Observational healthcare data offer the potential to estimate causal effects of medical products on a large scale. However, the confidence intervals and p-values produced by observational studies only account for random error and fail to account for systematic error. As a consequence, operating characteristics such as confidence interval coverage and Type I error rates often deviate sharply from their nominal values and render interpretation impossible. While there is longstanding awareness of systematic error in observational studies, analytic approaches to empirically account for systematic error are relatively new. Several authors have proposed approaches using negative controls (also known as "falsification hypotheses") and positive controls. The basic idea is to adjust confidence intervals and p-values in light of the bias (if any) detected in the analyses of the negative and positive control. In this work, we propose a Bayesian statistical procedure for posterior interval calibration that uses negative and positive controls. We show that the posterior interval calibration procedure restores nominal characteristics, such as 95% coverage of the true effect size by the 95% posterior interval.
Many large-scale machine learning problems involve estimating an unknown parameter $θ_{i}$ for each of many items. For example, a key problem in sponsored search is to estimate the click through rate (CTR) of each of billions of query-ad pairs. Most common methods, though, only give a point estimate of each $θ_{i}$. A posterior distribution for each $θ_{i}$ is usually more useful but harder to get. We present a simple post-processing technique that takes point estimates or scores $t_{i}$ (from any method) and estimates an approximate posterior for each $θ_{i}$. We build on the idea of calibrat…
In this paper, we aim to design robust estimation techniques based on the compound-Gaussian (CG) process and adapted for calibration of radio interferometers. The motivation beyond this is due to the presence of outliers leading to an unrealistic traditional Gaussian noise assumption. Consequently, to achieve robustness, we adopt a maximum a posteriori (MAP) approach which exploits Bayesian statistics and follows a sequential updating procedure here. The proposed algorithm is applied in a multi-frequency scenario in order to enhance the estimation and correction of perturbation effects. Numeri…
We study posterior contraction behaviors for parameters of interest in the context of Bayesian mixture modeling, where the number of mixing components is unknown while the model itself may or may not be correctly specified. Two representative types of prior specification will be considered: one requires explicitly a prior distribution on the number of mixture components, while the other places a nonparametric prior on the space of mixing distributions. The former is shown to yield an optimal rate of posterior contraction on the model parameters under minimal conditions, while the latter can be…
Evaluating Posterior Probabilities: Decision Theory, Proper Scoring Rules, and Calibration
Instituto de Ciencias de la Computación（计算机科学研究所）；Universidad de Buenos Aires - CONICET（布宜诺斯艾利斯大学 - 阿根廷国家科学与技术研究理事会）；Escuela Politécnica Superior, Universidad Autónoma de Madrid（马德里自治大学高等理工学院）
The current standard Bayesian approach to model calibration, which assigns a Gaussian process prior to the discrepancy term, often suffers from issues of unidentifiability and computational complexity and instability. When the goal is to quantify uncertainty in physical parameters for extrapolative prediction, then there is no need to perform inference on the discrepancy term. With this in mind, we introduce Gibbs posteriors as an alternative Bayesian method for model calibration, which updates the prior with a loss function connecting the data to the parameter. The target of inference is the …
Molecular dynamics (MD) simulations give access to equilibrium structures and dynamic properties given an ergodic sampling and an accurate force-field. The force-field parameters are calibrated to reproduce properties measured by experiments or simulations. The main contribution of this paper is an approximate Bayesian framework for the calibration and uncertainty quantification of the force-field parameters, without assuming parameter uncertainty to be Gaussian. To this aim, since the likelihood function of the MD simulation models are intractable in absence of Gaussianity assumption, we use …
Real-world problems, often couched as machine learning applications, involve quantities of interest that have real-world meaning, independent of any statistical model. To avoid potential model misspecification bias or over-complicating the problem formulation, a direct, model-free approach is desired. The traditional Bayesian framework relies on a model for the data-generating process so, apparently, the desired direct, model-free, posterior-probabilistic inference is out of reach. Fortunately, likelihood functions are not the only means of linking data and quantities of interest. Loss functio…
PAC-Bayesian algorithms and Gibbs posteriors are gaining popularity due to their robustness against model misspecification even when Bayesian inference is inconsistent. The PAC-Bayesian alpha-posterior is a generalization of the standard Bayes posterior which can be tempered with a parameter alpha to handle inconsistency. Data driven methods for tuning alpha have been proposed but are still few, and are often computationally heavy. Additionally, the adequacy of these methods in cases where we use variational approximations instead of exact alpha-posteriors is not clear. This narrows their usag…
University College London（伦敦大学学院）；dunnhumby Ltd（邓恩汉姆比有限公司）
Understanding the shopping motivations behind market baskets has high commercial value in the grocery retail industry. Analyzing shopping transactions demands techniques that can cope with the volume and dimensionality of grocery transactional data while keeping interpretable outcomes. Latent Dirichlet Allocation (LDA) provides a suitable framework to process grocery transactions and to discover a broad representation of customers' shopping motivations. However, summarizing the posterior distribution of an LDA model is challenging, while individual LDA draws may not be coherent and cannot capt…
University of Pittsburgh（匹兹堡大学）；Carnegie Mellon University（卡内基梅隆大学）；M*Modal（M*Modal公司）
Most classification models work by first predicting a posterior probability distribution over all classes and then selecting that class with the largest estimated probability. In many settings however, the quality of posterior probability itself (e.g., 65% chance having diabetes), gives more reliable information than the final predicted class alone. When these methods are shown to be poorly calibrated, most fixes to date have relied on posterior calibration, which rescales the predicted probabilities but often has little impact on final classifications. Here we propose an end-to-end training p…
Within the last few years, there has been a move towards using statistical models in conjunction with neural networks with the end goal of being able to better answer the question, "what do our models know?". From this trend, classical metrics such as Prediction Interval Coverage Probability (PICP) and new metrics such as calibration error have entered the general repertoire of model evaluation in order to gain better insight into how the uncertainty of our model compares to reality. One important component of uncertainty modeling is model uncertainty (epistemic uncertainty), a measurement of …
Modeling the relationship between vehicle speed and density on the road is a fundamental problem in traffic flow theory. Recent research found that using the least-squares (LS) method to calibrate single-regime speed-density models is biased because of the uneven distribution of samples. This paper explains the issue of the LS method from a statistical perspective: the biased calibration is caused by the correlations/dependencies in regression residuals. Based on this explanation, we propose a new calibration method for single-regime speed-density models by modeling the covariance of residuals…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；IBM Research（IBM研究院）
Bayesian decision theory provides an elegant framework for acting optimally under uncertainty when tractable posterior distributions are available. Modern Bayesian models, however, typically involve intractable posteriors that are approximated with, potentially crude, surrogates. This difficulty has engendered loss-calibrated techniques that aim to learn posterior approximations that favor high-utility decisions. In this paper, focusing on Bayesian neural networks, we develop methods for correcting approximate posterior predictive distributions encouraging them to prefer high-utility decisions…
Microsimulation models (MSMs) are used to predict population-level effects of health care policies by simulating individual-level outcomes. Simulated outcomes are governed by unknown parameters that are chosen so that the model accurately predicts specific targets, a process referred to as model calibration. Calibration targets can come from randomized controlled trials, observational studies, and expert opinion, and are typically summary statistics. A well calibrated model can reproduce a wide range of targets. MSM calibration generally involves searching a high dimensional parameter space an…
We develop a Bayesian approach called Bayesian projected calibration to address the problem of calibrating an imperfect computer model using observational data from a complex physical system. The calibration parameter and the physical system are parametrized in an identifiable fashion via $L_2$-projection. The physical process is assigned a Gaussian process prior, which naturally induces a prior distribution on the calibration parameter through the $L_2$-projection constraint. The calibration parameter is estimated through its posterior distribution, which provides a natural and non-asymptotic…
Bayesian calibration of black-box computer models offers an established framework to obtain a posterior distribution over model parameters. Traditional Bayesian calibration involves the emulation of the computer model and an additive model discrepancy term using Gaussian processes; inference is then carried out using MCMC. These choices pose computational and statistical challenges and limitations, which we overcome by proposing the use of approximate Deep Gaussian processes and variational inference techniques. The result is a practical and scalable framework for calibration, which obtains co…
There are many practical difficulties in the calibration of computer models to experimental data. One such complication is the fact that certain combinations of the calibration inputs can cause the code to output data lacking fundamental properties, or even to produce no output at all. In many cases the researchers want or need to exclude the possibility of these "failures" within their analyses. We propose a Bayesian (meta-)model in which the posterior distribution for the calibration parameters naturally excludes regions of the input space corresponding to failed runs. That is, we define a s…
Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require $O(10^5)$ model runs and are sensitive to internal variability noise, rendering them infeasible for climate models. Here we demonstrate an approac…
This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent stationary Gaussian processes with uncertain hyper-parameters describe uncertainties of the model structure and parameters while Karhunnen-Loeve expansion is adopted to spectrally represent these Gaussian processes. The Karhunnen-Loeve expansion of a prior Gaussian process is projected on a generalized Polynomial Chaos basis, whereas intrusive Galerkin projectio…
We introduce a Bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights. Building on the work of Ranjan, R. and Gneiting, T. (2010) and Gneiting, T. and Ranjan, R. (2013), we use infinite beta mixtures for the calibration. The proposed Bayesian nonparametric approach takes advantage of the flexibility of Dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions. The inference procedu…
CP4SBI: Local Conformal Calibration of Credible Sets in Simulation-Based Inference
Department of Statistics, Federal University of São Carlos（统计系，圣卡洛斯联邦大学）；Institute of Mathematics and Computer Science, University of São Paulo（数学与计算机科学学院，圣保罗大学）；Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK（格勒诺布尔阿尔卑斯大学，法国国家信息与自动化研究所，法国国家科学研究中心，格勒诺布尔INP，LJK）
Rapid evaluation and calibration of Bayesian group sequential designs via conjugate-mixture semi-simulation
The George Institute for Global Health, Imperial College London（乔治全球健康研究院，伦敦帝国理工学院）；Imperial Clinical Trials Unit, Imperial College London（伦敦帝国理工学院临床试验中心）；School of Mathematics, University of Bristol（布里斯托大学数学学院）；The George Institute for Global Health, University of New South Wales（乔治全球健康研究院，新南威尔士大学）
School of Computing, University of Glasgow（格拉斯哥大学计算学院）；Communication Systems Department, EURECOM（EURECOM通信系统部门）；Institute for Intelligent Networked Systems, Northeastern University London（伦敦东北大学智能网络系统研究所）
adabay: an R package for rapid evaluation and calibration of Bayesian group sequential designs across common endpoint types
The George Institute for Global Health, Imperial College London（全球健康乔治研究所，伦敦帝国理工学院）；Imperial Clinical Trials Unit, Imperial College London（伦敦帝国理工学院临床试验中心）；School of Mathematics, University of Bristol（布里斯托大学数学学院）
Making good predictions of a physical system using a computer code requires the inputs to be carefully specified. Some of these inputs called control variables have to reproduce physical conditions whereas other inputs, called parameters, are specific to the computer code and most often uncertain. The goal of statistical calibration consists in estimating these parameters with the help of a statistical model which links the code outputs with the field measurements. In a Bayesian setting, the posterior distribution of these parameters is normally sampled using MCMC methods. However, they are im…
This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm scales very badly with the number of data points and lacks online learning capability. The proposed FIPO-BC algorithm greatly improves the computational efficiency and enables the online calibration by executing the calibration on a set of predefined ind…
