Informative Bayesian Neural Network Priors for Weak Signals
Aalto University（阿尔托大学）；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Finnish Institute for Health and Welfare（芬兰健康与福利研究所）；Institute for Molecular Medicine Finland（芬兰分子医学研究所）；University of Manchester（曼彻斯特大学）
Tianyu Cui, Aki Havulinna, Pekka Marttinen, Samuel Kaski
Encoding domain knowledge into the prior over the high-dimensional weight space of a neural network is challenging but essential in applications with limited data and weak signals. Two types of domain knowledge are commonly available in scientific applications: 1. feature sparsity (fraction of features deemed relevant); 2. signal-to-noise ratio, quantified, for instance, as the proportion of variance explained (PVE). We show how to encode both types of domain knowledge into the widely used Gaussian scale mixture priors with Automatic Relevance Determination. Specifically, we propose a new joint prior over the local (i.e., feature-specific) scale parameters that encodes knowledge about feature sparsity, and a Stein gradient optimization to tune the hyperparameters in such a way that the distribution induced on the model's PVE matches the prior distribution. We show empirically that the new prior improves prediction accuracy, compared to existing neural network priors, on several publicly available datasets and in a genetics application where signals are weak and sparse, often outperforming even computationally intensive cross-validation for hyperparameter tuning.
We discuss Bayesian inference for parameters selected using the data. First, we provide a critical analysis of the existing positions in the literature regarding the correct Bayesian approach under selection. Second, we propose two types of non-informative priors for selection models. These priors may be employed to produce a posterior distribution in the absence of prior information as well as to provide well-calibrated frequentist inference for the selected parameter. We test the proposed priors empirically in several scenarios.
Using a collection of simulated an real benchmarks, we compare Bayesian and frequentist regularization approaches under a low informative constraint when the number of variables is almost equal to the number of observations on simulated and real datasets. This comparison includes new global noninformative approaches for Bayesian variable selection built on Zellner's g-priors that are similar to Liang et al. (2008). The interest of those calibration-free proposals is discussed. The numerical experiments we present highlight the appeal of Bayesian regularization methods, when compared with non-B…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Tübingen（图宾根大学）；RWTH Aachen University（亚琛工业大学）；IAV GmbH（IAV有限责任公司）
Reinforcement learning (RL) aims to find an optimal policy by interaction with an environment. Consequently, learning complex behavior requires a vast number of samples, which can be prohibitive in practice. Nevertheless, instead of systematically reasoning and actively choosing informative samples, policy gradients for local search are often obtained from random perturbations. These random samples yield high variance estimates and hence are sub-optimal in terms of sample complexity. Actively selecting informative samples is at the core of Bayesian optimization, which constructs a probabilisti…
The normal-normal hierarchical model (NNHM) constitutes a simple and widely used framework for meta-analysis. In the common case of only few studies contributing to the meta-analysis, standard approaches to inference tend to perform poorly, and Bayesian meta-analysis has been suggested as a potential solution. The Bayesian approach, however, requires the sensible specification of prior distributions. While non-informative priors are commonly used for the overall mean effect, the use of weakly informative priors has been suggested for the heterogeneity parameter, in particular in the setting of…
University of Adelaide（阿德莱德大学）；University of Liverpool（利物浦大学）
Deep learning models have demonstrated outstanding performance in several problems, but their training process tends to require immense amounts of computational and human resources for training and labeling, constraining the types of problems that can be tackled. Therefore, the design of effective training methods that require small labeled training sets is an important research direction that will allow a more effective use of resources.Among current approaches designed to address this issue, two are particularly interesting: data augmentation and active learning. Data augmentation achieves t…
Motivated by the analysis of the distribution of university grades, which is usually asymmetric, we discuss two informative priors for the shape parameter of the skew-normal distribution, showing that they lead to closed-form full-conditional posterior distributions, particularly useful in MCMC computation. Gibbs sampling algorithms are discussed for the joint vector of parameters, given independent prior distributions for the location and scale parameters. Simulation studies are performed to assess the performance of Gibbs samplers and to compare the choice of informative priors against a non…
The aim of this work is to study the problem of prior elicitation for the Mallows model with Spearman's distance, a popular distance-based model for rankings or permutation data. Previous Bayesian inference for such model has been limited to the use of the uniform prior over the space of permutations. We present a novel strategy to elicit subjective prior beliefs on the location parameter of the model, discussing the interpretation of hyper-parameters and the implication of prior choices for the posterior analysis.
Unit-level models for survey data offer many advantages over their area-level counterparts, such as potential for more precise estimates and a natural benchmarking property. However two main challenges occur in this context: accounting for an informative survey design and handling non-Gaussian data types. The pseudo-likelihood approach is one solution to the former, and conjugate multivariate distribution theory offers a solution to the latter. By combining these approaches, we attain a unit-level model for count data that accounts for informative sampling designs and includes fully Bayesian m…
ALINE: Joint Amortization for Bayesian Inference and Active Data Acquisition
ELLIS Institute Finland（芬兰ELLIS研究所）；Department of Computer Science, Aalto University, Finland（芬兰阿尔托大学计算机科学系）；Department of Computer Science, University of Helsinki, Finland（芬兰赫尔辛基大学计算机科学系）；Department of Computer Science, University of Manchester, UK（英国曼彻斯特大学计算机科学系）
In the case of informative sampling the sampling scheme explicitly or implicitly depends on the response variable. As a result, the sample distribution of response variable can- not be used for making inference about the population. In this research I investigate the problem of informative sampling from the Bayesian perspective. Application of the Bayesian approach permits solving the problems, which arise due to complexity of the models, being used for handling informative sampling. The main objective of the re- search is to combine the elements of the classical sampling theory and Bayesian a…
Computational convenience has led to widespread use of Bayesian inference with vague or flat priors to analyze state-space models in ecology. Vague priors are claimed to be objective and to let the data speak. Neither of these claims is valid. Statisticians have criticized the use of vague priors from philosophical to computational to pragmatic reasons. Ecologists, however, dismiss such criticisms as empty philosophical wonderings with no practical implications. We illustrate that use of vague priors in population viability analysis and occupancy models can have significant impact on the analy…
We investigate Bayesian predictive inference for finite population quantities when there are unequal probabilities of selection. Only limited information about the sample design is available; i.e., only the first-order selection probabilities corresponding to the sample units are known. Our methodology, unlike that of Chambers, Dorfman and Wang (1998), can be used to make inference for finite population quantities and provides measures of precision and intervals. Moreover, our methodology, using Markov chain Monte Carlo methods, avoids the necessity of using asymptotic closed form approximatio…
Additive Bayesian networks are types of graphical models that extend the usual Bayesian generalized linear model to multiple dependent variables through the factorisation of the joint probability distribution of the underlying variables. When fitting an ABN model, the choice of the prior of the parameters is of crucial importance. If an inadequate prior - like a too weakly informative one - is used, data separation and data sparsity lead to issues in the model selection process. In this work a simulation study between two weakly and a strongly informative priors is presented. As weakly informa…
Statistical estimates from survey samples have traditionally been obtained via design-based estimators. In many cases, these estimators tend to work well for quantities such as population totals or means, but can fall short as sample sizes become small. In today's "information age," there is a strong demand for more granular estimates. To meet this demand, using a Bayesian pseudo-likelihood, we propose a computationally efficient unit-level modeling approach for non-Gaussian data collected under informative sampling designs. Specifically, we focus on binary and multinomial data. Our approach i…
We study the accuracy of Bayesian supervised method used to cluster individuals into genetically homogeneous groups on the basis of dominant or codominant molecular markers. We provide a formula relating an error criterion the number of loci used and the number of clusters. This formula is exact and holds for arbitrary number of clusters and markers. Our work suggests that dominant markers studies can achieve an accuracy similar to that of codominant markers studies if the number of markers used in the former is about 1.7 times larger than in the latter.
Bayesian analysis is increasingly popular for use in social science and other application areas where the data are observations from an informative sample. An informative sampling design leads to inclusion probabilities that are correlated with the response variable of interest. Model inference performed on the observed sample taken from the population will be biased for the population generative model under informative sampling since the balance of information in the sample data is different from that for the population. Typical approaches to account for an informative sampling design under B…
While the choice of prior is one of the most critical parts of the Bayesian inference workflow, recent Bayesian deep learning models have often fallen back on vague priors, such as standard Gaussians. In this review, we highlight the importance of prior choices for Bayesian deep learning and present an overview of different priors that have been proposed for (deep) Gaussian processes, variational autoencoders, and Bayesian neural networks. We also outline different methods of learning priors for these models from data. We hope to motivate practitioners in Bayesian deep learning to think more c…
In this paper, the panel count data analysis for recurrent events is considered. Such analysis is useful for studying tumor or infection recurrences in both clinical trial and observational studies. A bivariate Gaussian Cox process model is proposed to jointly model the observation process and the recurrent event process. Bayesian nonparametric inference is proposed for simultaneously estimating regression parameters, bivariate frailty effects and baseline intensity functions. Inference is done through Markov chain Monte Carlo, with fully developed computational techniques. Predictive inferenc…
Functional data are often extremely high-dimensional and exhibit strong dependence structures but can often prove valuable for both prediction and inference. The literature on functional data analysis is well developed; however, there has been very little work involving functional data in complex survey settings. Motivated by physical activity monitor data from the National Health and Nutrition Examination Survey (NHANES), we develop a Bayesian model for functional covariates that can properly account for the survey design. Our approach is intended for non-Gaussian data and can be applied in m…
McGill University（麦吉尔大学）；University of Alberta（阿尔伯塔大学）
Bayesian priors offer a compact yet general means of incorporating domain knowledge into many learning tasks. The correctness of the Bayesian analysis and inference, however, largely depends on accuracy and correctness of these priors. PAC-Bayesian methods overcome this problem by providing bounds that hold regardless of the correctness of the prior distribution. This paper introduces the first PAC-Bayesian bound for the batch reinforcement learning problem with function approximation. We show how this bound can be used to perform model-selection in a transfer learning scenario. Our empirical …
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…
In recent years, Bayesian statistics methods in neuroscience have been showing important advances. In particular, detection of brain signals for studying the complexity of the brain is an active area of research. Functional magnetic resonance imagining (fMRI) is an important tool to determine which parts of the brain are activated by different types of physical behavior. According to recent results there is evidence that the values of the connectivity brain signal parameters are close to zero and due to the nature of time series fMRI data with high frequency behavior, Bayesian dynamic models f…
Bayesian estimation is increasingly popular for performing model based inference to support policymaking. These data are often collected from surveys under informative sampling designs where subject inclusion probabilities are designed to be correlated with the response variable of interest. Sampling weights constructed from marginal inclusion probabilities are typically used to form an exponentiated pseudo likelihood that adjusts the population likelihood for estimation on the sample due to ease-of-estimation. We propose an alternative adjustment based on a Bayes rule construction that simult…
Bayesian Semiparametric Model for Sequential Treatment Decisions with Informative Timing
Brown University（布朗大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Southern California（南加州大学）；Children’s Hospital of Philadelphia（费城儿童医院）；Rutgers University（罗格斯大学）
Centre for Wireless Communication, University of Oulu（奥卢大学无线通信中心）
This article deals with the problem of distributed machine learning, in which agents update their models based on their local datasets, and aggregate the updated models collaboratively and in a fully decentralized manner. In this paper, we tackle the problem of information heterogeneity arising in multi-agent networks where the placement of informative agents plays a crucial role in the learning dynamics. Specifically, we propose BayGo, a novel fully decentralized joint Bayesian learning and graph optimization framework with proven fast convergence over a sparse graph. Under our framework, age…
When random effects are correlated with sample design variables, the usual approach of employing individual survey weights (constructed to be inversely proportional to the unit survey inclusion probabilities) to form a pseudo-likelihood no longer produces asymptotically unbiased inference. We construct a weight-exponentiated formulation for the random effects distribution that achieves unbiased inference for generating hyperparameters of the random effects. We contrast our approach with frequentist methods that rely on numerical integration to reveal that only the Bayesian method achieves both…
Survey data are often collected under multistage sampling designs where units are binned to clusters that are sampled in a first stage. The unit-indexed population variables of interest are typically dependent within cluster. We propose a Fully Bayesian method that constructs an exact likelihood for the observed sample to incorporate unit-level marginal sampling weights for performing unbiased inference for population parameters while simultaneously accounting for the dependence induced by sampling clusters of units to produce correct uncertainty quantification. Our approach parameterizes clus…
Priors in which a large number of parameters are specified to be independent are dangerous; they make it hard to learn from data. I present a couple of examples from the literature and work through a bit of large sample theory to show what happens.
Bayesian likelihood-free inference, which is used to perform Bayesian inference when the likelihood is intractable, enjoys an increasing number of important scientific applications. However, many aspects of a Bayesian analysis become more challenging in the likelihood-free setting. One example of this is prior-data conflict checking, where the goal is to assess whether the information in the data and the prior are inconsistent. Conflicts of this kind are important to detect, since they may reveal problems in an investigator's understanding of what are relevant values of the parameters, and can…
Evaluating the effect of different non-informative prior specifications on the Bayesian proportional odds model in randomised controlled trials: a simulation study
In nonlinear regression models the Fisher information depends on the parameters of the model. Consequently, optimal designs maximizing some functional of the information matrix cannot be implemented directly but require some preliminary knowledge about the unknown parameters. Bayesian optimality criteria provide an attractive solution to this problem. These criteria depend sensitively on a reasonable specification of a prior distribution for the model parameters which might not be available in all applications. In this paper we investigate Bayesian optimality criteria with non-informative prio…
In problems with large amounts of missing data one must model two distinct data generating processes: the outcome process which generates the response and the missing data mechanism which determines the data we observe. Under the ignorability condition of Rubin (1976), however, likelihood-based inference for the outcome process does not depend on the missing data mechanism so that only the former needs to be estimated; partially because of this simplification, ignorability is often used as a baseline assumption. We study the implications of Bayesian ignorability in the presence of high-dimensi…
In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew-symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the Total Variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our noninformative priors induce posterior distributions with good frequentist pr…
We provide a framework for assessing the default nature of a prior distribution using the property of regular variation, which we study for global-local shrinkage priors. In particular, we demonstrate the horseshoe priors, originally designed to handle sparsity, also possess regular variation and thus are appropriate for default Bayesian analysis. To illustrate our methodology, we solve a problem of non-informative priors due to Efron (1973), who showed standard flat non-informative priors in high-dimensional normal means model can be highly informative for nonlinear parameters of interest. We…
An informative sampling design leads to unit inclusion probabilities that are correlated with the response variable of interest. However, multistage sampling designs may also induce higher order dependencies, which are typically ignored in the literature when establishing consistency of estimators for survey data under a condition requiring asymptotic independence among the unit inclusion probabilities. We refine and relax this condition of asymptotic independence or asymptotic factorization and demonstrate that consistency is still achieved in the presence of residual sampling dependence. A p…
Bayesian neural networks are theoretically well-understood only in the infinite-width limit, where Gaussian priors over network weights yield Gaussian priors over network outputs. Recent work has suggested that finite Bayesian networks may outperform their infinite counterparts, but their non-Gaussian function space priors have been characterized only though perturbative approaches. Here, we derive exact solutions for the function space priors for individual input examples of a class of finite fully-connected feedforward Bayesian neural networks. For deep linear networks, the prior has a simpl…
Bayesian optimization usually assumes that a Bayesian prior is given. However, the strong theoretical guarantees in Bayesian optimization are often regrettably compromised in practice because of unknown parameters in the prior. In this paper, we adopt a variant of empirical Bayes and show that, by estimating the Gaussian process prior from offline data sampled from the same prior and constructing unbiased estimators of the posterior, variants of both GP-UCB and probability of improvement achieve a near-zero regret bound, which decreases to a constant proportional to the observational noise as …
We explore the Cauchy and a new heavy tailed (Fuquene, Perez and Pericchi (2011)) priors to estimate proportions on small areas. Hierarchical models and the Binomial likelihood in the exponential family form are used. We believe that the heavy tailed priors in survey sampling settings could be more effective than the choice of noninformative priors to eliminate antipathy towards methods that involve subjective elements or assumptions. To illustrate the robust Bayesian approach, we apply this methodology in a popular example: "the clement problem". Finally, we recommend to use the Cauchy prior …
An informative sampling design leads to the selection of units whose inclusion probabilities are correlated with the response variable of interest. Model inference performed on the resulting observed sample will be biased for the population generative model. One approach that produces asymptotically unbiased inference employs marginal inclusion probabilities to form sampling weights used to exponentiate each likelihood contribution of a pseudo likelihood used to form a pseudo posterior distribution. Conditions for posterior consistency restrict applicable sampling designs to those under which …
The United States Bureau of Labor Statistics collects data using survey instruments under informative sampling designs that assign probabilities of inclusion to be correlated with the response. The bureau extensively uses Bayesian hierarchical models and posterior sampling to impute missing items in respondent-level data and to infer population parameters. Posterior sampling for survey data collected based on informative designs are computationally expensive and do not support production schedules of the bureau. Motivated by this problem, we propose a new method to scale Bayesian computations …
We are interested in the estimation and prediction of a parametric model on a short dataset upon which it is expected to overfit and perform badly. To overcome the lack of data (relatively to the dimension of the model) we propose the construction of an informative hierarchical Bayesian prior based upon another longer dataset which is assumed to share some similarities with the original, short dataset. We illustrate the performance of our prior on simulated dataset from three standard models. Then we apply the methodology to a working model for the electricity load forecasting on real datasets…
Eliciting informative prior distributions for Bayesian inference can often be complex and challenging. While popular methods rely on asking experts probability based questions to quantify uncertainty, these methods are not without their drawbacks and many alternative elicitation methods exist. This paper explores methods for eliciting informative priors categorized by type and briefly discusses their strengths and limitations. Most of the review literature in this field focuses on a particular type of elicitation approach. The primary aim of this work, however, is to provide a more complete ye…
