We study the computational complexity and variance of multilevel best linear unbiased estimators introduced in [D. Schaden and E. Ullmann, SIAM/ASA J. Uncert. Quantif., (2020)]. We specialize the results in this work to PDE-based models that are parameterized by a discretization quantity, e.g., the finite element mesh size. In particular, we investigate the asymptotic complexity of the so-called sample allocation optimal best linear unbiased estimators (SAOBs). These estimators have the smallest variance given a fixed computational budget. However, SAOBs are defined implicitly by solving an optimization problem and are difficult to analyze. Alternatively, we study a class of auxiliary estimators based on the Richardson extrapolation of the parametric model family. This allows us to provide an upper bound for the complexity of the SAOBs, showing that their complexity is optimal within a certain class of linear unbiased estimators. Moreover, the complexity of the SAOBs is not larger than the complexity of Multilevel Monte Carlo. The theoretical results are illustrated by numerical experiments with an elliptic PDE.
We present general principles for the design and analysis of unbiased Monte Carlo estimators in a wide range of settings. Our estimators posses finite work-normalized variance under mild regularity conditions. We apply our estimators to various settings of interest, including unbiased optimization in Sample Average Approximations, unbiased steady-state simulation of regenerative processes, quantile estimation and nested simulation problems.
Constructing unbiased estimators from Markov chain Monte Carlo (MCMC) outputs is a difficult problem that has recently received a lot of attention in the statistics and machine learning communities. However, the current unbiased MCMC framework only works when the quantity of interest is an expectation, which excludes many practical applications. In this paper, we propose a general method for constructing unbiased estimators for functions of expectations and extend it to construct unbiased estimators for nested expectations. Our approach combines and generalizes the unbiased MCMC and Multilevel…
Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes as special cases. New lower variance estimators are proposed, which are stratified versions of earlier unbiased schemes. Under general conditions, essentially when MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are shown to be asymptotically as efficient a…
In this short note we provide an unbiased multilevel Monte Carlo estimator of the log marginal likelihood and discuss its application to variational Bayes.
We introduce a class of unbiased Monte Carlo estimators for the multivariate density of max-stable fields generated by Gaussian processes. Our estimators take advantage of recent results on exact simulation of max-stable fields combined with identities studied in the Malliavin calculus literature and ideas developed in the multilevel Monte Carlo literature. Our approach allows estimating multivariate densities of max-stable fields with precision $\varepsilon $ at a computational cost of order $O\left( \varepsilon ^{-2}\log \log \log \left( 1/\varepsilon \right) \right) $.
To address the problem of multicollinearity in the logistic regression model, in this paper we propose a new estimator called Stochastic restricted almost unbiased logistic Liu-estimator (SRAULLE) when the prior information is available in the form of stochastic linear restrictions. A Monte Carlo simulation study was carried out to compare the performance of the proposed estimator with some existing estimators in the scalar mean squared error (SMSE) sense. Finally, a real data example was given to appraise the performance of the estimators.
We propose a new unbiased estimator for estimating the utility of the optimal stopping problem. The MUSE, short for Multilevel Unbiased Stopping Estimator, constructs the unbiased Multilevel Monte Carlo (MLMC) estimator at every stage of the optimal stopping problem in a backward recursive way. In contrast to traditional sequential methods, the MUSE can be implemented in parallel. We prove the MUSE has finite variance, finite computational complexity, and achieves $ε$-accuracy with $O(1/ε^2)$ computational cost under mild conditions. We demonstrate MUSE empirically in an option pricing problem…
An empirical best linear unbiased prediction (EBLUP) estimator is utilized for efficient inference in small-area estimation. To measure its uncertainty, we need to estimate its mean squared error (MSE) since the true MSE cannot generally be derived in a closed form. The "naive MSE estimator", one of the estimators available for small-area inference, is unlikely to be chosen, since it does not achieve the desired asymptotic property, namely second-order unbiasedness, although it maintains strict positivity and tractability. Therefore, users tend to choose the second-order unbiased MSE estimator…
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 cas…
ETH Zurich（苏黎世联邦理工学院）；The University of Tokyo（东京大学）
In this paper, we propose a new stochastic optimization algorithm for Bayesian inference based on multilevel Monte Carlo (MLMC) methods. In Bayesian statistics, biased estimators of the model evidence have been often used as stochastic objectives because the existing debiasing techniques are computationally costly to apply. To overcome this issue, we apply an MLMC sampling technique to construct low-variance unbiased estimators both for the model evidence and its gradient. In the theoretical analysis, we show that the computational cost required for our proposed MLMC estimator to estimate the …
The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit state function, which depends on the solution of a partial differential equation (PDE). Since numerical evaluations of PDEs are computationally expensive, estimating such probabilities of failure by Monte Carlo sampling is intractable. More efficient sampling methods from reliability analysis, such as Subset Simulation, are popular, but can still be impracticable if the PDE evaluations are very costly. In this article, we de…
We study the existence of algorithms generating almost surely nonnegative unbiased estimators. We show that given a nonconstant real-valued function $f$ and a sequence of unbiased estimators of $λ\in\mathbb{R}$, there is no algorithm yielding almost surely nonnegative unbiased estimators of $f(λ)\in\mathbb{R}^+$. The study is motivated by pseudo-marginal Monte Carlo algorithms that rely on such nonnegative unbiased estimators. These methods allow "exact inference" in intractable models, in the sense that integrals with respect to a target distribution can be estimated without any systematic er…
In the following article we consider the non-linear filtering problem in continuous-time and in particular the solution to Zakai's equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai's equation. That is, given access to only a first order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.
The Ewens sampling formula is a distribution related to the random partition of a positive integer. In this study, we investigate the issue of non-existence solutions in parameter estimation under the distribution. As a result, the first and second moments matching estimators to the uniformly minimum variance unbiased estimator are derived using the Ewens sampling formula in asymptotic sense. A Monte Carlo simulation study is performed to evaluate the efficiency of the resulting estimators.
Multilevel Sequential Monte Carlo Samplers for Normalizing Constants
Center INRIA Bordeaux Sud-Ouest & Institut de Mathematiques de Bordeaux, Universite de Bordeaux I（波尔多南西INRIA中心与波尔多数学研究所，波尔多第一大学）；Department of Statistics & Applied Probability, National University of Singapore（新加坡国立大学统计与应用概率系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory（橡树岭国家实验室计算机科学与数学部）
In this analytical study we derive the optimal unbiased value estimator (MVU) and compare its statistical risk to three well known value estimators: Temporal Difference learning (TD), Monte Carlo estimation (MC) and Least-Squares Temporal Difference Learning (LSTD). We demonstrate that LSTD is equivalent to the MVU if the Markov Reward Process (MRP) is acyclic and show that both differ for most cyclic MRPs as LSTD is then typically biased. More generally, we show that estimators that fulfill the Bellman equation can only be unbiased for special cyclic MRPs. The main reason being the probabilit…
Standard approaches for uncertainty quantification in cardiovascular modeling pose challenges due to the large number of uncertain inputs and the significant computational cost of realistic three-dimensional simulations. We propose an efficient uncertainty quantification framework utilizing a multilevel multifidelity Monte Carlo estimator to improve the accuracy of hemodynamic quantities of interest while maintaining reasonable computational cost. This is achieved by leveraging three cardiovascular model fidelities, each with varying spatial resolution to rigorously quantify the variability in…
The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest $θ$ is reduced on average by collecting relevant data $Y$. However, estimating the expected information gain has been considered computationally challenging since it is defined as a nested expectation with an outer expectation with respect to $Y$ and an inner expectation with respect to $θ$. In fact, the standard, nested Monte Carlo method requires a total computational cost of $O(\varepsilon^{-3})$ to achieve a r…
We propose a variance reduction framework for variational inference using the Multilevel Monte Carlo (MLMC) method. Our framework is built on reparameterized gradient estimators and "recycles" parameters obtained from past update history in optimization. In addition, our framework provides a new optimization algorithm based on stochastic gradient descent (SGD) that adaptively estimates the sample size used for gradient estimation according to the ratio of the gradient variance. We theoretically show that, with our method, the variance of the gradient estimator decreases as optimization proceed…
In this article we introduce two new estimates of the normalizing constant (or marginal likelihood) for partially observed diffusion (POD) processes, with discrete observations. One estimate is biased but non-negative and the other is unbiased but not almost surely non-negative. Our method uses the multilevel particle filter of Jasra et al (2015). We show that, under assumptions, for Euler discretized PODs and a given $\varepsilon>0$. in order to obtain a mean square error (MSE) of $\mathcal{O}(\varepsilon^2)$ one requires a work of $\mathcal{O}(\varepsilon^{-2.5})$ for our new estimates versu…
When an unbiased estimator of the likelihood is used within a Metropolis--Hastings chain, it is necessary to trade off the number of Monte Carlo samples used to construct this estimator against the asymptotic variances of averages computed under this chain. Many Monte Carlo samples will typically result in Metropolis--Hastings averages with lower asymptotic variances than the corresponding Metropolis--Hastings averages using fewer samples. However, the computing time required to construct the likelihood estimator increases with the number of Monte Carlo samples. Under the assumption that the d…
The expected value of partial perfect information (EVPPI) denotes the value of eliminating uncertainty on a subset of unknown parameters involved in a decision model. The EVPPI can be regarded as a decision-theoretic sensitivity index, and has been widely used for identifying relatively important unknown parameters. It follows from Jensen's inequality, however, that the standard nested Monte Carlo computation of the EVPPI results in biased estimates. In this paper we introduce two unbiased Monte Carlo estimators for the EVPPI based on multilevel Monte Carlo method, introduced by Heinrich (1998…
We consider the combined use of resampling and partial rejection control in sequential Monte Carlo methods, also known as particle filters. While the variance reducing properties of rejection control are known, there has not been (to the best of our knowledge) any work on unbiased estimation of the marginal likelihood (also known as the model evidence or the normalizing constant) in this type of particle filter. Being able to estimate the marginal likelihood without bias is highly relevant for model comparison, computation of interpretable and reliable confidence intervals, and in exact approx…
In many epidemiological contexts, disease occurrences and their rates are naturally modelled by counting processes and their intensities, allowing an analysis based on martingale methods. These methods lend themselves to extensions of nested case-control sampling designs where general methods of control selection can be easily incorporated. This same methodology allows for extensions of the Mantel-Haenszel estimator in two main directions. First, a variety of new sampling designs can be incorporated which can yield substantial efficiency gains over simple random sampling. Second, the extension…
Unbiased location- and scale-invariant `elemental' estimators for the GPD tail parameter are constructed. Each involves three log-spacings. The estimators are unbiased for finite sample sizes, even as small as N=3. It is shown that the elementals form a complete basis for unbiased location- and scale-invariant estimators constructed from linear combinations of log-spacings. Preliminary numerical evidence is presented which suggests that elemental combinations can be constructed which are consistent estimators of the tail parameter for samples drawn from the pure GPD family.
In this paper the complex-valued best linear unbiased estimator of an unknown constant mean of white noise was derived the ordinary least-squares estimator of an unknown constant mean of random field (arithmetic mean) charged by an imaginary error.
We introduce unbiased estimators for the Shannon entropy and the class number, in the situation that we are able to take sequences of independent samples of arbitrary length.
For classical estimation with an underlying linear model the best linear unbiased estimator (BLUE) is usually utilized for estimating the deterministic but unknown parameter vector. In the case of real valued parameter vectors but complex valued measurement matrices and noise vectors, the BLUE results in complex valued estimates, introducing a systematic error. In recent years widely linear estimators have been investigated for complex estimation. In this work a novel widely linear classical estimator is derived which incorporates the knowledge that the parameter vector is real valued. The pro…
Necessary and sufficient conditions for the square-integrability of recently proposed unbiased estimators are established. A geometric characterization of a distribution that optimizes the performance of these estimators is given. An algorithm based on convex hulls that finds the optimal distribution truncated to its first m terms in time linear in m is described. The algorithm exploits a connection with a recent randomized dimension reduction method and is illustrated via a numerical example.
We address the problem of existence of unbiased constrained parameter estimators. We show that if the constrained set of parameters is compact and the hypothesized distributions are absolutely continuous with respect to one another, then there exists no unbiased estimator. Weaker conditions for the absence of unbiased constrained estimators are also specified. We provide several examples which demonstrate the utility of these conditions.
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
An unbiased estimator for the ellipticity of an object in a noisy image is given in terms of the image moments. Three assumptions are made: i) the pixel noise is normally distributed, although with arbitrary covariance matrix, ii) the image moments are taken about a fixed centre, and iii) the point-spread function is known. The relevant combinations of image moments are then jointly normal and their covariance matrix can be computed. A particular estimator for the ratio of the means of jointly normal variates is constructed and used to provide the unbiased estimator for the ellipticity. Furthe…
We consider the problem of the estimation of the invariant distribution function of an ergodic diffusion process when the drift coefficient is unknown. The empirical distribution function is a natural estimator which is unbiased, uniformly consistent and efficient in different metrics. Here we study the properties of optimality for an other kind of estimator recently proposed. We consider a class of unbiased estimators and we show that they are also efficient in the sense that their asymptotic risk, defined as the integrated mean square error, attains an asymptotic minimax lower bound.
Based on observations of points uniformly distributed over a convex set in $\R^d$, a new estimator for the volume of the convex set is proposed. The estimator is minimax optimal and also efficient non-asymptotically: it is nearly unbiased with minimal variance among all unbiased oracle-type estimators. Our approach is based on a Poisson point process model and as an ingredient, we prove that the convex hull is a sufficient and complete statistic. No hypotheses on the boundary of the convex set are imposed. In a numerical study, we show that the estimator outperforms earlier estimators for the …
We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (URE) method, we build an estimator of the risk which allows to select an estimator in a collection of model. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology.
In this paper the problem of best linear unbiased estimation is investigated for continuous-time regression models. We prove several general statements concerning the explicit form of the best linear unbiased estimator (BLUE), in particular when the error process is a smooth process with one or several derivatives of the response process available for construction of the estimators. We derive the explicit form of the BLUE for many specific models including the cases of continuous autoregressive errors of order two and integrated error processes (such as integrated Brownian motion). The results…
In the framework of matrix valued observables with low rank means, Stein's unbiased risk estimate (SURE) can be useful for risk estimation and for tuning the amount of shrinkage towards low rank matrices. This was demonstrated by Candès et al. (2013) for singular value soft thresholding, which is a Lipschitz continuous estimator. SURE provides an unbiased risk estimate for an estimator whenever the differentiability requirements for Stein's lemma are satisfied. Lipschitz continuity of the estimator is sufficient, but it is emphasized that differentiability Lebesgue almost everywhere isn't. The…
A simple characterization of uniformly minimum variance unbiased estimators (UMVUEs) is provided (in the case when the sample space is finite) in terms of a linear independence condition on the likelihood functions corresponding to the possible samples. The crucial observation in the proof is that, if a UMVUE exists, then, after an appropriate cleaning of the parameter space, the nonzero likelihood functions are eigenvectors of an "artificial" matrix of Lagrange multipliers, and the values of the UMVUE are eigenvalues of that matrix. The characterization is then extended to best unbiased estim…
