This talk will report briey on some findings from the problem of picking the weights for a weighted function space in QMC. Then it will be mostly about importance sampling. We want to estimate the probability _ of a union of J rare events. The method uses n samples, each of which picks one of the rare events at random, samples conditionally on that rare event happening and counts the total number of rare events that happen. It was used by Naiman and Priebe for scan
statistics, Shi, Siegmund and Yakir for genomic scans and Adler, Blanchet and Liu for extrema of Gaussian processes. We call it ALOE, for `at least one event'. The ALOE estimate is unbiased and we find that it has a coefficient of variation no larger than p (J + J�1 � 2)=(4n). The coefficient of variation is also no larger than p (__=_ � 1)=n where __ is the union bound. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the J rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by 5772
constraints in 326 dimensions, with probability below 10�22, are estimated with a coefficient of variation of about 0:0024 with only n = 10;000 sample values. In a genomic context, the rare events become false discoveries. There we are interested in the possibility of a large number of simultaneous events, not just one or more. Some work with Kenneth Tay will be presented on that problem.
Joint with Yury Maximov and Michael Chertkov Los Alamos National Laboratory and Kenneth Tay, Stanford
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.
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.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
In this talk we present a framework for splitting data assimilation problems based upon the model dynamics. This is motivated by assimilation in the unstable subspace (AUS) and center manifold and inertial manifold techniques in dynamical systems. Recent efforts based upon the development of particle filters projected into the unstable subspace will be highlighted.
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.
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
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.
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.
Mean Absolute Percentage Error for regression models, presentation of the paper published in Neurocomputing, 2016.
http://www.sciencedirect.com/science/article/pii/S0925231216003325
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.
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.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
In this talk we present a framework for splitting data assimilation problems based upon the model dynamics. This is motivated by assimilation in the unstable subspace (AUS) and center manifold and inertial manifold techniques in dynamical systems. Recent efforts based upon the development of particle filters projected into the unstable subspace will be highlighted.
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.
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
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.
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.
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Similar to QMC: Transition Workshop - Importance Sampling the Union of Rare Events with an Application to Power Systems Analysis - Art Owen, May 9, 2018
Mean Absolute Percentage Error for regression models, presentation of the paper published in Neurocomputing, 2016.
http://www.sciencedirect.com/science/article/pii/S0925231216003325
Learning to discover monte carlo algorithm on spin ice manifoldKai-Wen Zhao
The global update Monte Carlo sampler can be discovered naturally by trained machine using policy gradient method on topologically constrained environment.
Distributed solution of stochastic optimal control problem on GPUsPantelis Sopasakis
Stochastic optimal control problems arise in many
applications and are, in principle,
large-scale involving up to millions of decision variables. Their
applicability in control applications is often limited by the
availability of algorithms that can solve them efficiently and within
the sampling time of the controlled system.
In this paper we propose a dual accelerated proximal
gradient algorithm which is amenable to parallelization and
demonstrate that its GPU implementation affords high speed-up
values (with respect to a CPU implementation) and greatly outperforms
well-established commercial optimizers such as Gurobi.
We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
To approximate the high-dimensional integral we use quasi-Monte Carlo (QMC) methods. These are deterministic or quasi-random quadrature rules that can be proven to be very efficient for the numerical integration of certain classes of high-dimensional functions. QMC methods have previously been applied to linear functionals of the solution of a similar elliptic source problem; however, because of the nonlinearity of eigenvalues the existing analysis of the integration error
does not hold in our case.
We show that the minimal eigenvalue belongs to the spaces required for QMC theory, outline the approximation algorithm and provide numerical results.
The main machine learning algorithms are built upon various mathematical foundations such as statistics, optimization, and probability. Will this also hold true for Artificial Intelligence? In this presentation, I will showcase some recent examples of interactions between machine learning and mathematics.
Colloquium @ CEREMADE (October 3, 2023)
Typically quantifying uncertainty requires many evaluations of a computational model or simulator. If a simulator is computationally expensive and/or high-dimensional, working directly with a simulator often proves intractable. Surrogates of expensive simulators are popular and powerful tools for overcoming these challenges. I will give an overview of surrogate approaches from an applied math perspective and from a statistics perspective with the goal of setting the stage for the "other" community.
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)
Accelerating Pseudo-Marginal MCMC using Gaussian ProcessesMatt Moores
The grouped independence Metropolis-Hastings (GIMH) and Markov chain within Metropolis (MCWM) algorithms are pseudo-marginal methods used to perform Bayesian inference in latent variable models. These methods replace intractable likelihood calculations with unbiased estimates within Markov chain Monte Carlo algorithms. The GIMH method has the posterior of interest as its limiting distribution, but suffers from poor mixing if it is too computationally intensive to obtain high-precision likelihood estimates. The MCWM algorithm has better mixing properties, but less theoretical support. In this paper we accelerate the GIMH method by using a Gaussian process (GP) approximation to the log-likelihood and train this GP using a short pilot run of the MCWM algorithm. Our new method, GP-GIMH, is illustrated on simulated data from a stochastic volatility and a gene network model. Our approach produces reasonable estimates of the univariate and bivariate posterior distributions, and the posterior correlation matrix in these examples with at least an order of magnitude improvement in computing time.
We apply tensor train (TT) data format to solve an elliptic PDE with uncertain coefficients. We reduce complexity and storage from exponential to linear. Post-processing in TT format is also provided.
We are interested in finding a permutation of the entries of a given square matrix so that the maximum number of its nonzero entries are moved to one of the corners in a L-shaped fashion.
If we interpret the nonzero entries of the matrix as the edges of a graph, this problem boils down to the so-called core–periphery structure, consisting of two sets: the core, a set of nodes that is highly connected across the whole graph, and the periphery, a set of nodes that is well connected only to the nodes that are in the core.
Matrix reordering problems have applications in sparse factorizations and preconditioning, while revealing core–periphery structures in networks has applications in economic, social and communication networks.
Adaptive Restore algorithm & importance Monte CarloChristian Robert
Talk given at the SMC 2024 workshop, ICMS, Edinburgh, Scotland
Similar to QMC: Transition Workshop - Importance Sampling the Union of Rare Events with an Application to Power Systems Analysis - Art Owen, May 9, 2018 (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)
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Digital Artifact 2 - Investigating Pavilion Designs
QMC: Transition Workshop - Importance Sampling the Union of Rare Events with an Application to Power Systems Analysis - Art Owen, May 9, 2018
1. SAMSI QMC Transition workshop 1
SAMSI QMC transition
workshop
Art Owen
Stanford University
May 9, 2018
2. SAMSI QMC Transition workshop 2
Importance sampling the union
of rare events, with bounded
error and an application to
power systems analysis and
another application to genomic
testing, plus something about
the weight working group
A. B. Owen + Yury Maximov + Michael Chertkov with Kenneth J. Tay and Yue Hui
May 9, 2018
3. SAMSI QMC Transition workshop 3
Acknowledgments
1) Mac Hyman, Ilse Ipsen & SAMSI staff
2) Fred, Frances, Pierre
May 9, 2018
4. SAMSI QMC Transition workshop 4
Weight working group
• We want µ = f(x) dx
• We use ˆµ = (1/n)
n
i=1 f(xi)
The RKHS theory is strong for f in a weighted Sobolev space
Which weighted space?
For most applications, f belongs to
• All of the RHKS, or
• none of them.
We had in mind a pipeline
f
1
→ weights
2
→ polynomial lattice rule
3
→ ˆµ
We got stuck on 2. Note from after the talk: chatting with Pierre L’Ecuyer it is clear
that the sticking point is more like step 1.5, choosing a figure of merit.
Lattice builder + SSJ: Godin, Jemel, L’Ecuyer, Marion, Munger
May 9, 2018
5. SAMSI QMC Transition workshop 5
Interleaving
This is a strategy by Dick & Baldeaux to build a new QMC input by smushing
together digits from two old ones.
You need a net in [0, 1]2s
to get points in [0, 1]s
.
Should we smush them all, or just some?
Yue Hui has looked into strategies for interleaving some but not all inputs to QMC
Preliminary results:
• small differences from which inputs get interleaved
• enormous gains for a variable with a large additive component
• lesser gains from variables that interact
Tentative finding: Variables with large closed Sobol’ index but not so large total
Sobol’ index are promising.
May 9, 2018
6. SAMSI QMC Transition workshop 6
Rare event sampling
Motivation: an electrical grid has N nodes. Power p1, p2, · · · , pN
• Random pi > 0, e.g., wind generation,
• Random pi < 0, e.g., consumption,
• Fixed pi for controllable nodes,
AC phase angles θi
• (θ1, . . . , θN ) = F(p1, . . . , pN )
• Constraints: |θi − θj| < ¯θ if i ∼ j
• Find P maxi∼j |θi − θj| ¯θ
Simplified model
• (p1, . . . , pN ) Gaussian
• θ linear in p1, . . . , pN
May 9, 2018
7. SAMSI QMC Transition workshop 7
Gaussian setup
For x ∼ N(0, Id), Hj = {x | ωT
j x τj} find µ = P(x ∈ ∪J
j=1Hj).
WLOG ωj = 1. Ordinarily τj > 0. d hundreds. J thousands.
−10 −5 0 5 10
−6−4−20246
c(−7,6)
q
q
q
q
q
q
Solid: deciles of x . Dashed: 10−3
· · · 10−7
. May 9, 2018
8. SAMSI QMC Transition workshop 8
Basic bounds
Let Pj ≡ P(ωT
j x τj) = Φ(−τj)
then µ ¯µ ≡
J
j=1
Pj (union bound)
and µ µ ≡ max
1 j J
Pj
Inclusion-exclusion
For u ⊆ 1:J = {1, 2, . . . , J}, let
Hu = ∪j∈uHj Hu(x) ≡ 1{x ∈ Hu} Hc
u(x) ≡ 1 − Hu(x)
Pu = P(Hu) = E(Hu(x))
µ =
|u|>0
(−1)|u|−1
Pu
May 9, 2018
9. SAMSI QMC Transition workshop 9
Importance sampling
For x ∼ p, seek η = Ep(f(x)) = f(x)p(x) dx. Take
ˆη =
1
n
n
i=1
f(xi)p(xi)
q(xi)
, xi
iid
∼ q
Unbiased if q(x) > 0 whenever f(x)p(x) = 0.
Variance
Var(ˆη) = σ2
q /n, where
σ2
q =
f2
p2
q
− µ2
= · · · =
(fp − µq)2
q
Num: seek q ≈ fp/µ, i.e., nearly proportional to fp
Den: watch out for small q.
May 9, 2018
10. SAMSI QMC Transition workshop 10
Mixture sampling
For αj 0, j αj = 1
ˆηα =
1
n
n
i=1
f(xi)p(xi)
j αjqj(xi)
, xi
iid
∼ qα ≡
j
αjqj
Defensive mixtures
Take q0 = p, α0 > 0. Get p/qα 1/α0. Hesterberg (1995)
Additional refs
Use qj = 1 as control variates. O & Zhou (2000).
Optimization over α is convex. He & O (2014).
Multiple IS. Veach & Guibas (1994). Veach’s Oscar.
Elvira, Martino, Luengo, Bugallo (2015) Generalizations.
May 9, 2018
12. SAMSI QMC Transition workshop 12
Mixture of conditional sampling
α1, . . . , αJ 0,
j
αj = 1, qα =
j
αjqj
µ = P(x ∈ ∪J
j=1Hj) = P(x ∈ H1:J ) = E H1:J (x)
Mixture IS
ˆµα =
1
n
n
i=1
H1:J (xi)p(xi)
J
j=1 αjqj(xi)
(where qj = pHj/Pj)
=
1
n
n
i=1
H1:J (xi)
J
j=1 αjHj(xi)/Pj
with H0 ≡ 1 and P0 = 1.
It would be nice to have αj/Pj constant.
May 9, 2018
13. SAMSI QMC Transition workshop 13
ALOE
At Least One (Rare) Event
Like AMIS of Cornuet, Marin, Mira, Robert (2012)
Take αj = α∗
j ∝ Pj. That is
α∗
j =
Pj
J
j =1 Pj
=
Pj
¯µ
, ( ¯µ is the union bound )
Then
ˆµα∗ =
1
n
n
i=1
H1:J (xi)
J
j=1 αjHj(xi)/Pj
= ¯µ ×
1
n
n
i=1
H1:J (xi)
J
j=1 Hj(xi)
If x ∼ qα∗ then H1:J (x) = 1. So
ˆµα∗ = ¯µ ×
1
n
n
i=1
1
S(xi)
, S(x) ≡
J
j=1
Hj(x)
May 9, 2018
14. SAMSI QMC Transition workshop 14
Some prior uses
Frigessi & Vercelis (1985)
Combinatorial enumeration.
Naiman & Priebe (2001)
Scan statistics, genetics and imaging.
Naiman, Priebe & Cope (2001)
Scan statistics, marked point processes (e.g., mine fields)
Shi, Siegmund, Yakir (2007)
Scan statistics, linkage analysis
Adler, Blanchet, Liu (2012)
Excursions of random fields over T ⊂ Rd
Nobody seems to have named it.
Thanks to: Bert Zwart, Jose Blanchet, David Siegmund for literature pointers
May 9, 2018
15. SAMSI QMC Transition workshop 15
Theorem
O, Maximov & Chertkov (2017)
Let H1, . . . , HJ be events defined by x ∼ p.
Let qj(x) = p(x)Hj(x)/Pj for Pj = P(x ∈ Hj).
Let xi
iid
∼ qα∗ =
J
j=1 α∗
j qj for α∗
j = Pj/¯µ.
Take
ˆµ = ¯µ ×
1
n
n
i=1
1
S(xi)
, S(xi) =
J
j=1
Hj(xi)
Then E(ˆµ) = µ and
Var(ˆµ) =
1
n
¯µ
J
s=1
Ts
s
− µ2 µ(¯µ − µ)
n
where Ts ≡ P(S(x) = s), x ∼ p.
The RHS follows because
J
s=1
Ts
s
J
s=1
Ts = µ.
May 9, 2018
16. SAMSI QMC Transition workshop 16
Remarks
With S(x) equal to the number of rare events at x,
ˆµ = ¯µ ×
1
n
n
i=1
1
S(xi)
1 S(x) J =⇒
¯µ
J
ˆµ ¯µ
Robustness
The usual problem with rare event estimation is getting no rare events in n tries.
Then ˆµ = 0.
Here the corresponding failure is never seeing S 2 rare events.
Then ˆµ = ¯µ, an upper bound and probably fairly accurate if all S(xi) = 1
Conditioning vs sampling from N(µj, I)
Avoids wasting samples outside the failure zone.
Avoids awkward likelihood ratios.
May 9, 2018
17. SAMSI QMC Transition workshop 17
General Gaussians
For y ∼ N(η, Σ) let the event be γT
j y κj.
Same as xT
ωj τj for x dim N(0, I) and
ωj =
γT
j Σ1/2
γT
j Σγj
and τj =
κj − γT
j η
γT
j Σγj
Optimizations
Later we will want to vary η.
That changes τj but not ωj.
May 9, 2018
18. SAMSI QMC Transition workshop 18
More bounds
For event count S(x) =
J
j=1 Hj(x) and x ∼ N(0, I):
E(S) =
J
j=1
Pj = ¯µ
P(S > 0) = E max
1 j J
Hj(x) = µ
E(S−1
| S > 0) =
J
s=1
P(S = s)
s
P(S > 0) =
J
s=1
P(S = s)
s
µ
Therefore
n × Var(ˆµ) = ¯µ
J
s=1
P(S = s)
s
− µ2
= ¯µ × µ × E(S−1
| S > 0) − µ2
= µ2
× E(S | S > 0) × E(S−1
| S > 0) − µ2
May 9, 2018
19. SAMSI QMC Transition workshop 19
Bounds continued
Var(ˆµ)
µ2
1
n
E(S | S > 0) × E(S−1
| S > 0) − 1
Lemma
Let S be a random variable in {1, 2, . . . , J} for J ∈ N. Then
E(S)E(S−1
)
J + J−1
+ 2
4
by Cauchy-Schwarz, with equality if and only if S ∼ U{1, J}.
Corollary
Var(ˆµ)
µ2
n
J + J−1
− 2
4
.
For J > 2 there must be a sharper bound for half-spaces and a Gaussian.
Thanks to Yanbo Tang and Jeffrey Negrea for an improved proof of the lemma.
May 9, 2018
20. SAMSI QMC Transition workshop 20
Numerical comparison
R function mvtnorm of Genz & Bretz (2009) gets
P a y b = P ∩j{aj yj bj} for y ∼ N(η, Σ)
Their code makes sophisticated use of quasi-Monte Carlo.
Adaptive, up to 25,000 evals in FORTRAN.
It was not designed for rare events.
It computes an intersection.
Usage
Pack ωj into Ω and τj into T
1 − µ = P(ΩT
x T ) = P(y T ), y ∼ N(0, ΩT
Ω)
It can handle up to 1000 inputs, i.e., J 1000
Upshot
ALOE works (much) better for rare events.
mvtnorm works better for non-rare events.
May 9, 2018
21. SAMSI QMC Transition workshop 21
Circumscribed polygon
P(J, τ) regular J sided polygon in R2
outside circle of radius τ > 0.
q
a priori bounds
µ = P(x ∈ Pc
) P(χ2
(2) τ2
) = exp(−τ2
/2)
This is pretty tight. A trigonometric argument gives
1
µ
exp(−τ2/2)
1 −
J tan π
J − π τ2
2π
.
= 1 −
π2
τ2
6J2
Let’s use J = 360
So µ
.
= exp(−τ2
/2) for reasonable τ.
May 9, 2018
22. SAMSI QMC Transition workshop 22
IS vs MVN
ALOE had n = 1000. MVN had up to 25,000. 100 random repeats.
τ µ E((ˆµALOE/µ − 1)2
) E((ˆµMVN/µ − 1)2
)
2 1.35×10−01
0.000399 9.42×10−08
3 1.11×10−02
0.000451 9.24×10−07
4 3.35×10−04
0.000549 2.37×10−02
5 3.73×10−06
0.000600 1.81×10+00
6 1.52×10−08
0.000543 4.39×10−01
7 2.29×10−11
0.000559 3.62×10−01
8 1.27×10−14
0.000540 1.34×10−01
For τ = 5 MVN had a few outliers.
May 9, 2018
23. SAMSI QMC Transition workshop 23
Polygon again
ALOE
Relative estimate
Frequency
0.96 1.00 1.04
0246810 Pmvnorm
Relative estimate
Frequency
1 2 3 4 5 6
0102030405060
Results of 100 estimates of the P(x ∈ P(360, 6)), divided by exp(−62
/2).
Left panel: ALOE. Right panel: pmvnorm.
May 9, 2018
24. SAMSI QMC Transition workshop 24
More examples
Similar results for high dimensions.
ALOE very effective on rare events.
MVN can go > 1000× the union bound.
May 9, 2018
25. SAMSI QMC Transition workshop 25
Power systems
Network with N nodes called busses and M edges.
Typically M/N is not very large.
Power pi at bus i. Each bus is Fixed or Random or Slack:
slack bus S has pS = − i=S pi.
p = (pT
F , pT
R, pS)T
∈ RN
Randomness driven by pR ∼ N(ηR, ΣRR)
Inductances Bij
A Laplacian Bij = 0 if i ∼ j in the network. Bii = − j=i Bij.
Taylor approximation
θ = B+
p (pseudo inverse)
Therefore θ is Gaussian as are all θi − θj for i ∼ j.
May 9, 2018
26. SAMSI QMC Transition workshop 26
Examples
Our examples come from MATPOWER (a Matlab toolbox).
Zimmernan, Murillo-S´anchez & Thomas (2011)
We used n = 10,000.
Polish winter peak grid
2383 busses, d = 326 random busses, J = 5772 phase constraints.
¯ω ˆµ se/ˆµ µ ¯µ
π/4 3.7 × 10−23
0.0024 3.6 × 10−23
4.2 × 10−23
π/5 2.6 × 10−12
0.0022 2.6 × 10−12
2.9 × 10−12
π/6 3.9 × 10−07
0.0024 3.9 × 10−07
4.4 × 10−07
π/7 2.0 × 10−03
0.0027 2.0 × 10−03
2.4 × 10−03
¯ω is the phase constraint, ˆµ is the ALOE estimate, se is the estimated standard
error, µ is the largest single event probability and ¯µ is the union bound. May 9, 2018
27. SAMSI QMC Transition workshop 27
Pegase 2869
Fliscounakis et al. (2013) “large part of the European system”
N = 2869 busses. d = 509 random busses. J = 7936 phase constraints.
Results
¯ω ˆµ se/ˆµ µ ¯µ
π/2 3.5 × 10−20
0∗
3.3 × 10−20
3.5 × 10−20
π/3 8.9 × 10−10
5.0 × 10−5
7.7 × 10−10
8.9 × 10−10
π/4 4.3 × 10−06
1.8 × 10−3
3.5 × 10−06
4.6 × 10−06
π/5 2.9 × 10−03
3.5 × 10−3
1.8 × 10−03
4.1 × 10−03
Notes
¯θ = π/2 is unrealistically large. We got ˆµ = ¯µ. All 10,000 samples had S = 1.
Some sort of Wilson interval or Bayesian approach could help.
One half space was sampled 9408 times, another 592 times. May 9, 2018
28. SAMSI QMC Transition workshop 28
Other models
IEEE case 14 and IEEE case 300 and Pegase 1354 were all dominated by one
failure mode so µ
.
= ¯µ and no sampling is needed.
Another model had random power corresponding to wind generators but phase
failures were not rare events in that model.
The Pegase 13659 model was too large for our computer. The Laplacian had
37,250 rows and columns.
The Pegase 9241 model was large and slow and phase failure was not a rare
event.
Caveats
We used a DC approximation to AC power flow (which is common) and the phase
estimates were based on a Taylor approximation.
May 9, 2018
29. SAMSI QMC Transition workshop 29
Genomics (in progress)
Similar problem comes up with false discoveries. Sketch
Measure G genes on n people.
Expression X = (X1 X2 · · · XG) ∈ Rn×G
And a phenotype: Y ∈ Rn
Exploration
Which genes (if any) correlate with Y ?
What could go wrong’?
Try G = 10,000 tests at α = 5% significance.
Expect at least 500 discoveries by chance.
Bonferroni
So · · · test at level α/G, e.g., 5 × 10−6
.
Seems conservative
May 9, 2018
30. SAMSI QMC Transition workshop 30
Multiple testing
FD = # False discoveries
TD = # True discoveries
False discovery proportion and rate
FDP =
FD
FD+TD
, denom not 0
0, else
FDR = E(FDP)
If Y Gaussian and independent of X,
then |XT
g Y | > τ brings a false discovery.
Benjamini-Hochberg
Controls FDR at e.g., 10% for independent tests
(and some dependent cases)
Correlations among genes =⇒ False discoveries come in bursts.
May 9, 2018
31. SAMSI QMC Transition workshop 31
Testing for association
Center and scale columns Xg.
Gene g is significantly associated with Y if
XT
g Y τ or XT
g Y −τ
For Gaussian Y we have similar geometry to the power grid problem
Using ALOE
We get P(FD k) X held fixed Y random
Power systems case was k = 1.
May 9, 2018
32. SAMSI QMC Transition workshop 32
Bursts from Benjamini-Hochberg
We will use actual X, (Duchenne Muscular dystrophy data)
Gaussian noise Y
Thanks to Jingshu Wang
Target false discovery rate 10%.
log2(1 + #FD) 0 1 2 3 4 6 7 8 9 10 11
3,756 genes, 30 subj 909 50 18 7 3 3 1 4 2 1 2
3,756 Indep. p-values 894 93 13
12,625 genes, 23 subj 929 30 19 6 3 1 1 3 1 5 2
12,625 Indep. p-values 902 85 12 1
Table 1: In 1000 simulations of BH with Y ∼ N(0, In), and real data X or indep
p-values.
May 9, 2018
34. SAMSI QMC Transition workshop 34
Duchenne data p = 0.005
3756 genes
Black: ALOE estimates (5 times)
Red: binomial reference
Source: Kenneth Tay May 9, 2018
35. SAMSI QMC Transition workshop 35
Duchenne data p = 5 × 10−5
3756 genes
Black: ALOE estimates
Red: binomial reference
Source: Kenneth Tay
May 9, 2018
36. SAMSI QMC Transition workshop 36
Complete null
In the complete null, no genes associate with Y .
What if some do?
Does that raise or lower the number of false discoveries?
May 9, 2018
37. SAMSI QMC Transition workshop 37
Null domination
P FD k | TD = 0 P FD k | TD > 0
Dudoit and van der Laan show t-tests have asymptotic null domination
K. Tay simulations show: not even close for small n and big G.
May 9, 2018
38. SAMSI QMC Transition workshop 38
Generalization I
Replace half spaces by convex sets
−10 −5 0 5 10
−6−4−20246
c(−7,6)
q
q
q
Algorithm will go through
Variance bounds too
Coefficient of variation: ok unless P(blue) P(red)
May 9, 2018
39. SAMSI QMC Transition workshop 39
Generalization II
Replace Gaussian by log concave density
−10 −5 0 5 10
−6−4−20246
c(−7,6)
q
q
q
We can find the instantons
We may have to do location shifts (instead of conditional sampling) May 9, 2018
40. SAMSI QMC Transition workshop 40
Generalization III
Sharper power systems models are nonlinear. Let
Hj(x) ≡ {x | gj(x) τj}
• Boundary not known a priori
• Sampling xi reveals gj(xi)
• Kriging for gj.
May 9, 2018
41. SAMSI QMC Transition workshop 41
Generalization IV
To minimize cost subject to P(Hu) ε we want to estimate
∂E(Hu)
∂τj
After some algebra
∂µ
∂τj
= −ϕ(τj) 1 − E(H−j(x) | ωT
j x = τj)
= −ϕ(τj)E(Hc
−j(x) | ωT
j x = τj)
= −ϕ(τj)P(No other events | Hj barely happened)
The goal is to get noisy estimates from the same data used to estimate µ.
Notes
1) For large τj most ωT
j x τj satisfy ωT
j x
.
= τj
2) Adapt the sampling to ensure that several Hj happen even if τj is large.
May 9, 2018
42. SAMSI QMC Transition workshop 42
Thanks
• Weight working group + Yue Hui
• Michael Chertkov and Yury Maximov, co-authors
• Center for Nonlinear Studies at LANL, for hospitality
• Kenneth Tay for genomic examples
• Jingshu Wang for Duchenne muscular dystrophy data
• Alan Genz for discussions on mvtnorm
• Bert Zwart, Jose Blanchet, David Siegmund for references
• Yanbo Tang and Jeffrey Negrea, improved Lemma proof
• NSF DMS-1407397 & DMS-1521145
• DOE/GMLC 2.0 Project: Emergency monitoring and controls through new
technologies and analytics
• SAMSI, Ilse Ipsen, David Banks
• Sue, Kerem, Thomas, Rita, Rebecca, Rick May 9, 2018