Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Error Analysis for Quasi-Monte Carlo Methods - Fred Hickernell, Aug 28, 2017
Multidimensional integrals may be approximated by weighted averages of integrand values. Quasi-Monte Carlo (QMC) methods are more accurate than simple Monte Carlo methods because they carefully choose where to evaluate the integrand. This tutorial focuses on how quickly QMC methods converge to the correct answer as the number of integrand values increases. The answer may depend on the smoothness of the integrand and the sophistication of the QMC method. QMC error analysis may assumes the integrand belongs to a reproducing kernel Hilbert space or may assume that the integrand is an instance of a stochastic process with known covariance structure. These two approaches have interesting parallels. This tutorial also explores how the computational cost of achieving a good approximation to the integral depends on the dimension of the domain of the integrand. Finally, this tutorial explores methods for determining how many integrand values are needed to satisfy the error tolerance. Relevant software is described.
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.
Many mathematical models use a large number of poorly-known parameters as inputs. Quantifying the influence of each of these parameters is one of the aims of sensitivity analysis. Global Sensitivity Analysis is an important paradigm for understanding model behavior, characterizing uncertainty, improving model calibration, etc. Inputs’ uncertainty is modeled by a probability distribution. There exist various measures built in that paradigm. This tutorial focuses on the so-called Sobol’ indices, based on functional variance analysis. Estimation procedures will be presented, and the choice of the designs of experiments these procedures are based on will be discussed. As Sobol’ indices have no clear interpretation in the presence of statistical dependences between inputs, it also seems promising to measure sensitivity with Shapley effects, based on the notion of Shapley value, which is a solution concept in cooperative game theory.
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
In this tutorial I will provide a survey of recent research efforts on the application of QMC methods to PDEs with random coefficients. Such PDE problems occur in the area of uncertainty quantification. A prime example is the flow of water through a disordered porous medium. There is a huge body of literature on this topic using a variety of methods. QMC methods are relatively new to this application area. The aim of this tutorial is to provide an entry point for QMC experts wanting to start research in this direction, for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods, and for anyone else who sees how to cross-fertilize the ideas to other application areas.
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.
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.
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
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.
Many mathematical models use a large number of poorly-known parameters as inputs. Quantifying the influence of each of these parameters is one of the aims of sensitivity analysis. Global Sensitivity Analysis is an important paradigm for understanding model behavior, characterizing uncertainty, improving model calibration, etc. Inputs’ uncertainty is modeled by a probability distribution. There exist various measures built in that paradigm. This tutorial focuses on the so-called Sobol’ indices, based on functional variance analysis. Estimation procedures will be presented, and the choice of the designs of experiments these procedures are based on will be discussed. As Sobol’ indices have no clear interpretation in the presence of statistical dependences between inputs, it also seems promising to measure sensitivity with Shapley effects, based on the notion of Shapley value, which is a solution concept in cooperative game theory.
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
In this tutorial I will provide a survey of recent research efforts on the application of QMC methods to PDEs with random coefficients. Such PDE problems occur in the area of uncertainty quantification. A prime example is the flow of water through a disordered porous medium. There is a huge body of literature on this topic using a variety of methods. QMC methods are relatively new to this application area. The aim of this tutorial is to provide an entry point for QMC experts wanting to start research in this direction, for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods, and for anyone else who sees how to cross-fertilize the ideas to other application areas.
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.
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.
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
One of the central tasks in computational mathematics and statistics is to accurately approximate unknown target functions. This is typically done with the help of data — samples of the unknown functions. The emergence of Big Data presents both opportunities and challenges. On one hand, big data introduces more information about the unknowns and, in principle, allows us to create more accurate models. On the other hand, data storage and processing become highly challenging. In this talk, we present a set of sequential algorithms for function approximation in high dimensions with large data sets. The algorithms are of iterative nature and involve only vector operations. They use one data sample at each step and can handle dynamic/stream data. We present both the numerical algorithms, which are easy to implement, as well as rigorous analysis for their theoretical foundation.
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.
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
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.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
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.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
One of the central tasks in computational mathematics and statistics is to accurately approximate unknown target functions. This is typically done with the help of data — samples of the unknown functions. The emergence of Big Data presents both opportunities and challenges. On one hand, big data introduces more information about the unknowns and, in principle, allows us to create more accurate models. On the other hand, data storage and processing become highly challenging. In this talk, we present a set of sequential algorithms for function approximation in high dimensions with large data sets. The algorithms are of iterative nature and involve only vector operations. They use one data sample at each step and can handle dynamic/stream data. We present both the numerical algorithms, which are easy to implement, as well as rigorous analysis for their theoretical foundation.
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.
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
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.
Classification with mixtures of curved Mahalanobis metricsFrank Nielsen
Presentation at ICIP 2016.
Slide 4, there is a typo, replace absolute value by parenthesis. The cross-ratio can be negative and we use the principal complex logarithm
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.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
A series of maximum entropy upper bounds of the differential entropy
Similar to Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Error Analysis for Quasi-Monte Carlo Methods - Fred Hickernell, Aug 28, 2017
1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
Tucker tensor analysis of Matern functions in spatial statistics Alexander Litvinenko
1. Motivation: improve statistical models
2. Motivation: disadvantages of matrices
3. Tools: Tucker tensor format
4. Tensor approximation of Matern covariance function via FFT
5. Typical statistical operations in Tucker tensor format
6. Numerical experiments
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Slides: On the Chi Square and Higher-Order Chi Distances for Approximating f-...Frank Nielsen
Slides for the paper:
On the Chi Square and Higher-Order Chi Distances for Approximating f-Divergences
published in IEEE SPL:
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6654274
This 20-minute talk was presented during the Operator Splitting Workshop
Similar to Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Error Analysis for Quasi-Monte Carlo Methods - Fred Hickernell, Aug 28, 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.
More from The Statistical and Applied Mathematical Sciences Institute (20)
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.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
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Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics Opening Workshop, Error Analysis for Quasi-Monte Carlo Methods - Fred Hickernell, Aug 28, 2017
1. Error Analysis for Quasi-Monte Carlo Methods
Fred J. Hickernell
Department of Applied Mathematics, 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 and the QMC Program organizers
SAMSI-QMC Opening Worskshop, August 28, 2017
2. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
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 = E[f(X)] =
X
f(x) ν(dx) ≈ ^µ =
n
i=1
f(xi)wi =
X
f(x) ^ν(dx) µ − ^µ = ?
2/22
3. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
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 = E[f(X)] =
X
f(x) ν(dx) ≈ ^µ =
n
i=1
f(xi)wi =
X
f(x) ^ν(dx) µ − ^µ = ?
2/22
4. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
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 = E[f(X)] =
X
f(x) ν(dx) ≈ ^µ =
n
i=1
f(xi)wi =
X
f(x) ^ν(dx) µ − ^µ = ?
2/22
5. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
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 = E[f(X)] =
X
f(x) ν(dx) ≈ ^µ =
n
i=1
f(xi)wi =
X
f(x) ^ν(dx) µ − ^µ = ?
2/22
6. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Error Analysis for Quasi-Monte Carlo Methods
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ =
X
f(x) (ν − ^ν)(dx) = ?
Reality Error Analysis (Theory)
What is What we think it should be by rigorous argument
Not certain that our assumptions hold
3/22
7. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Error Analysis for Quasi-Monte Carlo Methods
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ =
X
f(x) (ν − ^ν)(dx) = ?
Reality Error Analysis (Theory)
What is What we think it should be by rigorous argument
Not certain that our assumptions hold
Quasi-Monte Carlo Methods
x1, x2, . . . chosen more carefully that IID, dependent
3/22
8. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Error Decomposition in Reproducing Kernel Hilbert Spaces
Let the integrand lie in a Hilbert space, H, with reproducing kernel, K : X × X → R:
K(·, t) ∈ H, f(t) = K(·, t), f ∀t ∈ X, f ∈ H
The cubature error is given by the inner product with the representer, ηerr:
µ − ^µ =
X
f(x) (ν − ^ν)(dx) = ηerr, f , where ηerr(t) = K(·, t), ηerr =
X
K(x, t) (ν − ^ν)(dx)
The Cauchy-Schwartz inequality says that (H., 2000)
µ − ^µ = ηerr, f = cos(f, ηerr) DSC(ν − ^ν) f H discrepancy DSC(ν − ^ν) := ηerr H
DSC2
= ηerr, ηerr =
X×X
K(x, t) (ν − ^ν)(dx)(ν − ^ν)(dt)
=
X2
K(x, t) ν(dx)ν(dt) − 2
n
i=1
wi
X
K(xi, t) ν(dt) +
n
i,j=1
wiwjK(xi, xj)
= k0 − 2kT
w + wT
Kw w = K−1
k is optimal but expensive to compute
4/22
9. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
L2
-Discrepancy
ν = uniform on X = [0, 1]d
, ^ν = uniform on {xi}n
i=1, K(x, t) =
d
k=1
[2 − max(xk, tk)] (H., 1998)
f H = ∂u
f L2
u⊆1:d 2
, ∂u
f :=
∂|u|
f
∂xu xsu = 1
, 1:d := {1, . . . , d}, su := 1:d u
DSC2
=
4
3
d
−
2
n
n
i=1
d
k=1
3 − x2
ik
2
+
1
n2
n
i,j=1
d
k=1
[2 − max(xik, xjk)]
5/22
10. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
L2
-Discrepancy
ν = uniform on X = [0, 1]d
, ^ν = uniform on {xi}n
i=1, K(x, t) =
d
k=1
[2 − max(xk, tk)] (H., 1998)
f H = ∂u
f L2
u⊆1:d 2
, ∂u
f :=
∂|u|
f
∂xu xsu = 1
, 1:d := {1, . . . , d}, su := 1:d u
DSC2
=
4
3
d
−
2
n
n
i=1
d
k=1
3 − x2
ik
2
+
1
n2
n
i,j=1
d
k=1
[2 − max(xik, xjk)]
= ν([0, ·u]) − ^ν([0, ·u]) L2
∅=u⊆1:d
2
2
geometric interpretation
x = (. . . , 0.6, . . . , 0.4, . . .)
ν([0, x{5,8}]) − ^ν([0, x{5,8}])
= 0.24 − 7/32 = 0.02125
5/22
11. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
L2
-Discrepancy
ν = uniform on X = [0, 1]d
, ^ν = uniform on {xi}n
i=1, K(x, t) =
d
k=1
[2 − max(xk, tk)] (H., 1998)
f H = ∂u
f L2
u⊆1:d 2
, ∂u
f :=
∂|u|
f
∂xu xsu = 1
, 1:d := {1, . . . , d}, su := 1:d u
DSC2
=
4
3
d
−
2
n
n
i=1
d
k=1
3 − x2
ik
2
+
1
n2
n
i,j=1
d
k=1
[2 − max(xik, xjk)]
= ν([0, ·u]) − ^ν([0, ·u]) L2
∅=u⊆1:d
2
2
geometric interpretation
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f − f(1) H
DSC(ν − ^ν)
requires O(dn2
) operations to evaluate
= O(n−1/2
) on average for IID Monte Carlo
= O(n−1+
) for digital nets, integration lattices, ...
(Niederreiter, 1992; Dick and Pillichshammer, 2010)
O(n−1
) for all ^ν because this K has limited smoothness 5/22
12. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx, ^ν = uniform on {xi}n
i=1
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
DSC(ν−^ν) =
O(n−1/2
) for IID MC
O(n−1+
) for Sobol’
6/22
13. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx, ^ν = uniform on {xi}n
i=1
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
DSC(ν−^ν) =
O(n−1/2
) for IID MC
O(n−1+
) for Sobol’
6/22
14. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx, ^ν = uniform on {xi}n
i=1
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
DSC(ν−^ν) =
O(n−1/2
) for IID MC
O(n−1+
) for Sobol’
O(n−1.5+
) for Scr. Sobol’
w/ smoother kernel (Owen, 1997)
(H. and Yue, 2000; Heinrich et al., 2004)
6/22
15. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx, ^ν = uniform on {xi}n
i=1
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
DSC(ν−^ν) =
O(n−1/2
) for IID MC
O(n−1+
) for Sobol’
O(n−1.5+
) for Scr. Sobol’
w/ smoother kernel (Owen, 1997)
(H. and Yue, 2000; Heinrich et al., 2004)
For some typical choice of a, b, Σ, and d = 3
µ ≈ 0.6763
Low discrepancy sampling reduces cubature error;
randomization can help more
6/22
16. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Mean Square Discrepancy for Randomized Low Discrepancy Sets
Popular low discrepancy sets on [0, 1)d
—digital nets and lattice node sets—can be randomized while
preserving their low discrepancy properties. Let ψ : [0, 1)d
→ [0, 1)d
be a random mapping and
ψ(x) ∼ U[0, 1)d
, Kf(x, t) := E[K(ψ(x), ψ(t)] Kf1 = (1T
Kf,1)1, k0 = 1
where the vector Kf,1 is the first column of the matrix Kf, the analog of K with the original sequence but
for the filtered kernel, Kf. Taking equal weights w = 1/n,
E{[DSC(ν − ^ν; K)]2
} = E[k0 − 2kT
w + wT
Kw] = −1 +
1T
Kf,1
n
= [DSC(ν − ^ν; Kf)]2
7/22
17. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Mean Square Discrepancy for Randomized Low Discrepancy Sets
Popular low discrepancy sets on [0, 1)d
—digital nets and lattice node sets—can be randomized while
preserving their low discrepancy properties. Let ψ : [0, 1)d
→ [0, 1)d
be a random mapping and
ψ(x) ∼ U[0, 1)d
, Kf(x, t) := E[K(ψ(x), ψ(t)] Kf1 = (1T
Kf,1)1, k0 = 1
where the vector Kf,1 is the first column of the matrix Kf, the analog of K with the original sequence but
for the filtered kernel, Kf. Taking equal weights w = 1/n,
E{[DSC(ν − ^ν; K)]2
} = E[k0 − 2kT
w + wT
Kw] = −1 +
1T
Kf,1
n
= [DSC(ν − ^ν; Kf)]2
DSC(ν − ^ν; Kf) only requires O(n) operations to compute
Suggests using Hilbert spaces with reproducing kernels of the form Kf
Kf(x, t) =
l
λlφl(x)sφl(t),
φl(x) = Walsh functions for digital nets
φl(x) = exp(2πlT
x) for lattice nodesets
with λl chosen carefully so that the series sums in closed form and Kf has the desired smoothness
properties
7/22
18. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
The Effect of Dimension on the Decay of the Discrepancy
What happens when dimension, d, is large and but sample size, n, is modest?
L2
-discrepancy and variation:
DSC2
=
4
3
d
−
2
n
n
i=1
d
k=1
3 − x2
ik
2
+
1
n2
n
i,j=1
d
k=1
[2 − max(xik, xjk)]
f H =
∂|u|
f
∂xu xsu=1 L2 u⊆1:d 2
For Scrambled Sobol’ points
The onset of O(n−1+
) convergence of DSC is d dependent.
8/22
19. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Pricing an Asian Option
µ =
Rd
payoff(t)
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt =
[0,1]d
f(x) dx
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
Asian arithmetic mean option, d = 12, µ ≈ $13.1220
Error converges like O(n−1+
) for (scrambled)
Sobol’ even though discrepancy does not.
Phenomenon first seen by Paskov and Traub
(1995) for Collaterlized Mortgage Obligation
with d = 360
Scrambled Sobol’ does not achieve
O(n−3/2+
) convergence because f is not
smooth enough.
f is not even smooth enough for O(n−1+
)
convergence, except by a delicate argument
(Griebel et al., 2010; 2016+)
9/22
20. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Overcoming the Curse of Dimensionality with Weights
Sloan and Woźniakowski (1998) show how inserting decaying coordinate weights yields O(n−1+
)
convergence independent of dimension
Caflisch et al. (1997) explain how quasi-Monte Carlo methods work well for low effective dimension
Novak and Woźniakowski (2010) cover tractability comprehensively
K(x, t) =
d
k=1
[1 + γ2
k(1 − max(xk, tk))]
DSC2
=
d
k=1
1 +
γ2
k
3
−
2
n
n
i=1
d
k=1
1 +
γ2
k(1 − x2
ik)
2
+
1
n2
n
i,j=1
d
k=1
[1 + γ2
k(1 − max(xik, xjk))]
f H =
1
γu
∂|u|
f
∂xu xsu=1 L2
u⊆1:d 2
γu =
k∈u
γk
For Scrambled Sobol’ points
γ2
k = k−3
10/22
21. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Comments So Far
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
Cubature error represented as a trio identity (Meng, 2017+; H., 2017+); without the cos() term it is a
Koksma-Hlawka inequality
There are other versions than the deterministic one shown here
RKHS is an oft used tool—not only for cubature error (Fasshauer and McCourt, 2015)—and the analysis
may be extended to Banach spaces (H., 1998; 2000)
Error analysis can be attacked directly through Fourier Walsh or complex exponential series
If kernel has special form, then DSC(ν − ^ν) and a lower bound on f H can be computed in O(n log n)
operations; otherwise they are expensive
For this error decomposition the adaptive choice of the position of the next data site xn+1 does not
depend on f
11/22
22. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Comments So Far
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
The discrepancy measures the sample quality, and is independent of the integrand.
Decay rate may be improved by samples more carefully chosen than IID
Clever randomization low discrepancy points can reduce the discrepancy even further
Definition depends on the reproducing kernel (and the corresponding Hilbert space of integrands)
Working with filtered kernels may expedite computation of the discrepancy and of the optimal weights.
What is the appropriate filtered kernel for higher order nets?
Non-uniform low discrepancy points are often transformations of uniform low discrepancy points
There are other quality measures for points than the discrepancy, e.g., covering radius, minimum
distance between two points, the power function in RKHS approximation
11/22
23. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Comments So Far
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
The discrepancy measures the sample quality, and is independent of the integrand.
The strategy for choosing points that optimize the quality measure depends on whether you want to be
asymptotically optimal or pre-asymptotically optimal. See
WG IV Representative Points for Small-data and Big-data Problems
WG V Sampling and Analysis in High Dimensions When Samples Are Expensive
The size of the discrepancy may be affected by the dimension unless the space of integrands is defined
to de-emphasize higher dimensions. For how to choose the weights and sampling optimally for your
problem, see
WG AO Tuning QMC to the Specific Integrand of Interest
11/22
24. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Comments So Far
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
The term f H is often called the variation.
It can sometimes be made smaller by using a change of measure (variable transformation) to re-write
the integral with a different integrand.
11/22
25. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Comments So Far
µ := E[f(X)] :=
X
f(x) ν(dx) ≈ ^µ :=
n
i=1
f(xi)wi =
X
f(x) ^ν(dx)
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
The term f H is often called the variation.
It can sometimes be made smaller by using a change of measure (variable transformation) to re-write
the integral with a different integrand.
The term cos(f, ηerr) is called the confounding. It cannot exceed one in magnitude. It could be
quite small if your integrand is atypically nice.
11/22
26. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Bayesian Cubature
Random f postulated by Diaconis (1988), O’Hagan (1991), Ritter (2000), Rasmussen and Ghahramani
(2003) and others: f ∼ GP(0, s2
Cθ), a Gaussian process from the sample space F with zero mean and
covariance kernel, s2
Cθ, Cθ : X × X → R. Then
c0 =
X2
C(x, t) ν(dx)ν(dt), c =
X
C(xi, t) ν(dt)
n
i=1
C = C(xi, xj)
n
i,j=1
,
w = wi
n
i=1
= C−1
c is optimal
µ − ^µ = N(0, 1) × c0 − cTC−1c × s
The scale parameter, s, and shape parameter, θ, should be estimated, e.g. by maximum likelihood
estimation
Pr |µ − ^µ| 2.58 c0 − cTC−1c × s = 99% allows probabilistic error bounds
Requires O(n3
) operations to compute C−1
θ , but see Anitescu et al. (2016)
Ill-conditioning for smoother kernels (faster convergence)
DSC(ν − ^ν; K) = c0 − cTC−1c for optimal weights an d K = Cθ
See WG II Probabilistic Numerics 12/22
27. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx
µ − ^µ = N(0, 1) × c0 − cTC−1c × s
Use a product Matérn kernel with modest smoothness:
Cθ(x, t) =
d
k=1
(1 + θ |xk − tk|)e−θ|xk−tk|
Smaller error using Bayesian cubature with
scrambled Sobol’ data sites
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28. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx
µ − ^µ = N(0, 1) × c0 − cTC−1c × s
Use a product Matérn kernel with modest smoothness:
Cθ(x, t) =
d
k=1
(1 + θ |xk − tk|)e−θ|xk−tk|
Smaller error using Bayesian cubature with
scrambled Sobol’ data sites
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29. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Multivariate Normal Probability
µ =
[a,b]
exp −1
2 tT
Σ−1
t
(2π)d det(Σ)
dt
Genz (1993)
=
[0,1]d−1
f(x) dx
µ − ^µ = N(0, 1) × c0 − cTC−1c × s
Use a product Matérn kernel with modest smoothness:
Cθ(x, t) =
d
k=1
(1 + θ |xk − tk|)e−θ|xk−tk|
Smaller error using Bayesian cubature with
scrambled Sobol’ data sites
Confidence intervals succeed ≈ 83% of the time
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30. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
How Many Samples Are Needed?
We know that
µ − ^µ = cos(f, ηerr) DSC(ν − ^ν) f H
But, given an error tolerance, ε, how do we decide how many samples, n, are needed to make
|µ − ^µ| ε
Bayesian cubature provides data-based confidence intervals for ^µ
For digital net (e.g., Sobol’) and integration lattice sampling
Independent replications present a trade-off between number of replications and number of samples
But rigorous stopping criteria for a single sequence have been constructed in terms of the discrete
Fourier Walsh/complex exponential coefficients of f assuming that the true Fourier coefficients decay
steadily (H. and Jiménez Rugama, 2016; Jiménez Rugama and H., 2016; Li, 2016)
Functions of integrals and relative error criteria may also be handled (H. and Jiménez Rugama, 2016)
These automatic cubature rules are part of the Guaranteed Automatic Integration Library (GAIL) (Choi et
al., 2013–2017)
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31. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary 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 A Error Accuracy n Time (s)
1E−2 IID diff 2E−3 100% 6.1E7 33.000
1E−2 Scr. Sobol’ PCA 1E−3 100% 1.6E4 0.040
1E−2 Scr. Sob. cont. var. PCA 2E−3 100% 4.1E3 0.019
1E−2 Bayes. Latt. PCA 2E−3 99% 1.6E4 0.051
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33. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary 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
ε = 0.001, n = 9 000–17 000
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34. Introduction RKHS Dimension Effects Interlude Bayesian Cubature GAIL Summary References
Summary
Reproducing kernel Hilbert spaces are a powerful for characterizing the error of quasi-Monte Carlo
methods
Digital nets and nodesets of lattice have faster decaying discrepancies which suggests faster
decaying cubature errors
How well these bounds fit what is observed depends on whether the integrand is typical within the
space of intgrands
Practical stopping criteria are an area of active research
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References I
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effective dimension, J. Comput. Finance 1, 27–46.
Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, Ll. A. Jiménez Rugama, X. Tong, Y. Zhang, and X. Zhou. 2013–2017. GAIL:
Guaranteed Automatic Integration Library (versions 1.0–2.2).
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References II
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H., F. J. and Ll. A. Jiménez Rugama. 2016. Reliable adaptive cubature using digital sequences, Monte Carlo and quasi-Monte
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Owen, A. B. 1997. Scrambled net variance for integrals of smooth functions, Ann. Stat. 25, 1541–1562.
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Ritter, K. 2000. Average-case analysis of numerical problems, Lecture Notes in Mathematics, vol. 1733, Springer-Verlag, Berlin.
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