Incorporating a contrast in the Bayesian formula: What consequences for the MAP estimator and the posterior distribution? Applications in spatial statistics
S. Soubeyrand, F. Carpentier, N. Desassis, J. Chadœuf
In order to estimate model parameters and circumvent possible difficulties encountered with the likelihood function, we propose to replace the likelihood in the formula of the posterior distribution by a function depending on a contrast. The properties of the contrast-based (CB) posterior distribution and MAP estimator are studied to understand what the consequences of incorporating a contrast in the Bayesian formula are. We show that the proposed method can be used to make frequentist inference and allows the reduction of analytical calculations to get the limit variance matrix of the estimator. For specific contrasts, the CB--posterior distribution directly approximates the limit distribution of the estimator; the calculation of the limit variance matrix is then avoided. Moreover, for these contrasts, the CB--posterior distribution can also be used to make inference in the Bayesian way. The method is applied to three spatial data sets.
Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In this paper, we propose original connections between Hamilton--Jacobi (HJ) partial differential equations and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and log-concave prior. Whereas solutions to certain first-order HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimat…
We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributi…
Maximum a posteriori and Bayes estimators are two common methods of point estimation in Bayesian Statistics. It is commonly accepted that maximum a posteriori estimators are a limiting case of Bayes estimators with 0-1 loss. In this paper, we provide a counterexample which shows that in general this claim is false. We then correct the claim that by providing a level-set condition for posterior densities such that the result holds. Since both estimators are defined in terms of optimization problems, the tools of variational analysis find a natural application to Bayesian point estimation.
Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.
We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010) for the multivariate normal mean model in the situation that the mean vector is sparse in the nearly black sense. We assume the frequentist framework where the data is generated according to a fixed mean vector. We show that if the number of nonzero parameters of the mean vector is known, the horseshoe estimator attains the minimax $\ell_2$ risk, possibly up to a multiplicative constant. We provide conditions under which the horseshoe estimator combined with an empirical Bayes estimate of the number of nonzero means s…
The Bayesian approach is effective for inverse problems. The posterior density distribution provides useful information of the unknowns. However, for problems with non-unique solutions, the classical estimators such as the maximum a posterior (MAP) and conditional mean (CM) are not enough. We introduce two new estimators, the local maximum a posterior (LMAP) and local conditional mean (LCM). Their applications are demonstrated by three inverse problems: an inverse spectral problem, an inverse source problem, and an inverse medium problem.
We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators.
In frequentist inference, minimizing the Hellinger distance between a kernel density estimate and a parametric family produces estimators that are both robust to outliers and statistically efficienty when the parametric model is correct. This paper seeks to extend these results to the use of nonparametric Bayesian density estimators within disparity methods. We propose two estimators: one replaces the kernel density estimator with the expected posterior density from a random histogram prior; the other induces a posterior over parameters through the posterior for the random histogram. We show t…
Consider semiparametric models that display local asymptotic exponentiality (Ibragimov and Has'minskii (1981)), an asymptotic property of the likelihood associated with discontinuities of densities. Our interest goes to estimation of the location of such discontinuities while other aspects of the density form a nuisance parameter. It is shown that under certain conditions on model and prior, the posterior distribution displays Bernstein-von Mises-type asymptotic behaviour, with exponential distributions as the limiting sequence. In contrast to regular settings, the maximum likelihood estimator…
We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-param…
We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression and its plug-in property for differential operators. For a general class of kernels, we establish convergence rates of the posterior measure of the regression function and its derivatives, which are both minimax optimal up to a logarithmic factor for functions in certain classes. Our calculation shows that the rate-optimal estimation of the regression function and its derivatives share the same choice of hyperparameter, indicating that the Bayes procedure remarkably adapts to …
In practice, it is desired to have estimates that are invariant under reparameterization. The invariance property of the estimators helps to formulate a unified solution to the underlying estimation problem. In robust Bayesian analysis, a frequent criticism is that the optimal estimators are not invariant under smooth reparameterizations. This paper considers the problem of posterior regret gamma-minimax (PRGM) estimation of the natural parameter of the exponential family of distributions under intrinsic loss functions. We show that under the class of Jeffrey's Conjugate Prior (JCP) distributi…
The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable. One may sample from it using Markov chain Monte Carlo, but this is often too slow for big datasets. We consider instead variational approximations of the Gibbs posterior, which are fast to compute. We undertake a general study of the properties of such approximations. Our main finding is that such a variational approximation has often the same rate of converg…
Sampling-Based Accuracy Testing of Posterior Estimators for General Inference
Mila – Quebec AI Institute（米拉-魁北克人工智能研究所）；Université de Montréal（蒙特利尔大学）；CIELA Institute（CIELA研究所）；Flatiron Institute Center for Computational Mathematics（弗拉蒂龙研究所计算数学中心）
本文提出 TARP 覆盖率测试，通过随机点与采样估计生成式后验估计器的覆盖概率，无需后验评估，可在高维空间中识别现有方法无法发现的不准确推断。
This paper is concerned with Bayesian inferential methods for data from controlled branching processes that account for model robustness through the use of disparities. Under regularity conditions, we establish that estimators built on disparity-based posterior, such as expectation and maximum a posteriori estimates, are consistent and efficient under the posited model. Additionally, we show that the estimates are robust to model misspecification and presence of aberrant outliers. To this end, we develop several fundamental ideas relating minimum disparity estimators to Bayesian estimators bui…
Maximum a posteriori (MAP) estimation, like all Bayesian methods, depends on prior assumptions. These assumptions are often chosen to promote specific features in the recovered estimate. The form of the chosen prior determines the shape of the posterior distribution, thus the behavior of the estimator and complexity of the associated optimization problem. Here, we consider a family of Gaussian hierarchical models with generalized gamma hyperpriors designed to promote sparsity in linear inverse problems. By varying the hyperparameters, we move continuously between priors that act as smoothed $\…
National University of Singapore（新加坡国立大学）；Deakin University（迪肯大学）；University of Oxford（牛津大学）
Scientific experiments are usually expensive due to complex experimental preparation and processing. Experimental design is therefore involved with the task of finding the optimal experimental input that results in the desirable output by using as few experiments as possible. Experimenters can often acquire the knowledge about the location of the global optimum. However, they do not know how to exploit this knowledge to accelerate experimental design. In this paper, we adopt the technique of Bayesian optimization for experimental design since Bayesian optimization has established itself as an …
How can one perform Bayesian inference on stochastic simulators with intractable likelihoods? A recent approach is to learn the posterior from adaptively proposed simulations using neural network-based conditional density estimators. However, existing methods are limited to a narrow range of proposal distributions or require importance weighting that can limit performance in practice. Here we present automatic posterior transformation (APT), a new sequential neural posterior estimation method for simulation-based inference. APT can modify the posterior estimate using arbitrary, dynamically upd…
Nested error regression models are useful tools for analysis of grouped data, especially in the case of small area estimation. This paper suggests a nested error regression model using uncertain random effects in which the random effect in each area is expressed as a mixture of a normal distribution and a positive mass at $0$. For estimation of the model parameters and prediction of the random effects, an objective Bayesian inference is proposed by setting non-informative prior distributions on the model parameters. Under mild sufficient conditions, it is shown that the posterior distribution …
We study the Bayesian model averaging approach to learning Bayesian network structures (DAGs) from data. We develop new algorithms including the first algorithm that is able to efficiently sample DAGs according to the exact structure posterior. The DAG samples can then be used to construct estimators for the posterior of any feature. We theoretically prove good properties of our estimators and empirically show that our estimators considerably outperform the estimators from the previous state-of-the-art methods.
Imperial College London（帝国理工学院）；University of Bristol（布里斯托大学）；University of Cambridge（剑桥大学）；ETH Zürich（苏黎世联邦理工学院）
Bayesian neural networks that incorporate data augmentation implicitly use a ``randomly perturbed log-likelihood [which] does not have a clean interpretation as a valid likelihood function'' (Izmailov et al. 2021). Here, we provide several approaches to developing principled Bayesian neural networks incorporating data augmentation. We introduce a ``finite orbit'' setting which allows likelihoods to be computed exactly, and give tight multi-sample bounds in the more usual ``full orbit'' setting. These models cast light on the origin of the cold posterior effect. In particular, we find that the …
We consider the sparse high-dimensional linear regression model $Y=Xb+ε$ where $b$ is a sparse vector. For the Bayesian approach to this problem, many authors have considered the behavior of the posterior distribution when, in truth, $Y=Xβ+ε$ for some given $β$. There have been numerous results about the rate at which the posterior distribution concentrates around $β$, but few results about the shape of that posterior distribution. We propose a prior distribution for $b$ such that the marginal posterior distribution of an individual coordinate $b_i$ is asymptotically normal centered around an …
In this paper, we study a class of non-parametric density estimators under Bayesian settings. The estimators are piecewise constant functions on binary partitions. We analyze the concentration rate of the posterior distribution under a suitable prior, and demonstrate that the rate does not directly depend on the dimension of the problem. This paper can be viewed as an extension of a parallel work where the convergence rate of a related sieve MLE was established. Compared to the sieve MLE, the main advantage of the Bayesian method is that it can adapt to the unknown complexity of the true densi…
A wide variety of phenomena of engineering and scientific interest are of a continuous-time nature and can be modeled by stochastic differential equations (SDEs), which represent the evolution of the uncertainty in the states of a system. For systems of this class, some parameters of the SDE might be unknown and the measured data often includes noise, so state and parameter estimators are needed to perform inference and further analysis using the system state path. The distributions of SDEs which are nonlinear or subject to non-Gaussian measurement noise do not admit tractable analytic express…
The ordinary Bayes estimator based on the posterior density suffers from the potential problems of non-robustness under data contamination or outliers. In this paper, we consider the general set-up of independent but non-homogeneous (INH) observations and study a robustified pseudo-posterior based estimation for such parametric INH models. In particular, we focus on the $R^{(α)}$-posterior developed by Ghosh and Basu (2016) for IID data and later extended by Ghosh and Basu (2017) for INH set-up, where its usefulness and desirable properties have been numerically illustrated. In this paper, we …
