D. Romeres, G. Prando, G. Pillonetto, A. Chiuso
This paper compares classical parametric methods with recently developed Bayesian methods for system identification. A Full Bayes solution is considered together with one of the standard approximations based on the Empirical Bayes paradigm. Results regarding point estimators for the impulse response as well as for confidence regions are reported.
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Model estimates obtained from traditional subspace identification methods may be subject to significant variance. This elevated variance is aggravated in the cases of large models or of a limited sample size. Common solutions to reduce the effect of variance are regularized estimators, shrinkage estimators and Bayesian estimation. In the current work we investigate the latter two solutions, which have not yet been applied to subspace identification. Our experimental results show that our proposed estimators may reduce the estimation risk up to $40\%$ of that of traditional subspace methods.
Bayes-assisted Confidence Regions: Focal Point Estimator and Bounded-influence Priors
提出FAB置信区域框架，证明高斯似然下后验均值包含于FAB-CR，并引入幂律尾条件实现鲁棒FAB-CR，提出满足条件的收缩先验类，建立与贝叶斯收缩方法的联系。
For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz-Thompson estima…
We propose a variational Bayesian inference procedure for online nonlinear system identification. For each output observation, a set of parameter posterior distributions is updated, which is then used to form a posterior predictive distribution for future outputs. We focus on the class of polynomial NARMAX models, which we cast into probabilistic form and represent in terms of a Forney-style factor graph. Inference in this graph is efficiently performed by a variational message passing algorithm. We show empirically that our variational Bayesian estimator outperforms an online recursive least-…
Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.
Error Bounds Revisited, and How to Use Bayesian Statistics While Remaining a Frequentist
本文探讨了似然函数存在多峰时点估计器的可靠性问题，提出将先验重新解释为加权函数，使贝叶斯估计器在频率学派语境中具有意义，并以到达时间（TOA）问题为例，推导了不同于置信区域的基于区域的估计器理论。
Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the ma…
We propose a robust inferential procedure for assessing uncertainties of parameter estimation in high-dimensional linear models, where the dimension $p$ can grow exponentially fast with the sample size $n$. Our method combines the de-biasing technique with the composite quantile function to construct an estimator that is asymptotically normal. Hence it can be used to construct valid confidence intervals and conduct hypothesis tests. Our estimator is robust and does not require the existence of first or second moment of the noise distribution. It also preserves efficiency in the sense that the …
Smaller Confidence Intervals From IPW Estimators via Data-Dependent Coarsening
本文提出数据依赖的粗化 IPW（CIPW）估计量，通过合并协变量在倾向得分存在误差时仍使置信区间按 ε+1/√n 缩放，解决了现有 IPW 估计量对倾向得分不准确和极端倾向得分不稳健的问题。
We develop a novel procedure for constructing confidence bands for components of a sparse additive model. Our procedure is based on a new kernel-sieve hybrid estimator that combines two most popular nonparametric estimation methods in the literature, the kernel regression and the spline method, and is of interest in its own right. Existing methods for fitting sparse additive model are primarily based on sieve estimators, while the literature on confidence bands for nonparametric models are primarily based upon kernel or local polynomial estimators. Our kernel-sieve hybrid estimator combines th…
Diffusion source identification on networks is a problem of fundamental importance in a broad class of applications, including rumor controlling and virus identification. Though this problem has received significant recent attention, most studies have focused only on very restrictive settings and lack theoretical guarantees for more realistic networks. We introduce a statistical framework for the study of diffusion source identification and develop a confidence set inference approach inspired by hypothesis testing. Our method efficiently produces a small subset of nodes, which provably covers …
We investigate and provide new insights on the sampling rule called Top-Two Thompson Sampling (TTTS). In particular, we justify its use for fixed-confidence best-arm identification. We further propose a variant of TTTS called Top-Two Transportation Cost (T3C), which disposes of the computational burden of TTTS. As our main contribution, we provide the first sample complexity analysis of TTTS and T3C when coupled with a very natural Bayesian stopping rule, for bandits with Gaussian rewards, solving one of the open questions raised by Russo (2016). We also provide new posterior convergence resul…
We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors $k$ may depend on and diverge with sample size $n$. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically,…
An $\varepsilon$-Best-Arm Identification Algorithm for Fixed-Confidence and Beyond
该研究提出用于随机多臂老虎机ε-最优臂识别的任意时间采样规则EB-TCε，给出多类理论保证并通过仿真验证其性能优于现有算法。
Fixed Confidence Best Arm Identification in the Bayesian Setting
该研究针对贝叶斯场景下的固定置信度最优臂识别问题，证明传统频率论FC-BAI算法在该场景下性能次优，推导了期望采样次数下界，提出匹配该下界（仅差对数因子）的连续消除变体算法并通过仿真验证。
We propose an estimator and confidence interval for computing the value of a policy from off-policy data in the contextual bandit setting. To this end we apply empirical likelihood techniques to formulate our estimator and confidence interval as simple convex optimization problems. Using the lower bound of our confidence interval, we then propose an off-policy policy optimization algorithm that searches for policies with large reward lower bound. We empirically find that both our estimator and confidence interval improve over previous proposals in finite sample regimes. Finally, the policy opt…
In the setting of high-dimensional linear regression models, we propose two frameworks for constructing pointwise and group confidence sets for penalized estimators which incorporate prior knowledge about the organization of the non-zero coefficients. This is done by desparsifying the estimator as in van de Geer et al. [18] and van de Geer and Stucky [17], then using an appropriate estimator for the precision matrix $Θ$. In order to estimate the precision matrix a corresponding structured matrix norm penalty has to be introduced. After normalization the result is an asymptotic pivot. The asymp…
Fixed-Confidence Best-Arm Identification for Causal Mediation Analysis
针对因果中介分析中最大化期望自然直接潜在结果的处理识别问题，基于Track-and-Stop框架结合割集法开发固定置信度最优臂识别算法，实现高概率保正的高效样本识别与渐近最优性。
Elimination algorithms for bandit identification, which prune the plausible correct answers sequentially until only one remains, are computationally convenient since they reduce the problem size over time. However, existing elimination strategies are often not fully adaptive (they update their sampling rule infrequently) and are not easy to extend to combinatorial settings, where the set of answers is exponentially large in the problem dimension. On the other hand, most existing fully-adaptive strategies to tackle general identification problems are computationally demanding since they repeate…
We propose a new procedure for inference on optimal treatment regimes in the model-free setting, which does not require to specify an outcome regression model. Existing model-free estimators for optimal treatment regimes are usually not suitable for the purpose of inference, because they either have nonstandard asymptotic distributions or do not necessarily guarantee consistent estimation of the parameter indexing the Bayes rule due to the use of surrogate loss. We first study a smoothed robust estimator that directly targets the parameter corresponding to the Bayes decision rule for optimal t…
Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others, as particu…
This paper introduces a class of k-nearest neighbor ($k$-NN) estimators called bipartite plug-in (BPI) estimators for estimating integrals of non-linear functions of a probability density, such as Shannon entropy and Rényi entropy. The density is assumed to be smooth, have bounded support, and be uniformly bounded from below on this set. Unlike previous $k$-NN estimators of non-linear density functionals, the proposed estimator uses data-splitting and boundary correction to achieve lower mean square error. Specifically, we assume that $T$ i.i.d. samples ${X}_i \in \mathbb{R}^d$ from the densit…
We study the problem of finding the most mutually correlated arms among many arms. We show that adaptive arms sampling strategies can have significant advantages over the non-adaptive uniform sampling strategy. Our proposed algorithms rely on a novel correlation estimator. The use of this accurate estimator allows us to get improved results for a wide range of problem instances.
We give a complete characterization of the complexity of best-arm identification in one-parameter bandit problems. We prove a new, tight lower bound on the sample complexity. We propose the `Track-and-Stop' strategy, which we prove to be asymptotically optimal. It consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.
Fixed-Confidence Multiple Change Point Identification under Bandit Feedback
研究固定置信度分段常数bandit问题中的多变点识别，基于实例依赖下界设计渐近最优的Track-and-Stop变体，通过集中采样变点附近实现高效识别。
Simulator-Based Inference with Waldo: Confidence Regions by Leveraging Prediction Algorithms and Posterior Estimators for Inverse Problems
针对模拟器推断中预测算法和后验估计器产生有偏或过度自信参数区域的问题，提出WALDO方法，通过重构Wald统计量并采用回归式Neyman反演，构建有限样本条件有效的置信区域，并在高能物理问题中验证其纠正能力。
We give a complete characterization of the sampling complexity of best Markovian arm identification in one-parameter Markovian bandit models. We derive instance specific nonasymptotic and asymptotic lower bounds which generalize those of the IID setting. We analyze the Track-and-Stop strategy, initially proposed for the IID setting, and we prove that asymptotically it is at most a factor of four apart from the lower bound. Our one-parameter Markovian bandit model is based on the notion of an exponential family of stochastic matrices for which we establish many useful properties. For the analys…
The V-fold jackknife for semiparametric inference: variance estimation, confidence intervals, and simultaneous confidence bands
研究针对半参数推断开发V折刀切法，它计算高效且理论合理，仅需V次留一法重拟合，利用伪值经验离散度量化不确定性，给出有效置信区间和同时置信带，扩展理论到特定估计量，模拟证实其可靠推断。
Optimal Confidence Band for Kernel Gradient Flow Estimator
本文研究了核回归方法中核梯度流的 supremum-范数泛化误差和统一推断，基于容量-源条件框架，推导了连续和离散核梯度流的收敛速率，并构建了同时置信带，其宽度最优。
Likelihood Ratio Confidence Sets for Sequential Decision Making
本文提出基于似然比构建任意时刻有效的模型无关置信序列，可证明选择最优估计量序列并设计重加权方案适配非参数场景，在多种序列决策任务中验证了实际优势。
We present a new finite-time analysis of the estimation error of the Ordinary Least Squares (OLS) estimator for stable linear time-invariant systems. We characterize the number of observed samples (the length of the observed trajectory) sufficient for the OLS estimator to be $(\varepsilon,δ)$-PAC, i.e., to yield an estimation error less than $\varepsilon$ with probability at least $1-δ$. We show that this number matches existing sample complexity lower bounds [1,2] up to universal multiplicative factors (independent of ($\varepsilon,δ)$ and of the system). This paper hence establishes the opti…
This paper studies the high-dimensional mixed linear regression (MLR) where the output variable comes from one of the two linear regression models with an unknown mixing proportion and an unknown covariance structure of the random covariates. Building upon a high-dimensional EM algorithm, we propose an iterative procedure for estimating the two regression vectors and establish their rates of convergence. Based on the iterative estimators, we further construct debiased estimators and establish their asymptotic normality. For individual coordinates, confidence intervals centered at the debiased …
We propose Causal Interaction Trees for identifying subgroups of participants that have enhanced treatment effects using observational data. We extend the Classification and Regression Tree algorithm by using splitting criteria that focus on maximizing between-group treatment effect heterogeneity based on subgroup-specific treatment effect estimators to dictate decision-making in the algorithm. We derive properties of three subgroup-specific treatment effect estimators that account for the observational nature of the data -- inverse probability weighting, g-formula and doubly robust estimators…
The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several nonparametric divergence estimators exist, relatively few have known convergence properties. In particular, even for those estimators whose MSE convergence rates are known, the asymptotic distributions are unknown. We establish the asymptotic normality of a recently proposed ensemble estimator of f-divergence between two distributions from a finite number of samples. This estimator has MSE convergence rate of O(1/T), is simple to implement, and performs well …
One of the most commonly used methods for forming confidence intervals for statistical inference is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role, its theoretical properties are still not well understood for non-asymptotically normal estimators. In this paper, under stability conditions, we establish the limiting distribution of the empirical bootstrap estimator, derive tight conditions for it to be asymptotically consistent, and quantify the speed of convergence. Moreover, we propose thre…
We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node's state variable is a convex comb…
This paper develops robust confidence intervals in high-dimensional and left-censored regression. Type-I censored regression models are extremely common in practice, where a competing event makes the variable of interest unobservable. However, techniques developed for entirely observed data do not directly apply to the censored observations. In this paper, we develop smoothed estimating equations that augment the de-biasing method, such that the resulting estimator is adaptive to censoring and is more robust to the misspecification of the error distribution. We propose a unified class of robus…
Adaptive experiment designs can dramatically improve statistical efficiency in randomized trials, but they also complicate statistical inference. For example, it is now well known that the sample mean is biased in adaptive trials. Inferential challenges are exacerbated when our parameter of interest differs from the parameter the trial was designed to target, such as when we are interested in estimating the value of a sub-optimal treatment after running a trial to determine the optimal treatment using a stochastic bandit design. In this context, typical estimators that use inverse propensity w…
Optimal Best-Arm Identification under Fixed Confidence with Multiple Optima
在固定置信度下，本文提出改进的Track-and-Stop算法，通过考虑平局情况，实现多最优臂识别的最优性能保证。
Nearly Optimal Fixed-Confidence Best-Arm Identification with 1-Bit Feedback
针对1比特反馈下的固定置信度最佳臂识别，提出基于随机阈值查询的均值估计原语，并嵌入候选-挑战者算法，实现间隙自适应样本复杂度，且证明对数惩罚下界。
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