Frequentist coverage of $(1-α)$-highest posterior density (HPD) credible sets is studied in a signal plus noise model under a large class of noise distributions. We consider a specific class of spike-and-slab prior distributions. Different regimes are identified and we derive closed form expressions for the $(1-α)$-HPD on each of these regimes. Similar to the earlier work by Marchand and Strawderman, it is shown that under suitable conditions, the frequentist coverage can drop to $1-3α/2.$
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry--Esseen type bound on quasi-posterior dis…
For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a s…
We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of "polished tail" parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessar…
We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called "general polished tail" condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abst…
Bayesian methods provide a natural means for uncertainty quantification, that is, credible sets can be easily obtained from the posterior distribution. But is this uncertainty quantification valid in the sense that the posterior credible sets attain the nominal frequentist coverage probability? This paper investigates the frequentist validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. In particular, we show that our marginal posterior credible intervals achieve the nominal frequentist coverage probability under conditions sligh…
For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-α)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\frac{3α}{2}$ and $1-\frac{3α}{2}+\frac{α^2}{1+α}$; with the lower bound $1-\frac{3α}{2}$ improving (for $α\leq 1/3$) on the previously established ([…
We investigate the relation between frequentist and Bayesian approaches. Namely, we find the "frequentist" Bayes prior π_{f}(λ,x_{obs}) = -\frac{\int_{-\infty}^{x_{obs}}\frac{\partial f(x,λ)}{\partial λ}dx}{f(x_{obs},λ)} (here f(x,λ) is the probability density) for which the results of frequentist and Bayes approaches to the determination of confidence intervals coincide. In many cases (but not always) the "frequentist" prior which reproduces frequentist results coincides with the Jeffreys prior.
Posterior distributions for community structure in sparse planted bi-section models are shown to achieve exact (resp. almost-exact) recovery, with sharp bounds for the sparsity regimes where edge probabilities decrease as $O(\log(n)/n)$ (resp. $O(1/n)$). Assuming posterior recovery, one may interpret credible sets (resp. enlarged credible sets) as asymptotically consistent confidence sets; the diameters of those credible sets are controlled by the rate of posterior concentration. If credible levels are chosen to grow to one quickly enough, corresponding credible sets can be interpreted as freq…
To the frequentist who computes posteriors, not all priors are useful asymptotically: in this paper Schwartz's 1965 Kullback-Leibler condition is generalised to enable frequentist interpretation of convergence of posterior distributions with the complex models and often dependent datasets in present-day statistical applications. We prove four simple and fully general frequentist theorems, for posterior consistency; for posterior rates of convergence; for consistency of the Bayes factor in hypothesis testing or model selection; and a theorem to obtain confidence sets from credible sets. The lat…
In this paper we develop and study adaptive empirical Bayesian smoothing splines. These are smoothing splines with both smoothing parameter and penalty order determined via the empirical Bayes method from the marginal likelihood of the model. The selected order and smoothing parameter are used to construct adaptive credible sets with good frequentist coverage for the underlying regression function. We use these credible sets as a proxy to show the superior performance of adaptive empirical Bayesian smoothing splines compared to frequentist smoothing splines.
Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop a Bernstein von-Mises type result under supremum norm in random design GP regression. Identifying both the mean and covariance function of the posterior distribution of the Gaussian process as regularized $M$-estimators, we show that the sampling distribution of the posterior me…
We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. We put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an "immersion map" from the space of unrestricted functions to that of multivariate monotone functions. This allows maintaining the natural conjugacy for posterior sampling. A natural…
Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to Hölder smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing paramet…
In a sparse stochastic block model with two communities of unequal sizes we derive two posterior concentration inequalities, that imply (1) posterior (almost-)exact recovery of the community structure under sparsity bounds comparable to well-known sharp bounds in the planted bi-section model; (2) a construction of confidence sets for the community assignment from credible sets, with finite graph sizes. The latter enables exact frequentist uncertain quantification with Bayesian credible sets at non-asymptotic graph sizes, where posteriors can be simulated well. There turns out to be no proporti…
For a general statistical model, we introduce the notion of data dependent measure (DDM) on the model parameter. Typical examples of DDM are the posterior distributions. Like for posteriors, the quality of a DDM is characterized by the contraction rate which we allow to be local, i.e., depending on the parameter. We construct confidence sets as DDM-credible sets and address the issue of optimality of such sets, via a trade-off between its "size" (the local radial rate) and its coverage probability. In the mildly ill-posed inverse signal-in-white-noise model, we construct a DDM as empirical Bay…
In the sparse normal means model, coverage of adaptive Bayesian posterior credible sets associated to spike and slab prior distributions is considered. The key sparsity hyperparameter is calibrated via marginal maximum likelihood empirical Bayes. First, adaptive posterior contraction rates are derived with respect to $d_q$--type--distances for $q\leq 2$. Next, under a type of so-called excessive-bias conditions, credible sets are constructed that have coverage of the true parameter at prescribed $1-α$ confidence level and at the same time are of optimal diameter. We also prove that the previou…
We investigate the credible sets and marginal credible intervals resulting from the horseshoe prior in the sparse multivariate normal means model. We do so in an adaptive setting without assuming knowledge of the sparsity level (number of signals). We consider both the hierarchical Bayes method of putting a prior on the unknown sparsity level and the empirical Bayes method with the sparsity level estimated by maximum marginal likelihood. We show that credible balls and marginal credible intervals have good frequentist coverage and optimal size if the sparsity level of the prior is set correctl…
We investigate the frequentist coverage properties of credible sets resulting in from Gaussian process priors with squared exponential covariance kernel. First we show that by selecting the scaling hyper-parameter using the maximum marginal likelihood estimator in the (slightly modified) squared exponential covariance kernel the corresponding credible sets will provide overconfident, misleading uncertainty statements for a large, representative subclass of the functional parameters in context of the Gaussian white noise model. Then we show that by either blowing up the credible sets with a log…
In the setting of nonparametric multivariate regression with unknown error variance, we study asymptotic properties of a Bayesian method for estimating a regression function f and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for f, and the error variance is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, L2 and supremum norm posterior contraction rates for f and its mixed partial derivatives, and show that they coinc…
We study a Bayesian approach to recovering the initial condition for the heat equation from noisy observations of the solution at a later time. We consider a class of prior distributions indexed by a parameter quantifying "smoothness" and show that the corresponding posterior distributions contract around the true parameter at a rate that depends on the smoothness of the true initial condition and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the optimal minimax rate. One type of priors leads to a rate-adaptive Bayesian procedure. The frequentist …
We consider the problem of constructing Bayesian based confidence sets for linear functionals in the inverse Gaussian white noise model. We work with a scale of Gaussian priors indexed by a regularity hyper-parameter and apply the data-driven (slightly modified) marginal likelihood empirical Bayes method for the choice of this hyper-parameter. We show by theory and simulations that the credible sets constructed by this method have sub-optimal behaviour in general. However, by assuming "self-similarity" the credible sets have rate-adaptive size and optimal coverage. As an application of these r…
We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance parameters can be finite-, high-, or infinite-dimensional. A mixture of point masses at zero and continuous distributions is used for the prior distribution on sparse regression coefficients, and appropriate prior distributions are used for nuisance parameters. The optimal posterior contraction of sparse regression coefficients, hampered by the presence of nuisance parameters, is also examined and discussed. It is shown that t…
In a general class of Bayesian nonparametric models, we prove that the posterior distribution can be asymptotically approximated by a Gaussian process. Our results apply to nonparametric exponential family that contains both Gaussian and non-Gaussian regression, and also hold for both efficient (root-n) and inefficient (non root-n) estimation. Our general approximation theorem does not rely on posterior conjugacy, and can be verified in a class of Gaussian process priors that has a smoothing spline interpretation [59, 44]. In particular, the limiting posterior measure becomes prior-free under …
Consider a linear regression model with regression parameter beta and normally distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where c and t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri, 2009, JSPI, describe a new frequentist 1-alpha confidence interval for theta that utilizes this uncertain prior information. We compare this confidence interval with Bayesian 1-alpha equi-tailed and shortest credible in…
An advantage of methods that base inference on a posterior distribution is that credible regions are readily obtained. Except in well-specified situations, however, there is no guarantee that such regions will achieve the nominal frequentist coverage probability, even approximately. To overcome this difficulty, we propose a general strategy that introduces an additional scalar tuning parameter to control the posterior spread, and we develop an algorithm that chooses this parameter so that the corresponding credible region achieves the nominal coverage probability.
The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the pr…
For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-α)%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the restricted parameter space. Namely, they establish the attractive lower bound of $\frac{1-α}{1+α}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-α}{1+α}$ is applicable for a substantially more general setting with underlying distributional symmetry, and obtain various other properties. The deriv…
We propose a new, two-step empirical Bayes-type of approach for neural networks. We show in context of the nonparametric regression model that the procedure (up to a logarithmic factor) provides optimal recovery of the underlying functional parameter of interest and provides Bayesian credible sets with frequentist coverage guarantees. The approach requires fitting the neural network only once, hence it is substantially faster than Bootstrapping type approaches. We demonstrate the applicability of our method over synthetic data, observing good estimation properties and reliable uncertainty quan…
We investigate Bernstein-von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a vi…
Reacting against the limitation of statistics to decision procedures, R. A. Fisher proposed for inductive reasoning the use of the fiducial distribution, a parameter-space distribution of epistemological probability transferred directly from limiting relative frequencies rather than computed according to the Bayes update rule. The proposal is developed as follows using the confidence measure of a scalar parameter of interest. (With the restriction to one-dimensional parameter space, a confidence measure is essentially a fiducial probability distribution free of complications involving ancillar…
Many recently developed Bayesian methods have focused on sparse signal detection. However, much less work has been done addressing the natural follow-up question: how to make valid inferences for the magnitude of those signals after selection. Ordinary Bayesian credible intervals suffer from selection bias, owing to the fact that the target of inference is chosen adaptively. Existing Bayesian approaches for correcting this bias produce credible intervals with poor frequentist properties, while existing frequentist approaches require sacrificing the benefits of shrinkage typical in Bayesian met…
The primary objective of this paper is to revisit and make a case for the merits of R.A. Fisher's objections to the decision-theoretic framing of frequentist inference. It is argued that this framing is congruent with the Bayesian but incongruent with the frequentist inference. It provides the Bayesian approach with a theory of optimal inference, but it misrepresents the theory of optimal frequentist inference by framing inferences solely in terms of the universal quantifier `for all values of theta in the parameter space'. This framing is at odds with the primary objective of model-based freq…
The prediction interval has been increasingly used in meta-analyses as a useful measure for assessing the magnitude of treatment effect and between-studies heterogeneity. In calculations of the prediction interval, although the Higgins-Thompson-Spiegelhalter method is used most often in practice, it might not have adequate coverage probability for the true treatment effect of a future study under realistic situations. An effective alternative candidate is the Bayesian prediction interval, which has also been widely used in general prediction problems. However, these prediction intervals are co…
In statistical practice, whether a Bayesian or frequentist approach is used in inference depends not only on the availability of prior information but also on the attitude taken toward partial prior information, with frequentists tending to be more cautious than Bayesians. The proposed framework defines that attitude in terms of a specified amount of caution, thereby enabling data analysis at the level of caution desired and on the basis of any prior information. The caution parameter represents the attitude toward partial prior information in much the same way as a loss function represents th…
We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally $β$-Hölder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under…
We develop a set of scalable Bayesian inference procedures for a general class of nonparametric regression models. Specifically, nonparametric Bayesian inferences are separately performed on each subset randomly split from a massive dataset, and then the obtained local results are aggregated into global counterparts. This aggregation step is explicit without involving any additional computation cost. By a careful partition, we show that our aggregated inference results obtain an oracle rule in the sense that they are equivalent to those obtained directly from the entire data (which are computa…
Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be understood only very recently. In this work, we continue the theoretical investigation of BART initiated by Rockova and van der Pas (2017). In particular, we study the Bernstein-von Mises (BvM) phenomenon (i.e. asymptotic normality) for smooth linear functionals of the regression surface within the framework of non-parametric regression with fixed covar…
In this paper, we consider Bayesian inference on a class of multivariate median and the multivariate quantile functionals of a joint distribution using a Dirichlet process prior. Since, unlike univariate quantiles, the exact posterior distribution of multivariate median and multivariate quantiles are not obtainable explicitly, we study these distributions asymptotically. We derive a Bernstein-von Mises theorem for the multivariate $\ell_1$-median with respect to general $\ell_p$-norm, which in particular shows that its posterior concentrates around its true value at $n^{-1/2}$-rate and its cre…
We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply as devices for producing frequentist prediction regions. Here we analyse the $O(n^{-1})$ term in the expansion of the coverage probability of a Bayesian prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike the situation for parametric matching, asymptotic predictive matching priors may depend on the level $α$. We investigate uniformly predictive match…
This paper develops some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, and the ratio of the conditional variance of one variable given the other to the marginal variance of the other variable. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. The paper uses various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior densit…
