Lutz Duembgen, Andre Huesler, Kaspar Rufibach
We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indidate an EM algorithm to treat arbitrarily censored or binned data.
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We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.
Nonparametric Estimation for a Log-concave Distribution Function with Interval-censored Data
针对混合案例区间删失数据下的潜在事件时间估计问题，提出对数凹分布函数假设下的非参数极大似然方法，建立估计量性质并开发R包，相比现有方法更具灵活性与平衡性能。
This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the…
Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures, especially in the high-dimensional problems. This paper makes some progress in this dire…
Chain of Log-Concave Markov Chains
本文提出由对数凹马尔可夫链构成的非马尔可夫采样框架，以固定高斯平滑和经验贝叶斯推广 walk-jump sampling，并通过 2-Wasserstein 指标验证其多模态隧穿能力。
Improving score-based sampling via affine post-processing
针对仅可访问对数密度的采样问题，提出通过仿射后处理整合多个分数估计以降低偏差和方差，在困难密度上优于计算匹配的基线方法。
We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume a model which includes models previously considered for interval-censored variables with log-concave distributions as special cases. The resulting log-likelihood is concave, which we use to establish asymptotic normality of its maximizer as the number of observations $n$ tends to infinity with the number of parameters $d$ fixed, and rates of convergence of $…
Convergence of coordinate ascent variational inference for log-concave measures via optimal transport
本文针对对数凹测度下坐标上升变分推断（CAVI）收敛性不明确的问题，借助最优传输中的位移凸性性质，证明了CAVI的收敛性，并分别给出了对数凹密度带Lipschitz梯度及强对数凹情形下的线性与指数收敛速率。
In this paper, we provide non-asymptotic upper bounds on the error of sampling from a target density using three schemes of discretized Langevin diffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, the Euler discretization of the Langevin diffusion. The second and the third schemes are, respectively, the kinetic Langevin Monte Carlo (KLMC) for differentiable potentials and the kinetic Langevin Monte Carlo for twice-differentiable potentials (KLMC2). The main focus is on the target densities that are smooth and log-concave on $\mathbb R^p$, but not necessarily strongly log-…
Phylogenetic trees are key data objects in biology, and the method of phylogenetic reconstruction has been highly developed. The space of phylogenetic trees is a nonpositively curved metric space. Recently, statistical methods to analyze the set of trees on this space are being developed utilizing this property. Meanwhile, in Euclidean space, the log-concave maximum likelihood method has emerged as a new nonparametric method for probability density estimation. In this paper, we derive a sufficient condition for the existence and uniqueness of the log-concave maximum likelihood estimator on tre…
This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that exploits log-concavity to construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau-Yoshida regularisations for convex functions and uses proximity mappings instead of grad…
Monte Carlo methods are often necessary for the implementation of optimal Bayesian estimators. A fundamental technique that can be used to generate samples from virtually any target probability distribution is the so-called rejection sampling method, which generates candidate samples from a proposal distribution and then accepts them or not by testing the ratio of the target and proposal densities. The class of adaptive rejection sampling (ARS) algorithms is particularly interesting because they can achieve high acceptance rates. However, the standard ARS method can only be used with log-conca…
Finite Mixture Cox Model for Heterogeneous Time-dependent Right-Censored Data
针对异质患者群体生存分析中单一Cox PH模型的不足，提出结合SCAD/MCP动态惩罚函数的有限混合Cox模型，通过改进EM算法实现亚组识别与成分选择，提升风险预测精度。
Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on $\mathbb R^p$. The present work focuses on this framework and studies the behavior of Monte Carlo algorithms based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is, in the overdamped regime, optimal in terms of its dependence on the …
High-dimensional regression and regression with a left-censored response are each well-studied topics. In spite of this, few methods have been proposed which deal with both of these complications simultaneously. The Tobit model -- long the standard method for censored regression in economics -- has not been adapted for high-dimensional regression at all. To fill this gap and bring up-to-date techniques from high-dimensional statistics to the field of high-dimensional left-censored regression, we propose several penalized Tobit models. We develop a fast algorithm which combines quadratic minimi…
Parallelized Midpoint Randomization for Langevin Monte Carlo
本文针对光滑强对数凹密度，在可并行评估梯度的设定下分析并行化随机中点 Langevin Monte Carlo，给出 Wasserstein 距离上界并证明并行处理带来显著运行时间提升。
We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $π= \exp(-U)/Z$ with respect to the Lebesgue measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm (High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm, A. Durmus and E. Moulines). Polynomial bounds in the dimension $d$ are obtained with an exponent that depends on the assumptions made on $U$. The algorithm also provides a theoretically grounded choice of the annealing sequence of…
Variable Selection in Ultra-high Dimensional Feature Space for the Cox Model with Interval-Censored Data
本文针对区间删失数据下的超高维 Cox 模型，提出基于 lasso、adaptive lasso、SCAD 和 MCP 惩罚的变量选择方法，证明其具有 oracle 性质，并通过模拟和儿童龋齿全基因组关联研究验证了方法效果。
In this article, we revisit the problem of fitting a mixture model under the assumption that the mixture components are symmetric and log-concave. To this end, we first study the nonparametric maximum likelihood estimation (NPMLE) of a monotone log-concave probability density. By following the arguments of Rufibach (2006), we show that the NPMLE is uniformly consistent with respect to the supremum norm on compact subsets of the interior of the support. To fit the mixture model, we propose a semiparametric EM (SEM) algorithm, which can be adapted to other semiparametric mixture models. In our n…
Robust Subgroup Analysis for Heterogeneous Censored Data
针对异质删失数据，提出结合逆概率加权、M估计和凹成对融合惩罚的方法，开发RISA - ADMM算法，推导估计量理论性质，经模拟和应用验证，可有效识别子组并估计协变量效应。
The present paper deals with a nonparametric M-estimation for right censored regression model with stationary ergodic data. Defined as an implicit function, a kernel type estimator of a family of robust regression is considered when the covariate take its values in R^d (d >= 1) and the data are sampled from stationary ergodic process. The strong consistency (with rate) and the asymptotic distribution of the estimator are established under mild assumptions. Moreover, a usable confidence interval is provided which does not depend on any unknown quantity. Our results hold without any mixing condi…
Nonparametric statistics for distribution functions F or densities f=F' under qualitative shape constraints provides an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: F is said to be bi-log-concave, if both log(F) and log(1 - F) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f. F with log-concave density f = F' is bi-log-concave. But in contrast to the latter constraint, bi-log-concavity allows for multimodal densities. We prov…
In survival studies, classical inferences for left-truncated data require quasi-independence, a property that the joint density of truncation time and failure time is factorizable into their marginal densities in the observable region. The quasi-independence hypothesis is testable; many authors have developed tests for left-truncated data with or without right-censoring. In this paper, we propose a class of test statistics for testing the quasi-independence which unifies the existing methods and generates new useful statistics such as conditional Spearman's rank correlation coefficient. Asympt…
Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters, have become very popular in signal processing over the last years. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as rejection sampling (RS). Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of…
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. In this paper, we remove this limitation, providing polynomial-time convergence guarantees for a variant of LMC in the setting of nonsmooth log-concave distributions. At a high level, our results follow by leveraging the implic…
Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we are able to reformulate the issue of computation in terms of a non-differentiable convex optimisation problem, and thus combine techniques of computational geometry with Shor's…
On the Addams family of discrete frailty distributions for modelling multivariate case I interval-censored data
本文研究了Addams族离散脆弱性分布的实现与解释，针对多元情形I区间删失数据提出了共享脆弱性模型下的参数估计方法，并利用其支撑集依赖于参数的特性实现了非齐次协变量效应建模。
We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\l…
Motivated by studies in biological sciences to detect differentially expressed genes, a semiparametric two-component mixture model with one known component is being studied in this paper. Assuming the density of the unknown component to be log-concave, which contains a very broad family of densities, we develop a semiparametric maximum likelihood estimator and propose an EM algorithm to compute it. Our new estimation method finds the mixing proportions and the distribution of the unknown component simultaneously. We establish the identifiability of the proposed semiparametric mixture model and…
Log-concave density estimation in undirected graphical models
本文提出了一种在无向图模型中对数凹密度估计的方法，通过有限维凸优化问题求解MLE，并探讨了其存在性、唯一性和自洽性条件。
In recent years, log-concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one or more bandwidths. The purpose of this article is to describe some of the properties of the class of log-concave densities on $\mathbb{R}^d$ which make it so attractive from a statistical perspective, and to outline the latest methodological, theoretical and computational advances in the area.
High-Order Langevin Monte Carlo Algorithms
针对大规模采样问题，提出基于P阶（P≥3）朗之万动力学离散化的蒙特卡洛算法，通过分裂与精确积分法设计，获对数凹光滑分布采样的Wasserstein收敛保证，混合时间对d和ε的依赖性随P增大更优，数值实验验证其效率。
Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.
Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on $\mathbb{R}^d$ that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fac…
Doubly robust estimation and inference for a log-concave counterfactual density
针对含二元/离散处理的观测数据因果推断问题，提出带对数凹形状约束的反事实密度双重鲁棒估计量，证明其一致性并构建逐点置信区间与置信带，为分布差异分析提供工具。
High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by modern applications in high-throughput genomic data analysis and credit risk analysis. In this article, we propose a class of regularization methods for simultaneous variable selection and estimation in the additive hazards model, by combining the nonconcave penalized likelihood approach and the pseudoscore method. In a high-dimensional setting where the dimensionality can grow fast, polynomially or nonpolynomially, with the sample size, we establish the weak oracle property and ora…
A smooth ROC curve estimator based on log-concave density estimates
本文提出基于对数凹密度估计的平滑ROC曲线估计量，证明其渐近等价于经验ROC曲线，并在有限样本下具有效率增益、稳健性和更短的自举置信区间，计算高效。
Log-Laplace Nuggets for Fully Bayesian Fitting of Spatial Extremes Models to Threshold Exceedances
本文提出一种乘法对数拉普拉斯 nugget 方法，通过在截断似然中引入条件独立性，实现高维空间极值模型的高效贝叶斯推断，同时保持极值依赖结构。
Smooth tail index estimation
本文提出用基于对数凹密度估计的分布函数分位数替代次序统计量，构造平滑尾部指数估计量，以改善小样本估计性能，并通过Monte Carlo模拟验证其在均方误差上显著优于非平滑估计量。
We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on $\mathbb{R}^p$ and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are $K$-homothetic (i.e. scalar multiples of a convex body $K \subseteq \mathbb{R}^p$). When $K$ is known, we prove that the $K$-homothetic log-concave maximum likelihood estimator based on $n$ independent observations from such a density has a …
Based on independently distributed $X_1 \sim N_p(θ_1, σ^2_1 I_p)$ and $X_2 \sim N_p(θ_2, σ^2_2 I_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(θ_1, σ^2_Y I_p)$, with the additional information $θ_1 - θ_2 \in A$ and known $σ^2_1, σ^2_2, σ^2_Y$. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for $α-$divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss …
SLGP: An R Package for Spatial Conditional Density Estimation
SLGP包通过空间逻辑高斯过程实现条件密度场的非参数估计，采用随机傅里叶特征和凸优化，提供MAP、拉普拉斯和MCMC三种模式，并在斐济-汤加地震数据上验证。
Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries
针对未知噪声分布的位置估计问题，提出一种形状无关的估计器，通过多尺度中位数摘要达到实例最优速率，适用于更广的对称单峰密度类，且计算复杂度为O(log n)。
Density functions that represent sample data are often multimodal, i.e. they exhibit more than one maximum. Typically this behavior is taken to indicate that the underlying data deserves a more detailed representation as a mixture of densities with individually simpler structure. The usual specification of a component density is quite restrictive, with log-concave the most general case considered in the literature, and Gaussian the overwhelmingly typical case. It is also necessary to determine the number of mixture components \emph{a priori}, and much art is devoted to this. Here, we introduce…
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