Bayesian methods are useful for statistical inference. However, real-world problems can be challenging using Bayesian methods when the data analyst has only limited prior knowledge. In this paper we consider a class of problems, called Partial Bayes problems, in which the prior information is only partially available. Taking the recently proposed Inferential Model approach, we develop a general inference framework for Partial Bayes problems, and derive both exact and efficient solutions. In addition to the theoretical investigation, numerical results and real applications are used to demonstrate the superior performance of the proposed method.
Unlike standard linear regression, quantile regression captures the relationship between covariates and the conditional response distribution as a whole, rather than only the relationship between covariates and the expected value of the conditional response. However, while there are well-established quantile regression methods for continuous variables and some forms of discrete data, there is no widely accepted method for ordinal variables, despite their importance in many medical contexts. In this work, we describe two existing ordinal quantile regression methods and demonstrate their weaknes…
We generalize the results of \cite{SPU, SJPU} by showing how the Gaussian aggregator may be computed in a setting where parameter estimation is not required. We proceed to provide an explicit formula for a "one-shot" aggregation problem with two forecasters.
In causal estimation problems, the parameter of interest is often only partially identified, implying that the parameter cannot be recovered exactly, even with infinite data. Here, we study Bayesian inference for partially identified treatment effects in multi-treatment causal inference problems with unobserved confounding. In principle, inferring the partially identified treatment effects is natural under the Bayesian paradigm, but the results can be highly sensitive to parameterization and prior specification, often in surprising ways. It is thus essential to understand which aspects of the …
We study the problem of sequential experimental design to estimate the parametric component of a partially linear model with a Gaussian process prior. We consider an active learning setting where an experimenter adaptively decides which data to collect to achieve their goal efficiently. The experimenter's goals may vary, such as reducing the classification error probability or improving the accuracy of estimating the parameters of the data generating process. This study aims to improve the accuracy of estimating the parametric component of a partially linear model. Under some assumptions, the …
We consider the problem of model choice for stochastic epidemic models given partial observation of a disease outbreak through time. Our main focus is on the use of Bayes factors. Although Bayes factors have appeared in the epidemic modelling literature before, they can be hard to compute and little attention has been given to fundamental questions concerning their utility. In this paper we derive analytic expressions for Bayes factors given complete observation through time, which suggest practical guidelines for model choice problems. We extend the power posterior method for computing Bayes …
Following an extensive simulation study comparing the operating characteristics of three different procedures used for establishing equivalence (the frequentist "TOST", the Bayesian "HDI-ROPE", and the Bayes factor interval null procedure), Linde et al. (2021) conclude with the recommendation that "researchers rely more on the Bayes factor interval null approach for quantifying evidence for equivalence." We redo the simulation study of Linde et al. (2021) in its entirety but with the different procedures calibrated to have the same predetermined maximum type 1 error rate. Our results suggest t…
Preferential sampling is a common feature in geostatistics and occurs when the locations to be sampled are chosen based on information about the phenomena under study. In this case, point pattern models are commonly used as the probability law for the distribution of the locations. However, analytic intractability of the point process likelihood prevents its direct calculation. Many Bayesian (and non-Bayesian) approaches in non-parametric model specifications handle this difficulty with approximation-based methods. These approximations involve errors that are difficult to quantify and can lead…
We present a new functional Bayes classifier that uses principal component (PC) or partial least squares (PLS) scores from the common covariance function, that is, the covariance function marginalized over groups. When the groups have different covariance functions, the PC or PLS scores need not be independent or even uncorrelated. We use copulas to model the dependence. Our method is semiparametric; the marginal densities are estimated nonparametrically by kernel smoothing and the copula is modeled parametrically. We focus on Gaussian and t-copulas, but other copulas could be used. The strong…
Approximate Bayesian Computation (ABC) has become increasingly prominent as a method for conducting parameter inference in a range of challenging statistical problems, most notably those characterized by an intractable likelihood function. In this paper, we focus on the use of ABC not as a tool for parametric inference, but as a means of generating probabilistic forecasts; or for conducting what we refer to as `approximate Bayesian forecasting'. The four key issues explored are: i) the link between the theoretical behavior of the ABC posterior and that of the ABC-based predictive; ii) the use …
Partial Ordering Bayesian Logistic Regression Model for Phase I Combination Trials and Computationally Efficient Approach to Operational Prior Specification
MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计学单元）
We provide a flexible framework for selecting among a class of additive partial linear models that allows both linear and nonlinear additive components. In practice, it is challenging to determine which additive components should be excluded from the model while simultaneously determining whether nonzero additive components should be represented as linear or non-linear components in the final model. In this paper, we propose a Bayesian model selection method that is facilitated by a carefully specified class of models, including the choice of a prior distribution and the nonparametric model us…
The l1-regularization is very popular in high dimensional statistics -- it changes a combinatorial problem of choosing which subset of the parameter are zero, into a simple continuous optimization. Using a continuous prior concentrated near zero, the Bayesian counterparts are successful in quantifying the uncertainty in the variable selection problems; nevertheless, the lack of exact zeros makes it difficult for broader problems such as the change-point detection and rank selection. Inspired by the duality of the l1-regularization as a constraint onto an l1-ball, we propose a new prior by proj…
This paper proposes a new methodology to perform Bayesian inference for a class of multidimensional Cox processes in which the intensity function is piecewise constant. Poisson processes with piecewise constant intensity functions are believed to be suitable to model a variety of point process phenomena and, given its simpler structure, are expected to provide more precise inference when compared to processes with non-parametric and continuously varying intensity functions. The partition of the space domain is flexibly determined by a level-set function of a latent Gaussian process. Despite th…
Statistical modeling of point patterns is an important and common problem in several areas. The Poisson process is the most common process used for this purpose, in particular, its generalization that considers the intensity function to be stochastic. This is called a Cox process and different choices to model the dynamics of the intensity gives rise to a wide range of models. We present a new class of unidimensional Cox process models in which the intensity function assumes parametric functional forms that switch among them according to a continuous-time Markov chain. A novel methodology is p…
We explore various Bayesian approaches to estimate partial Gaussian graphical models. Our hierarchical structures enable to deal with single-output as well as multiple-output linear regressions, in small or high dimension, enforcing either no sparsity, sparsity, group sparsity or even sparse-group sparsity for a bi-level selection through partial correlations (direct links) between predictors and responses, thanks to spike-and-slab priors corresponding to each setting. Adaptative and global shrinkages are also incorporated in the Bayesian modeling of the direct links. An existing result for mo…
This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously est…
In this paper we consider Bayesian parameter inference for partially observed fractional Brownian motion (fBM) models. The approach we follow is to time-discretize the hidden process and then to design Markov chain Monte Carlo (MCMC) algorithms to sample from the posterior density on the parameters given data. We rely on a novel representation of the time discretization, which seeks to sample from an approximation of the posterior and then corrects via importance sampling; the approximation reduces the time (in terms of total observation time T) by O(T). This method is extended by using a mult…
We present new methodologies for Bayesian inference on the rate parameters of a discretely observed continuous-time Markov jump processes with a countably infinite state space. The usual method of choice for inference, particle Markov chain Monte Carlo (particle MCMC), struggles when the observation noise is small. We consider the most challenging regime of exact observations and provide two new methodologies for inference in this case: the minimal extended state space algorithm (MESA) and the nearly minimal extended state space algorithm (nMESA). By extending the Markov chain Monte Carlo stat…
We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal-gamma prior distribution. In this context, we exhibit a closed-form expression of the marginal likelihood which allows to infer an optimal number of components. We also propose a heuristic based on the expected shape of the marginal likelihood curve in order to choose the hyperparameters. In non-asymptotic frameworks, we show on simulated data that this exac…
The cyclical and heterogeneous nature of many substance use disorders highlights the need to adapt the type or the dose of treatment to accommodate the specific and changing needs of individuals. The Adaptive Treatment for Alcohol and Cocaine Dependence study (ENGAGE) is a multi-stage randomized trial that aimed to provide longitudinal data for constructing treatment strategies to improve patients' engagement in therapy. However, the high rate of noncompliance and lack of analytic tools to account for noncompliance have impeded researchers from using the data to achieve the main goal of the tr…
We consider exact algorithms for Bayesian inference with model selection priors (including spike-and-slab priors) in the sparse normal sequence model. Because the best existing exact algorithm becomes numerically unstable for sample sizes over n=500, there has been much attention for alternative approaches like approximate algorithms (Gibbs sampling, variational Bayes, etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO) or empirical Bayesian methods. However, by introducing algorithmic ideas from online sequential prediction, we show that exact calculations are feas…
Computer models are widely used across a range of scientific disciplines to describe various complex physical systems, however to perform full uncertainty quantification we often need to employ emulators. An emulator is a fast statistical construct that mimics the slow to evaluate computer model, and greatly aids the vastly more computationally intensive uncertainty quantification calculations that an important scientific analysis often requires. We examine the problem of emulating computer models that possess multiple, partial discontinuities occurring at known non-linear location. We introdu…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse problems whilst accounting, in a statistically principled way, for the impact of discretisation error due to numerical solution of the PDE. In particular, the approach confers robustness to failure of the numerical PDE solver, with statistical inferences driven to be more conservative in the presence of substantial discretisation error. Going further, the p…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. Th…
We develop a new Bayesian modelling framework for the class of higher-order, variable-memory Markov chains, and introduce an associated collection of methodological tools for exact inference with discrete time series. We show that a version of the context tree weighting algorithm can compute the prior predictive likelihood exactly (averaged over both models and parameters), and two related algorithms are introduced, which identify the a posteriori most likely models and compute their exact posterior probabilities. All three algorithms are deterministic and have linear-time complexity. A family…
