Efficient Monte Carlo Estimators Research Articles

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy.

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.

Network dependence in multi-indexed data on international trade flows

Faced with the problem that conventional multidimensional fixed effects models only focus on unobserved heterogeneity, but ignore any potential cross-sectional dependence due to network interactions, we introduce a model of trade flows between countries over time that allows for network dependence in flows, based on sociocultural connectivity structures. We show that conventional multidimensional fixed effects model specifications exhibit cross-sectional dependence between countries that should be modeled to avoid simultaneity bias. Given that the source of network interaction is unknown, we propose a panel gravity model that examines multiple network interaction structures, using Bayesian model probabilities to determine those most consistent with the sample data. This is accomplished with the use of computationally efficient Markov Chain Monte Carlo estimation methods that produce a Monte Carlo integration estimate of the log-marginal likelihood that can be used for model comparison. Application of the model to a panel of trade flows points to network spillover effects, suggesting the presence of network dependence and biased estimates from conventional trade flow specifications. The most important sources of network dependence were found to be membership in trade organizations, historical colonial ties, common currency, and spatial proximity of countries.

Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to \textit{catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $\beta=1$ for the standard case (without IS), to $\beta=1+\delta$, where $0<\delta<1$ is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2}\right)$, which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

Drift-preserving numerical integrators for stochastic Hamiltonian systems

The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

Multilevel Monte Carlo Estimation of the Expected Value of Sample Information

We study Monte Carlo estimation of the expected value of sample information (EVSI) which measures the expected benefit of gaining additional information for decision making under uncertainty. EVSI is defined as a nested expectation in which an outer expectation is taken with respect to one random variable $Y$ and an inner conditional expectation with respect to the other random variable $\theta$. Although the nested (Markov chain) Monte Carlo estimator has been often used in this context, a root-mean-square accuracy of $\varepsilon$ is achieved notoriously at a cost of $O(\varepsilon^{-2-1/\alpha})$, where $\alpha$ denotes the order of convergence of the bias and is typically between $1/2$ and $1$. In this article we propose a novel efficient Monte Carlo estimator of EVSI by applying a multilevel Monte Carlo (MLMC) method. Instead of fixing the number of inner samples for $\theta$ as done in the nested Monte Carlo estimator, we consider a geometric progression on the number of inner samples, which yields a hierarchy of estimators on the inner conditional expectation with increasing approximation levels. Based on an elementary telescoping sum, our MLMC estimator is given by a sum of the Monte Carlo estimates of the differences between successive approximation levels on the inner conditional expectation. We show, under a set of assumptions on decision and information models, that successive approximation levels are tightly coupled, which directly proves that our MLMC estimator improves the necessary computational cost to optimal $O(\varepsilon^{-2})$. Numerical experiments confirm the considerable computational savings as compared to the nested Monte Carlo estimator.

Evaluating two-electron-repulsion integrals over arbitrary orbitals using zero variance Monte Carlo: Application to full configuration interaction calculations with Slater-type orbitals

A Monte Carlo method for evaluating multicenter two-electron-repulsion integrals over any types of orbitals (Slater, Sturmian, finite-range, numerical, etc.) is presented. The approach is based on a simple and universal (orbital-independent) gaussian sampling of the two-electron configuration space and on the use of efficient zero-variance Monte Carlo estimators. Quite remarkably, it is shown that the high level of accuracy required on two-electron integrals to make Hartree-Fock (HF) and configuration interaction (CI) calculations feasible can be achieved. A first zero-variance estimator is built by introducing a gaussian approximation of the orbitals and by evaluating the two-electron integrals using a correlated sampling scheme for the difference between exact and approximate orbitals. A second one is based on the introduction of a general coordinate transformation. The price to pay for this simple and general Monte Carlo scheme is the high computational cost required. However, we argue that the great simplicity of the algorithm, its embarrassingly parallel nature, its ideal adaptation to modern computational platforms and, most importantly, the possibility of using more compact and physically meaningful basis sets nevertheless make the method attractive. HF and near full CI (FCI) calculations using Slater-type orbitals (STOs) are reported for Be, CH4, and [H2N(CH)NH2]+ (a simple model of cyanine). To the best of our knowledge, our largest FCI calculation involving 18 active electrons distributed among 90 orbitals for the cyanine molecule is the most extensive molecular calculation performed so far using pure STOs (no gaussian approximation, even for the challenging four-center two-electron integrals).

Scalable Virtual Ray Lights Rendering for Participating Media

AbstractVirtual ray lights (VRL) are a powerful representation for multiple‐scattered light transport in volumetric participating media. While efficient Monte Carlo estimators can importance sample the contribution of a VRL along an entire sensor subpath, render time still scales linearly in the number of VRLs. We present a new scalable hierarchial VRL method that preferentially samples VRLs according to their image contribution. Similar to Lightcuts‐based approaches, we derive a tight upper bound on the potential contribution of a VRL that is efficient to compute. Our bound takes into account the sampling probability densities used when estimating VRL contribution. Ours is the first such upper bound formulation, leading to an efficient and scalable rendering technique with only a few intuitive user parameters. We benchmark our approach in scenes with many VRLs, demonstrating improved scalability compared to existing state‐of‐the‐art techniques.

Efficient MCMC estimation of inflated beta regression models

This paper introduces a new and computationally efficient Markov chain Monte Carlo (MCMC) estimation algorithm for the Bayesian analysis of zero, one, and zero and one inflated beta regression models. The algorithm is computationally efficient in the sense that it has low MCMC autocorrelations and computational time. A simulation study shows that the proposed algorithm outperforms the slice sampling and random walk Metropolis–Hastings algorithms in both small and large sample settings. An empirical illustration on a loss given default banking model demonstrates the usefulness of the proposed algorithm.

Likelihood Ratio Method and Algorithmic Differentiation: Fast Second Order Greeks

We show how Adjoint Algorithmic Differentiation can be combined with the so-called Pathwise Derivative and Likelihood Ratio Method to construct efficient Monte Carlo estimators of second order price sensitivities of derivative portfolios. We demonstrate with a numerical example how the proposed technique can be straightforwardly implemented to greatly reduce the computation time of second order risk.

Multilevel Monte Carlo Quadrature of Discontinuous Payoffs in the Generalized Heston Model Using Malliavin Integration by Parts

In this article, we establish an integration by parts formula for the quadrature of discontinuous payoffs in a multidimensional Heston model. For its derivation we use Malliavin calculus techniques and work under mild integrability conditions on the payoff and under the assumption of a strictly positive volatility. The integration by parts procedure smoothes the original functional, and thus our formula in combination with a payoff splitting allows us to construct efficient multilevel Monte Carlo estimators. This is confirmed by our numerical analysis and is illustrated by several numerical examples.

Likelihood Ratio Method and Algorithmic Differentiation: Fast Second Order Greeks

We show how Adjoint Algorithmic Differentiation can be combined with the so-called Pathwise Derivative and Likelihood Ratio Method to construct efficient Monte Carlo estimators of second order price sensitivities of derivative portfolios. We demonstrate with a numerical example how the proposed technique can be straightforwardly implemented to greatly reduce the computation time of second order risk.

Efficient Monte Carlo characterization of quantum operations for qudits

For qubits, Monte Carlo estimation of the average fidelity of Clifford unitaries is efficient -- it requires a number of experiments that is independent of the number $n$ of qubits and classical computational resources that scale only polynomially in $n$. Here, we identify the requirements for efficient Monte Carlo estimation and the corresponding properties of the measurement operator basis when replacing two-level qubits by $p$-level qudits. Our analysis illuminates the intimate connection between mutually unbiased measurements and the existence of unitaries that can be characterized efficiently. It allows us to propose a 'hierarchy' of generalizations of the standard Pauli basis from qubits to qudits according to the associated scaling of resources required in Monte Carlo estimation of the average fidelity.

Efficient Monte Carlo counterparty credit risk pricing and measurement

Counterparty credit risk (CCR), a key driver of the 2007-8 credit crisis, has become one of the main focuses of major global and US regulatory standards. Financial institutions invest large amounts of resources employing Monte Carlo simulation to measure and price their counterparty credit risk. We develop efficient Monte Carlo CCR estimation frameworks by focusing on the most widely used and regulatory-driven CCR measures: expected positive exposure, credit value adjustment, and effective expected positive exposure. Our numerical examples illustrate that our proposed efficient Monte Carlo estimators outperform the existing crude estimators of these CCR measures substantially in terms of mean square error (MSE).We also demonstrate that the two widely used sampling methods, the so-called path dependent simulation and direct jump to simulation date, are not equivalent in that they lead to Monte Carlo CCR estimators which are drastically different in terms of their MSE.

On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

In this paper, we consider the extreme behavior of a Gaussian random field $f(t)$ living on a compact set $T$. In particular, we are interested in tail events associated with the integral $\int_{T}e^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_{T}e^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_{T}e^{f(t)}\,dt$ is asymptotically equivalent to the tail event of $\sup_{T}\gamma(t)$ where $\gamma(t)$ is a Gaussian process and it is an affine function of $f(t)$ and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of $\log b$ to compute the probability $P(\int_{T}e^{f(t)}\,dt>b)$ with a prescribed relative accuracy.

Implicit Estimation for the Stochastic Volatility Model

Efficient, accurate, and fast Markov Chain Monte Carlo estimation methods based on the Implicit approach are proposed. In this article, we introduced the notion of Implicit method for the estimation of parameters in Stochastic Volatility models. Implicit estimation offers a substantial computational advantage for learning from observations without prior knowledge and thus provides a good alternative to classical inference in Bayesian method when priors are missing. Both Implicit and Bayesian approach are illustrated using simulated data and are applied to analyze daily stock returns data on CAC40 index.

Efficient Monte Carlo Counterparty Credit Risk Pricing and Measurement

Counterparty credit risk (CCR), a key driver of the 2007-08 credit crisis, has become one of the main focuses of the major global and U.S. regulatory standards. Financial institutions invest large amounts of resources employing Monte Carlo simulation to measure and price their counterparty credit risk. We develop efficient Monte Carlo CCR estimation frameworks by focusing on the most widely used and regulatory-driven CCR measures: expected positive exposure (EPE), credit value adjustment (CVA), and effective expected positive exposure (EEPE). Our numerical examples illustrate that our proposed efficient Monte Carlo estimators outperform the existing crude estimators of these CCR measures substantially in terms of mean square error (MSE). We also demonstrate that the two widely used sampling methods, the so-called Path Dependent Simulation (PDS) and Direct Jump to Simulation date (DJS), are not equivalent in that they lead to Monte Carlo CCR estimators which are drastically different in terms of their MSE.

Effi cient Monte Carlo Counterparty Credit Risk Pricing and Measurement

Counterparty credit risk (CCR), a key driver of the 2007-08 credit crisis, has become one of the main focuses of the major global and U.S. regulatory standards. Financial institutions invest large amounts of resources employing Monte Carlo simulation to measure and price their counterparty credit risk. We develop efficient Monte Carlo CCR estimation frameworks by focusing on the most widely used and regulatory-driven CCR measures: expected positive exposure (EPE), credit value adjustment (CVA), and effective expected positive exposure (EEPE). Our numerical examples illustrate that our proposed efficient Monte Carlo estimators outperform the existing crude estimators of these CCR measures substantially in terms of mean square error (MSE). We also demonstrate that the two widely used sampling methods, the so-called Path Dependent Simulation (PDS) and Direct Jump to Simulation date (DJS), are not equivalent in that they lead to Monte Carlo CCR estimators which are drastically different in terms of their MSE.

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

A sequential Monte Carlo approach to computing tail probabilities in stochastic models

Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential Monte Carlo estimators, we show how resampling weights can be chosen to yield logarithmically efficient Monte Carlo estimates of large deviation probabilities for multidimensional Markov random walks.