Inspired by the pioneering work of Rubin (1978), we employ the potential outcomes framework to develop a finite-population Bayesian causal inference framework for randomized controlled $2^K$ factorial designs with binary outcomes, which are common in medical research. As demonstrated by simulated and empirical examples, the proposed framework corrects the well-known variance over-estimation issue of the classic "Neymanian" inference framework, under various settings.
Natural image statistics exhibit hierarchical dependencies across multiple scales. Representing such prior knowledge in non-factorial latent tree models can boost performance of image denoising, inpainting, deconvolution or reconstruction substantially, beyond standard factorial "sparse" methodology. We derive a large scale approximate Bayesian inference algorithm for linear models with non-factorial (latent tree-structured) scale mixture priors. Experimental results on a range of denoising and inpainting problems demonstrate substantially improved performance compared to MAP estimation or to …
For unstructured experimental units, the minimum aberration due to Fries and Hunter (1980) is a popular criterion for choosing regular fractional factorial designs. Following which, many related studies have focused on multi-stratum factorial designs, in which multiple error terms arise from the complicated structures of experimental units. Chang and Cheng (2018) proposed a Bayesian-inspired aberration criterion for selecting multi-stratum factorial designs, which can be considered as a generalized version of that in Fries and Hunter (1980). However, they did not propose algorithms for searchi…
Factorial hidden Markov models (FHMMs) are powerful tools of modeling sequential data. Learning FHMMs yields a challenging simultaneous model selection issue, i.e., selecting the number of multiple Markov chains and the dimensionality of each chain. Our main contribution is to address this model selection issue by extending Factorized Asymptotic Bayesian (FAB) inference to FHMMs. First, we offer a better approximation of marginal log-likelihood than the previous FAB inference. Our key idea is to integrate out transition probabilities, yet still apply the Laplace approximation to emission proba…
Ordinal cumulative probability models (CPMs) -- also known as cumulative link models -- such as the proportional odds regression model are typically used for discrete ordered outcomes, but can accommodate both continuous and mixed discrete/continuous outcomes since these are also ordered. Recent papers describe ordinal CPMs in this setting using non-parametric maximum likelihood estimation. We formulate a Bayesian CPM for continuous or mixed outcome data. Bayesian CPMs inherit many of the benefits of frequentist CPMs and have advantages with regard to interpretation, flexibility, and exact inf…
Ideally, a meta-analysis will summarize data from several unbiased studies. Here we consider the less than ideal situation in which contributing studies may be compromised by measurement error. Measurement error affects every study design, from randomized controlled trials to retrospective observational studies. We outline a flexible Bayesian framework for continuous outcome data which allows one to obtain appropriate point and interval estimates with varying degrees of prior knowledge about the magnitude of the measurement error. We also demonstrate how, if individual-participant data (IPD) a…
To achieve the goal of providing the best possible care to each patient, physicians need to customize treatments for patients with the same diagnosis, especially when treating diseases that can progress further and require additional treatments, such as cancer. Making decisions at multiple stages as a disease progresses can be formalized as a dynamic treatment regime (DTR). Most of the existing optimization approaches for estimating dynamic treatment regimes including the popular method of Q-learning were developed in a frequentist context. Recently, a general Bayesian machine learning framewo…
$2^K$ factorial designs are widely adopted by statisticians and the broader scientific community. In this short note, under the potential outcomes framework (Neyman, 1923; Rubin, 1974), we adopt the partial identification approach and derive the sharp lower bound of the sampling variance of the estimated factorial effects, which leads to an "improved" Neymanian variance estimator that mitigates the over-estimation issue suffered by the classic Neymanian variance estimator by Dasgupta et al. (2015).
We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data ana…
With the arrival of the R packages nlme and lme4, linear mixed models (LMMs) have come to be widely used in experimentally-driven areas like psychology, linguistics, and cognitive science. This tutorial provides a practical introduction to fitting LMMs in a Bayesian framework using the probabilistic programming language Stan. We choose Stan (rather than WinBUGS or JAGS) because it provides an elegant and scalable framework for fitting models in most of the standard applications of LMMs. We ease the reader into fitting increasingly complex LMMs, first using a two-condition repeated measures sel…
Psychiatric and social epidemiology often involves assessing the effects of environmental exposure on outcomes that are difficult to measure directly. To address this problem, it is common to measure outcomes using a comprehensive battery of different tests thought to be related to a common, underlying construct of interest. In the application that motivates our work, for example, researchers wanted to assess the impact of in utero alcohol exposure on child cognition and neuropsychological development, which were evaluated using a range of different tests. Statistical analysis of the resulting…
The problem we consider considers estimating a multivariate longitudinal panel data model whose outcomes can be a combination of discrete and continuous variables. This problem is challenging because the likelihood is usually analytically intractable. Our article makes both a methodological contribution and also a substantive contribution to the application. The methodological contribution is to introduce into the panel data literature a particle Metropolis within Gibbs method to carry out Bayesian inference, using a Hamiltonian Monte Carlo (Neal 2011} proposal for sampling the vector of unkno…
Trial-based economic evaluations are typically performed on cross-sectional variables, derived from the responses for only the completers in the study, using methods that ignore the complexities of utility and cost data (e.g. skewness and spikes). We present an alternative and more efficient Bayesian parametric approach to handle missing longitudinal outcomes in economic evaluations, while accounting for the complexities of the data. We specify a flexible parametric model for the observed data and partially identify the distribution of the missing data with partial identifying restrictions and…
In medical research, a scenario often entertained is randomized controlled $2^2$ factorial design with a binary outcome. By utilizing the concept of potential outcomes, Dasgupta et al. (2015) proposed a randomization-based causal inference framework, allowing flexible and simultaneous estimations and inferences of the factorial effects. However, a fundamental challenge that Dasgupta et al. (2015)'s proposed methodology faces is that the sampling variance of the randomization-based factorial effect estimator is unidentifiable, rendering the corresponding classic "Neymanian" variance estimator s…
The paper presents a hierarchical Bayesian model for simultaneous inference of tournament graphs and informant error. From multiple informant reports or measurement instrument outputs, the model estimates the structure of a criterion (i.e., true) tournament graph with possibly tied outcomes. Tournament graphs with possibly tied outcomes are graphs in which there are three possible states for each unordered pair of graph nodes: node i wins and node j loses; node j wins and node i loses; neither node wins (i.e., tied outcome). The model also estimates the rates at which individual informants (or…
Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Recently those models have been investigated in a Bayesian nonparametric approach to species sampling problems as alternatives to the Dirichlet and the Pitman-Yor process priors. Here we derive marginals of conditional and unconditional multivariate distributions arising from exchangeable Gibbs partitions to obtain explicit formulas for joint falling factorial moments of corresponding c…
We propose a deep generative factor analysis model with beta process prior that can approximate complex non-factorial distributions over the latent codes. We outline a stochastic EM algorithm for scalable inference in a specific instantiation of this model and present some preliminary results.
The goal of causal inference is to understand the outcome of alternative courses of action. However, all causal inference requires assumptions. Such assumptions can be more influential than in typical tasks for probabilistic modeling, and testing those assumptions is important to assess the validity of causal inference. We develop model criticism for Bayesian causal inference, building on the idea of posterior predictive checks to assess model fit. Our approach involves decomposing the problem, separately criticizing the model of treatment assignments and the model of outcomes. Conditioned on …
We focus on causal inference for longitudinal treatments, where units are assigned to treatments at multiple time points, aiming to assess the effect of different treatment sequences on an outcome observed at a final point. A common assumption in similar studies is Sequential Ignorability (SI): treatment assignment at each time point is assumed independent of future potential outcomes given past observed outcomes and covariates. SI is questionable when treatment participation depends on individual choices, and treatment assignment may depend on unobservable quantities associated with future ou…
Joint modeling of spatially-oriented dependent variables is commonplace in the environmental sciences, where scientists seek to estimate the relationships among a set of environmental outcomes accounting for dependence among these outcomes and the spatial dependence for each outcome. Such modeling is now sought for massive data sets with variables measured at a very large number of locations. Bayesian inference, while attractive for accommodating uncertainties through hierarchical structures, can become computationally onerous for modeling massive spatial data sets because of its reliance on i…
This paper proposes a flexible Bayesian approach to multiple imputation using conditional Gaussian mixtures. We introduce novel shrinkage priors for covariate-dependent mixing proportions in the mixture models to automatically select the suitable number of components used in the imputation step. We develop an efficient sampling algorithm for posterior computation and multiple imputation via Markov Chain Monte Carlo methods. The proposed method can be easily extended to the situation where the data contains not only continuous variables but also discrete variables such as binary and count value…
Clinical outcome prediction from high-dimensional data is problematic in the common setting where there is only a relatively small number of samples. The imbalance causes data overfitting, and outcome prediction becomes computationally expensive or even impossible. We propose a Bayesian outcome prediction method that can be applied to data of arbitrary dimension d, from 2 outcome classes, and reduces overfitting without any approximations at parameter level. This is achieved by avoiding numerical integration or approximation, and solving the Bayesian integrals analytically. We thereby reduce t…
Humans are routinely exposed to mixtures of chemical and other environmental factors, making the quantification of health effects associated with environmental mixtures a critical goal for establishing environmental policy sufficiently protective of human health. The quantification of the effects of exposure to an environmental mixture poses several statistical challenges. It is often the case that exposure to multiple pollutants interact with each other to affect an outcome. Further, the exposure-response relationship between an outcome and some exposures, such as some metals, can exhibit com…
Optimal treatment regimes (OTR) are individualised treatment assignment strategies that identify a medical treatment as optimal given all background information available on the individual. We discuss Bayes optimal treatment regimes estimated using a loss function defined on the bivariate distribution of dichotomous potential outcomes. The proposed approach allows considering more general objectives for the OTR than maximization of an expected outcome (e.g., survival probability) by taking into account, for example, unnecessary treatment burden. As a motivating example we consider the case of …
Greater understanding of the pathways through which an environmental mixture operates is important to design effective interventions. We present new methodology to estimate natural direct and indirect effects and controlled direct effects of a complex mixture exposure on an outcome through a mediator variable. We implement Bayesian Kernel Machine Regression (BKMR) to allow for all possible interactions and nonlinear effects of (1) the co-exposures on the mediator, (2) the co-exposures and mediator on the outcome, and (3) selected covariates on the mediator and/or outcome. From the posterior pr…
Recent advances in bioinformatics have made high-throughput microbiome data widely available, and new statistical tools are required to maximize the information gained from these data. For example, analysis of high-dimensional microbiome data from designed experiments remains an open area in microbiome research. Contemporary analyses work on metrics that summarize collective properties of the microbiome, but such reductions preclude inference on the fine-scale effects of environmental stimuli on individual microbial taxa. Other approaches model the proportions or counts of individual taxa as r…
Longitudinal Outcomes Truncated by Death: Causal Estimands and Bayesian Estimators
Amsterdam UMC（阿姆斯特丹大学医学中心）；Singapore University of Technology and Design（新加坡科技设计大学）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）
One of the main goals of sequential, multiple assignment, randomized trials (SMART) is to find the most efficacious design embedded dynamic treatment regimes. The analysis method known as multiple comparisons with the best (MCB) allows comparison between dynamic treatment regimes and identification of a set of optimal regimes in the frequentist setting for continuous outcomes, thereby, directly addressing the main goal of a SMART. In this paper, we develop a Bayesian generalization to MCB for SMARTs with binary outcomes. Furthermore, we show how to choose the sample size so that the inferior e…
We consider the design and analysis of multi-factor experiments using fractional factorial and incomplete designs within the potential outcome framework. These designs are particularly useful when limited resources make running a full factorial design infeasible. We connect our design-based methods to standard regression methods. We further motivate the usefulness of these designs in multi-factor observational studies, where certain treatment combinations may be so rare that there are no measured outcomes in the observed data corresponding to them. Therefore, conceptualizing a hypothetical fra…
Under the potential outcomes framework, we introduce matched-pair factorial designs, and propose the matched-pair estimator of the factorial effects. We also calculate the randomization-based covariance matrix of the matched-pair estimator, and provide the "Neymanian" estimator of the covariance matrix.
Robustly estimating heterogeneity in factorial data using Rashomon Partitions
Department of Statistics, University of Washington, USA（美国华盛顿大学统计学系）；Department of Economics, Stanford University, USA（美国斯坦福大学经济学系）；J-PAL, NBER, USA（美国J-PAL和NBER）；Department of Sociology, University of Washington, USA（美国华盛顿大学社会学系）
We consider Bayesian optimization of expensive-to-evaluate experiments that generate vector-valued outcomes over which a decision-maker (DM) has preferences. These preferences are encoded by a utility function that is not known in closed form but can be estimated by asking the DM to express preferences over pairs of outcome vectors. To address this problem, we develop Bayesian optimization with preference exploration, a novel framework that alternates between interactive real-time preference learning with the DM via pairwise comparisons between outcomes, and Bayesian optimization with a learne…
