have their advantages and disadvantages concerning the resolution of the data in space and time, the directness of measurement of the neural activity and which organisms they can be applied to. For some of these modes and for some organisms, significant amounts of data are now available in large standardized open-source datasets. I will report on our efforts to apply causal discovery algorithms to, among others, fMRI data from the Human Connectome Project, and to lightsheet microscopy data from zebrafish larvae. In particular, I will focus on the challenges we have faced both in terms of the nature of the data and the computational features of the discovery algorithms, as well as the modeling of experimental interventions.
Bayesian Additive Regression Trees (BART) has been shown to be an effective framework for modeling nonlinear regression functions, with strong predictive performance in a variety of contexts. The BART prior over a regression function is defined by independent prior distributions on tree structure and leaf or end-node parameters. In observational data settings, Bayesian Causal Forests (BCF) has successfully adapted BART for estimating heterogeneous treatment effects, particularly in cases where standard methods yield biased estimates due to strong confounding.
We introduce BART with Targeted Smoothing, an extension which induces smoothness over a single covariate by replacing independent Gaussian leaf priors with smooth functions. We then introduce a new version of the Bayesian Causal Forest prior, which incorporates targeted smoothing for modeling heterogeneous treatment effects which vary smoothly over a target covariate. We demonstrate the utility of this approach by applying our model to a timely women's health and policy problem: comparing two dosing regimens for an early medical abortion protocol, where the outcome of interest is the probability of a successful early medical abortion procedure at varying gestational ages, conditional on patient covariates. We discuss the benefits of this approach in other women’s health and obstetrics modeling problems where gestational age is a typical covariate.
Difference-in-differences is a widely used evaluation strategy that draws causal inference from observational panel data. Its causal identification relies on the assumption of parallel trends, which is scale-dependent and may be questionable in some applications. A common alternative is a regression model that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes. In the context of linear models, Angrist and Pischke (2009) show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect. We show that the same bracketing relationship holds in general nonparametric (model-free) settings. We also extend the result to semiparametric estimation based on inverse probability weighting.
We develop sensitivity analyses for weak nulls in matched observational studies while allowing unit-level treatment effects to vary. In contrast to randomized experiments and paired observational studies, we show for general matched designs that over a large class of test statistics, any valid sensitivity analysis for the weak null must be unnecessarily conservative if Fisher's sharp null of no treatment effect for any individual also holds. We present a sensitivity analysis valid for the weak null, and illustrate why it is conservative if the sharp null holds through connections to inverse probability weighted estimators. An alternative procedure is presented that is asymptotically sharp if treatment effects are constant, and is valid for the weak null under additional assumptions which may be deemed reasonable by practitioners. The methods may be applied to matched observational studies constructed using any optimal without-replacement matching algorithm, allowing practitioners to assess robustness to hidden bias while allowing for treatment effect heterogeneity.
The world of health care is full of policy interventions: a state expands eligibility rules for its Medicaid program, a medical society changes its recommendations for screening frequency, a hospital implements a new care coordination program. After a policy change, we often want to know, “Did it work?” This is a causal question; we want to know whether the policy CAUSED outcomes to change. One popular way of estimating causal effects of policy interventions is a difference-in-differences study. In this controlled pre-post design, we measure the change in outcomes of people who are exposed to the new policy, comparing average outcomes before and after the policy is implemented. We contrast that change to the change over the same time period in people who were not exposed to the new policy. The differential change in the treated group’s outcomes, compared to the change in the comparison group’s outcomes, may be interpreted as the causal effect of the policy. To do so, we must assume that the comparison group’s outcome change is a good proxy for the treated group’s (counterfactual) outcome change in the absence of the policy. This conceptual simplicity and wide applicability in policy settings makes difference-in-differences an appealing study design. However, the apparent simplicity belies a thicket of conceptual, causal, and statistical complexity. In this talk, I will introduce the fundamentals of difference-in-differences studies and discuss recent innovations including key assumptions and ways to assess their plausibility, estimation, inference, and robustness checks.
We present recent advances and statistical developments for evaluating Dynamic Treatment Regimes (DTR), which allow the treatment to be dynamically tailored according to evolving subject-level data. Identification of an optimal DTR is a key component for precision medicine and personalized health care. Specific topics covered in this talk include several recent projects with robust and flexible methods developed for the above research area. We will first introduce a dynamic statistical learning method, adaptive contrast weighted learning (ACWL), which combines doubly robust semiparametric regression estimators with flexible machine learning methods. We will further develop a tree-based reinforcement learning (T-RL) method, which builds an unsupervised decision tree that maintains the nature of batch-mode reinforcement learning. Unlike ACWL, T-RL handles the optimization problem with multiple treatment comparisons directly through a purity measure constructed with augmented inverse probability weighted estimators. T-RL is robust, efficient and easy to interpret for the identification of optimal DTRs. However, ACWL seems more robust against tree-type misspecification than T-RL when the true optimal DTR is non-tree-type. At the end of this talk, we will also present a new Stochastic-Tree Search method called ST-RL for evaluating optimal DTRs.
A fundamental feature of evaluating causal health effects of air quality regulations is that air pollution moves through space, rendering health outcomes at a particular population location dependent upon regulatory actions taken at multiple, possibly distant, pollution sources. Motivated by studies of the public-health impacts of power plant regulations in the U.S., this talk introduces the novel setting of bipartite causal inference with interference, which arises when 1) treatments are defined on observational units that are distinct from those at which outcomes are measured and 2) there is interference between units in the sense that outcomes for some units depend on the treatments assigned to many other units. Interference in this setting arises due to complex exposure patterns dictated by physical-chemical atmospheric processes of pollution transport, with intervention effects framed as propagating across a bipartite network of power plants and residential zip codes. New causal estimands are introduced for the bipartite setting, along with an estimation approach based on generalized propensity scores for treatments on a network. The new methods are deployed to estimate how emission-reduction technologies implemented at coal-fired power plants causally affect health outcomes among Medicare beneficiaries in the U.S.
Laine Thomas presented information about how causal inference is being used to determine the cost/benefit of the two most common surgical surgical treatments for women - hysterectomy and myomectomy.
We provide an overview of some recent developments in machine learning tools for dynamic treatment regime discovery in precision medicine. The first development is a new off-policy reinforcement learning tool for continual learning in mobile health to enable patients with type 1 diabetes to exercise safely. The second development is a new inverse reinforcement learning tools which enables use of observational data to learn how clinicians balance competing priorities for treating depression and mania in patients with bipolar disorder. Both practical and technical challenges are discussed.
The method of differences-in-differences (DID) is widely used to estimate causal effects. The primary advantage of DID is that it can account for time-invariant bias from unobserved confounders. However, the standard DID estimator will be biased if there is an interaction between history in the after period and the groups. That is, bias will be present if an event besides the treatment occurs at the same time and affects the treated group in a differential fashion. We present a method of bounds based on DID that accounts for an unmeasured confounder that has a differential effect in the post-treatment time period. These DID bracketing bounds are simple to implement and only require partitioning the controls into two separate groups. We also develop two key extensions for DID bracketing bounds. First, we develop a new falsification test to probe the key assumption that is necessary for the bounds estimator to provide consistent estimates of the treatment effect. Next, we develop a method of sensitivity analysis that adjusts the bounds for possible bias based on differences between the treated and control units from the pretreatment period. We apply these DID bracketing bounds and the new methods we develop to an application on the effect of voter identification laws on turnout. Specifically, we focus estimating whether the enactment of voter identification laws in Georgia and Indiana had an effect on voter turnout.
We study experimental design in large-scale stochastic systems with substantial uncertainty and structured cross-unit interference. We consider the problem of a platform that seeks to optimize supply-side payments p in a centralized marketplace where different suppliers interact via their effects on the overall supply-demand equilibrium, and propose a class of local experimentation schemes that can be used to optimize these payments without perturbing the overall market equilibrium. We show that, as the system size grows, our scheme can estimate the gradient of the platform’s utility with respect to p while perturbing the overall market equilibrium by only a vanishingly small amount. We can then use these gradient estimates to optimize p via any stochastic first-order optimization method. These results stem from the insight that, while the system involves a large number of interacting units, any interference can only be channeled through a small number of key statistics, and this structure allows us to accurately predict feedback effects that arise from global system changes using only information collected while remaining in equilibrium.
We discuss a general roadmap for generating causal inference based on observational studies used to general real world evidence. We review targeted minimum loss estimation (TMLE), which provides a general template for the construction of asymptotically efficient plug-in estimators of a target estimand for realistic (i.e, infinite dimensional) statistical models. TMLE is a two stage procedure that first involves using ensemble machine learning termed super-learning to estimate the relevant stochastic relations between the treatment, censoring, covariates and outcome of interest. The super-learner allows one to fully utilize all the advances in machine learning (in addition to more conventional parametric model based estimators) to build a single most powerful ensemble machine learning algorithm. We present Highly Adaptive Lasso as an important machine learning algorithm to include.
In the second step, the TMLE involves maximizing a parametric likelihood along a so-called least favorable parametric model through the super-learner fit of the relevant stochastic relations in the observed data. This second step bridges the state of the art in machine learning to estimators of target estimands for which statistical inference is available (i.e, confidence intervals, p-values etc). We also review recent advances in collaborative TMLE in which the fit of the treatment and censoring mechanism is tailored w.r.t. performance of TMLE. We also discuss asymptotically valid bootstrap based inference. Simulations and data analyses are provided as demonstrations.
We describe different approaches for specifying models and prior distributions for estimating heterogeneous treatment effects using Bayesian nonparametric models. We make an affirmative case for direct, informative (or partially informative) prior distributions on heterogeneous treatment effects, especially when treatment effect size and treatment effect variation is small relative to other sources of variability. We also consider how to provide scientifically meaningful summaries of complicated, high-dimensional posterior distributions over heterogeneous treatment effects with appropriate measures of uncertainty.
Climate change mitigation has traditionally been analyzed as some version of a public goods game (PGG) in which a group is most successful if everybody contributes, but players are best off individually by not contributing anything (i.e., “free-riding”)—thereby creating a social dilemma. Analysis of climate change using the PGG and its variants has helped explain why global cooperation on GHG reductions is so difficult, as nations have an incentive to free-ride on the reductions of others. Rather than inspire collective action, it seems that the lack of progress in addressing the climate crisis is driving the search for a “quick fix” technological solution that circumvents the need for cooperation.
This seminar discussed ways in which to produce professional academic writing, from academic papers to research proposals or technical writing in general.
Machine learning (including deep and reinforcement learning) and blockchain are two of the most noticeable technologies in recent years. The first one is the foundation of artificial intelligence and big data, and the second one has significantly disrupted the financial industry. Both technologies are data-driven, and thus there are rapidly growing interests in integrating them for more secure and efficient data sharing and analysis. In this paper, we review the research on combining blockchain and machine learning technologies and demonstrate that they can collaborate efficiently and effectively. In the end, we point out some future directions and expect more researches on deeper integration of the two promising technologies.
In this talk, we discuss QuTrack, a Blockchain-based approach to track experiment and model changes primarily for AI and ML models. In addition, we discuss how change analytics can be used for process improvement and to enhance the model development and deployment processes.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop, When to Stop Sampling: Answers and Further Questions - Fred Hickernell, Dec 12, 2017
1. When to Stop Sampling: Answers and Further Questions
Fred J. Hickernell
Department of Applied Mathematics & Center for Interdisciplinary Scientific Computation
Illinois Institute of Technology
hickernell@iit.edu mypages.iit.edu/~hickernell
Thanks to the Guaranteed Automatic Integration Library (GAIL) team and friends
Supported by NSF-DMS-1522687
Thanks to SAMSI, the QMC Program organizers, and the Worskshop Organizers
Trends and Advances in Monte Carlo Sampling Algorithms, December 12, 2017
2. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Multidimensional Integration Examples
µ =
Rd
f(x) ν(dx) = option price
f(x) = discounted payoff determined by market forces x
µ =
X
g(x) dx = P(X ∈ X) = probability
g = probability density function
µj
µ0
= X
bjL(b|data) (b) db
X
L(b|data) (b) db
= Bayesian posterior expectation of βj
µ :=
X
g(x)dx =
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n
i=1
f(xi) =
X
f(x) ^νn(dx)
How big should n be so that |µ − µn| ε ?
2/21
3. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Multidimensional Integration Examples
µ =
Rd
f(x) ν(dx) = option price
f(x) = discounted payoff determined by market forces x
µ =
X
g(x) dx = P(X ∈ X) = probability
g = probability density function
µj
µ0
= X
bjL(b|data) (b) db
X
L(b|data) (b) db
= Bayesian posterior expectation of βj
µ :=
X
g(x)dx =
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n
i=1
f(xi) =
X
f(x) ^νn(dx)
How big should n be so that |µ − µn| ε ?
2/21
4. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Error of (Different Kinds of) Monte Carlo Methods
µ :=
X
g(x) dx =
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n
i=1
f(xi) =
X
f(x) ^νn(dx)
errn(f) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
∀f
errn(f)
∀ reasonable f
errn(f)
data-driven choose n large enough
ε
errn(f) = size(f)
unknown
× size(ν − ^νn)
maybe known
(H., 2017+)
may choose ^νn more carefully than IID
for random algorithms/integrands, the inequalities above should hold with high probability
algorithms described here in the Guaranteed Automatic Integration Library (GAIL) (Choi et al.,
2013–2017)
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5. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping IID Sampling
µ :=
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n
i=1
f(xi), xi
IID
∼ ν, errn(f) = |µ − µn|
The Central Limit Theorem says that
P [errn(f) errn(f)] ≈
n→∞
99%, errn(f) =
2.58σ
√
n
, σ2
= var(f(X))
4/21
6. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping IID Sampling
µ :=
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n+nσ
i=1+nσ
f(xi), xi
IID
∼ ν, errn(f) = |µ − µn|
The Central Limit Theorem says that
P [errn(f) errn(f)] ≈
n→∞
99%, errn(f) =
2.58σ
√
n
errn(f) =
2.58 × 1.2^σ
√
n
, ^σ2
=
1
nσ − 1
nσ
i=1
f(xi)
4/21
7. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping IID Sampling
µ :=
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n+nσ
i=1+nσ
f(xi), xi
IID
∼ ν, errn(f) = |µ − µn|
The Central Limit Theorem says that
P [errn(f) errn(f)] ≈
n→∞
99%, errn(f) =
2.58σ
√
n
errn(f) =
2.58 × 1.2^σ
√
n
, ^σ2
=
1
nσ − 1
nσ
i=1
f(xi)
Choose n =
2.58 × 1.2^σ
ε
2
to make errn(f) ε
4/21
8. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping IID Sampling
µ :=
X
f(x) ν(dx) =: E[f(X)] ≈ µn :=
1
n
n+nσ
i=1+nσ
f(xi), xi
IID
∼ ν, errn(f) = |µ − µn|
H. et al. (2013) (see also Bayer et al. (2014), H. et al. (2017+)) assume that the integrand belongs to the
cone of reasonable integrands: C = {f ∈ L4
(X, ν) : kurt(f(X)) κup}. Then we may replace n→∞ by
using Berry-Esseen inequalities (Nefedova and Shevtsova, 2012; Shevtsova, 2014) with a more
complicated formula for errn(f). Bound on the kurtosis also allows us to be highly confident that
σ 1.2^σ via Cantelli’s inequality.
P [errn(f) errn(f)] 99.5%, P(σ 1.2^σ) 99.5%, ^σ2
=
1
nσ − 1
nσ
i=1
f(xi)
errn(f) = min δ > 0 : Φ −
√
nδ/σ +
1
√
n
min 0.3328(κ
3/4
up + 0.429),
18.1139κ
3/4
up
1 +
√
nδ/σ
3
0.25% ,
errn(f) = same as errn(f) but with σ replaced by 1.2^σ
Choose n to make errn(f) ε
4/21
9. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Option Pricing
µ = fair price =
Rd
e−rT
max
1
d
d
j=1
Sj − K, 0
e−zT
z/2
(2π)d/2
dz ≈ $13.12
Sj = S0e(r−σ2
/2)jT/d+σxj
= stock price at time jT/d,
x = Az, AAT
= Σ = min(i, j)T/d
d
i,j=1
, T = 1/4, d = 13 here
Abs. Error Median Worst 10% Worst 10%
Tolerance Method Error Accuracy n Time (s)
1E−2 IID 2E−3 100% 6.1E7 33.000
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10. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
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11. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
12. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
13. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
14. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
15. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
16. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
17. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Faster Convergence with Quasi-Monte Carlo Sampling
err(f, n) := |µ − µn| =
X
f(x)
make small by
importance sampling
(ν − ^νn)
make small by
clever sampling
(dx)
6/21
19. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping Quasi-Monte Carlo Sampling
µ =
[0,1]d
f(x) dx = f(0) ≈ µn = fn(0), f(x) =
k
f(k)φk(x), f(k) =
[0,1]d
f(x)φk(x) dx
φk are complex exponentials (for lattices) or Walsh functions (for nets)
· · ·
· · ·
errn(f) errn(f) :=
k∈dual set{0}
f(k) , dual set = {k : φk(xi) = φk(x1), i = 1, . . . , n
φk looks like a constant
}
discrete transform, fn(k) =
1
n
n
i=1
f(xi)φk(xi) ∀k, can be computed in O(n log(n)) operations
7/21
20. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Stopping Quasi-Monte Carlo Sampling
µ =
[0,1]d
f(x) dx = f(0) ≈ µn = fn(0), f(x) =
k
f(k)φk(x), f(k) =
[0,1]d
f(x)φk(x) dx
φk are complex exponentials (for lattices) or Walsh functions (for nets)
errn(f) errn(f) :=
k∈dual set{0}
f(k) , dual set = {k : φk(xi) = φk(x1), i = 1, . . . , n
φk looks like a constant
}
discrete transform, fn(k) =
1
n
n
i=1
f(xi)φk(xi) ∀k, can be computed in O(n log(n)) operations
H. and Jiménez Rugama (2016), Jiménez Rugama and H. (2016) and (see also H. et al. (2017+)) define
a cone, C, of reasonable integrands whose true Fourier coefficients do not decay too erratically. Then
errn(f) errn(f) errn(f) := C(n)
moderatek
fn(k) ∀f ∈ C
Control variates are possible (H. et al., 2017+).
7/21
21. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
A Reasonable and an Unreasonable Integrand
8/21
22. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Option Pricing
µ = fair price =
Rd
e−rT
max
1
d
d
j=1
Sj − K, 0
e−zT
z/2
(2π)d/2
dz ≈ $13.12
Sj = S0e(r−σ2
/2)jT/d+σxj
= stock price at time jT/d,
x = Az, AAT
= Σ = min(i, j)T/d
d
i,j=1
, T = 1/4, d = 13 here
Abs. Error Median Worst 10% Worst 10%
Tolerance Method Error Accuracy n Time (s)
1E−2 IID 2E−3 100% 6.1E7 33.000
1E−2 Scr. Sobol’ 1E−3 100% 1.6E4 0.040
9/21
23. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Option Pricing
µ = fair price =
Rd
e−rT
max
1
d
d
j=1
Sj − K, 0
e−zT
z/2
(2π)d/2
dz ≈ $13.12
Sj = S0e(r−σ2
/2)jT/d+σxj
= stock price at time jT/d,
x = Az, AAT
= Σ = min(i, j)T/d
d
i,j=1
, T = 1/4, d = 13 here
Abs. Error Median Worst 10% Worst 10%
Tolerance Method Error Accuracy n Time (s)
1E−2 IID 2E−3 100% 6.1E7 33.000
1E−2 Scr. Sobol’ 1E−3 100% 1.6E4 0.040
1E−2 Scr. Sob. cont. var. 2E−3 100% 4.1E3 0.019
10/21
24. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Bayesan Cubature with Nice Covariance Kernels
f ∼ GP(m, s2
Cθ), Cθ : [0, 1]d
× [0, 1]d
→ R, m, s, θ to be inferred by MLE,
C = Cθ(xi, xj)
n
i,j=1
=
1
n
WΛWH
, Λ = diag(λ1, . . . , λn), C1 = λ11,
[0,1]d
Cθ(x, t) dt = 1 ∀x
Using maximum likelihood estimation (MLE) gives
mMLE = µn =
1
n
n
i=1
f(xi), s2
MLE =
1
n2
n
i=2
|yi|2
λi
, y = WT
f(x1)
...
f(xn)
fast transform, O(n log n)
,
θMLE = argmin
θ
log
n
i=2
|yi|2
λi
+
1
n
n
i=1
log(λi)
Then µ is a Gaussian random variable, and
P
|µ − µn| 2.58 1 −
n
λ1
1
n2
n
i=2
|yi|2
λ2
i
= 99% for reasonable f
11/21
25. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Option Pricing
µ = fair price =
Rd
e−rT
max
1
d
d
j=1
Sj − K, 0
e−zT
z/2
(2π)d/2
dz ≈ $13.12
Sj = S0e(r−σ2
/2)jT/d+σxj
= stock price at time jT/d,
x = Az, AAT
= Σ = min(i, j)T/d
d
i,j=1
, T = 1/4, d = 13 here
Abs. Error Median Worst 10% Worst 10%
Tolerance Method Error Accuracy n Time (s)
1E−2 IID 2E−3 100% 6.1E7 33.000
1E−2 Scr. Sobol’ 1E−3 100% 1.6E4 0.040
1E−2 Scr. Sob. cont. var. 2E−3 100% 4.1E3 0.019
1E−2 Bayes. Latt. 2E−3 99% 1.6E4 0.051
12/21
26. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
More General Error Criteria
Let’s change the problem: find µn satisfying
|µ − µn| max(ε, εr |µ|), given |µ − µn| errn(f)
H. et al. (2017+) show that if n is large enough to make
errn(f)
max(ε, εr µn + errn(f) ) + max(ε, εr µn − errn(f) )
2
then a satisfactory choice is
µn =
(µn − errn(f)) max(ε, εr µn + errn(f) ) + (µn + errn(f)) max(ε, εr µn − errn(f) )
max(ε, εr µn + errn(f) ) + max(ε, εr µn − errn(f) )
=
µn, εr = 0
max(µ2
n − errn(f)2
, 0)
µn
, ε = 0
H. et al. (2017+) extend this to the case where the solution is v(µ):
|v(µ) − ˜v| max(ε, εr |v(µ)|), given µ ∈ [µ − errn(f), µ + errn(f)]
13/21
28. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Bayesian Inference for Logistic Regression
µj
µ0
= X
bjL(b|data) (b) db
X
L(b|data) (b) db
= Bayesian posterior expectation of βj
yi ∼ Ber
exp(β1 + β2xi)
1 + exp(β1 + β2xi)
, (b) =
exp −(b2
1 + b2
2)/2
2π
Importance sample via an
appropriately chosen normal
ε = 0.001
n = 9 000–17 000
15/21
29. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Sobol’ Indices
Y = g(X), where X ∼ U[0, 1]d
; Sobol’ Indexj(µ) describes how much coordinate j of input X influences
output Y (Sobol’, 1990; 2001):
Sobol’ Indexj(µ) :=
µ1
µ2 − µ2
3
, j = 1, . . . , d
µ1 :=
[0,1)2d
[g(x) − g(xj, x −j)]g(x ) dx dx
µ2 :=
[0,1)d
[g(x)]2
dx, µ3 :=
[0,1)d
g(x) dx
Jiménez Rugama and Gilquin (2017+) have used adaptive Sobol’ sampling to compute Sobol’ indices:
g(x) = −x1 + x1x2 − x1x2x3 + · · · + x1x2x3x4x5x6 (Bratley et al., 1992)
ε = 1E−3, εr = 0 j 1 2 3 4 5 6
n 65 536 32 768 16 384 16 384 2 048 2 048
Sobol’ Indexj 0.6529 0.1791 0.0370 0.0133 0.0015 0.0015
Sobol’ Indexj 0.6528 0.1792 0.0363 0.0126 0.0010 0.0012
Sobol’ Indexj(µn) 0.6492 0.1758 0.0308 0.0083 0.0018 0.0039
16/21
30. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Summary
Established conservative, pre-asymptotic confidence intervals for reasonable
integrands
for IID sampling
for Quasi-Monte Carlo (low discrepancy) sampling, and
via Bayesian cubature
to fit even general error criteria
Quasi-Monte Carlo sampling can give faster convergence
Bayesian cubature is practical if the sample matches the covariance kernel so that
the vector-matrix operations are fast enough
Algorithms implemented in open-source GAIL (Choi et al., 2013–2017), which is
undergoing continual development
17/21
31. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
Further Questions
Are the cones of reasonable integrands reasonable in practice?
What data-driven necessary conditions are there for f to be reasonable (f ∈ C)?
Can we get higher order convergence with higher order nets?
What can be done for Markov Chain Monte Carlo?
What can be done for multi-level Monte Carlo (Giles, 2015)?
18/21
32. Slides available at speakerdeck.com/fjhickernell/
samsi-qmc-december-2017-trends-workshop
33. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
References I
Bayer, C., H. Hoel, E. von Schwerin, and R. Tempone. 2014. On nonasymptotic optimal stopping criteria in Monte Carlo
Simulations, SIAM J. Sci. Comput. 36, A869–A885.
Bratley, P., B. L. Fox, and H. Niederreiter. 1992. Implementation and tests of low-discrepancy sequences, ACM Trans. Model.
Comput. Simul. 2, 195–213.
Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, Ll. A. Jiménez Rugama, D. Li, J. Rathinavel, X. Tong, K. Zhang, Y. Zhang, and X. Zhou.
2013–2017. GAIL: Guaranteed Automatic Integration Library (versions 1.0–2.2).
Genz, A. 1993. Comparison of methods for the computation of multivariate normal probabilities, Computing Science and
Statistics 25, 400–405.
Giles, M. 2015. Multilevel Monte Carlo methods, Acta Numer. 24, 259–328.
H., F. J. 2017+. The trio identity for quasi-Monte Carlo error analysis, Monte Carlo and quasi-Monte Carlo methods: MCQMC,
Stanford, usa, August 2016. submitted for publication, arXiv:1702.01487.
H., F. J., S.-C. T. Choi, L. Jiang, and Ll. A. Jiménez Rugama. 2017+. Monte Carlo simulation, automatic stopping criteria for,
Wiley statsref-statistics reference online. to appear.
H., F. J., L. Jiang, Y. Liu, and A. B. Owen. 2013. Guaranteed conservative fixed width confidence intervals via Monte Carlo
sampling, Monte Carlo and quasi-Monte Carlo methods 2012, pp. 105–128.
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34. Introduction IID Sampling QMC Sampling Bayesian Cubature General Error Criteria Concluding Remarks References
References II
H., F. J. and Ll. A. Jiménez Rugama. 2016. Reliable adaptive cubature using digital sequences, Monte Carlo and quasi-Monte
Carlo methods: MCQMC, Leuven, Belgium, April 2014, pp. 367–383. arXiv:1410.8615 [math.NA].
H., F. J., Ll. A. Jiménez Rugama, and D. Li. 2017+. Adaptive quasi-Monte Carlo methods for cubature, Contemporary
computational mathematics — a celebration of the 80th birthday of ian sloan. to appear, arXiv:1702.01491 [math.NA].
Jiménez Rugama, Ll. A. and L. Gilquin. 2017+. Reliable error estimation for Sobol’ indices, Statistics and Computing. in press.
Jiménez Rugama, Ll. A. and F. J. H. 2016. Adaptive multidimensional integration based on rank-1 lattices, Monte Carlo and
quasi-Monte Carlo methods: MCQMC, Leuven, Belgium, April 2014, pp. 407–422. arXiv:1411.1966.
Nefedova, Yu. S. and I. G. Shevtsova. 2012. On non-uniform convergence rate estimates in the central limit theorem, Theory
Probab. Appl. 57, 62–97.
Shevtsova, I. 2014. On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands, Dokl.
Akad. Nauk 456, 650–654.
Sobol’, I. M. 1990. On sensitivity estimation for nonlinear mathematical models, Matem. Mod. 2, no. 1, 112–118.
. 2001. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Math. Comput.
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