arXiv subjects

Andrew Gelman

Publications and source records attributed to Andrew Gelman.

At least 37 records · Page 2Linked to original sources

arXiv2024

Model validation for aggregate inferences in out-of-sample prediction

Generalization to new samples is a fundamental rationale for statistical modeling. For this purpose, model validation is particularly important, but recent work in survey inference has suggested that simple aggregation of individual prediction scores does not give a good measure of the score for population aggregate estimates. In this manuscript we explain why this occurs, propose two scoring metrics designed specifically for this problem, and demonstrate their use in three different ways. We show that these scoring metrics correctly order models when compared to the true score, although they do underestimate the magnitude of the score. We demonstrate with a problem in survey research, where multilevel regression and poststratification (MRP) has been used extensively to adjust convenience and low-response surveys to make population and subpopulation estimates.

stat.ME↗

arXiv2024

Pareto Smoothed Importance Sampling

Importance weighting is a general way to adjust Monte Carlo integration to account for draws from the wrong distribution, but the resulting estimate can be highly variable when the importance ratios have a heavy right tail. This routinely occurs when there are aspects of the target distribution that are not well captured by the approximating distribution, in which case more stable estimates can be obtained by modifying extreme importance ratios. We present a new method for stabilizing importance weights using a generalized Pareto distribution fit to the upper tail of the distribution of the simulated importance ratios. The method, which empirically performs better than existing methods for stabilizing importance sampling estimates, includes stabilized effective sample size estimates, Monte Carlo error estimates, and convergence diagnostics. The presented Pareto $\hat{k}$ finite sample convergence rate diagnostic is useful for any Monte Carlo estimator.

stat.CO↗

arXiv2024

For how many iterations should we run Markov chain Monte Carlo?

Standard Markov chain Monte Carlo (MCMC) admits three fundamental control parameters: the number of chains, the length of the warmup phase, and the length of the sampling phase. These control parameters play a large role in determining the amount of computation we deploy. In practice, we need to walk a line between achieving sufficient precision and not wasting precious computational resources and time. We review general strategies to check the length of the warmup and sampling phases, and examine the three control parameters of MCMC in the contexts of CPU- and GPU-based hardware. Our discussion centers around three tasks: (1) inference about a latent variable, (2) computation of expectation values and quantiles, and (3) diagnostics to check the reliability of the estimators. This chapter begins with general recommendations on the control parameters of MCMC, which have been battle-tested over the years and often motivate defaults in Bayesian statistical software. Usually we do not know ahead of time how a sampler will interact with a target distribution, and so the choice of MCMC algorithm and its control parameters, tend to be based on experience, re-evaluated after simulations have been obtained and analyzed. The second part of this chapter provides a theoretical motivation for our recommended approach, with pointers to some concerns and open problems. We also examine recent developments on the algorithmic and hardware fronts, which motivate new computational approaches to MCMC.

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arXiv2023

Simulation-Based Calibration Checking for Bayesian Computation: The Choice of Test Quantities Shapes Sensitivity

Simulation-based calibration checking (SBC) is a practical method to validate computationally-derived posterior distributions or their approximations. In this paper, we introduce a new variant of SBC to alleviate several known problems. Our variant allows the user to in principle detect any possible issue with the posterior, while previously reported implementations could never detect large classes of problems including when the posterior is equal to the prior. This is made possible by including additional data-dependent test quantities when running SBC. We argue and demonstrate that the joint likelihood of the data is an especially useful test quantity. Some other types of test quantities and their theoretical and practical benefits are also investigated. We provide theoretical analysis of SBC, thereby providing a more complete understanding of the underlying statistical mechanisms. We also bring attention to a relatively common mistake in the literature and clarify the difference between SBC and checks based on the data-averaged posterior. We support our recommendations with numerical case studies on a multivariate normal example and a case study in implementing an ordered simplex data type for use with Hamiltonian Monte Carlo. The SBC variant introduced in this paper is implemented in the $\mathtt{SBC}$ R package.

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arXiv2023

Artificial Intelligence and Aesthetic Judgment

Generative AIs produce creative outputs in the style of human expression. We argue that encounters with the outputs of modern generative AI models are mediated by the same kinds of aesthetic judgments that organize our interactions with artwork. The interpretation procedure we use on art we find in museums is not an innate human faculty, but one developed over history by disciplines such as art history and art criticism to fulfill certain social functions. This gives us pause when considering our reactions to generative AI, how we should approach this new medium, and why generative AI seems to incite so much fear about the future. We naturally inherit a conundrum of causal inference from the history of art: a work can be read as a symptom of the cultural conditions that influenced its creation while simultaneously being framed as a timeless, seemingly acausal distillation of an eternal human condition. In this essay, we focus on an unresolved tension when we bring this dilemma to bear in the context of generative AI: are we looking for proof that generated media reflects something about the conditions that created it or some eternal human essence? Are current modes of interpretation sufficient for this task? Historically, new forms of art have changed how art is interpreted, with such influence used as evidence that a work of art has touched some essential human truth. As generative AI influences contemporary aesthetic judgment we outline some of the pitfalls and traps in attempting to scrutinize what AI generated media means.

cs.CY↗

arXiv2023

Causal quartets: Different ways to attain the same average treatment effect

The average causal effect can often be best understood in the context of its variation. We demonstrate with two sets of four graphs, all of which represent the same average effect but with much different patterns of heterogeneity. As with the famous correlation quartet of Anscombe (1973), these graphs dramatize the way in which real-world variation can be more complex than simple numerical summaries. The graphs also give insight into why the average effect is often much smaller than anticipated.

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arXiv2023

Federated Learning as Variational Inference: A Scalable Expectation Propagation Approach

The canonical formulation of federated learning treats it as a distributed optimization problem where the model parameters are optimized against a global loss function that decomposes across client loss functions. A recent alternative formulation instead treats federated learning as a distributed inference problem, where the goal is to infer a global posterior from partitioned client data (Al-Shedivat et al., 2021). This paper extends the inference view and describes a variational inference formulation of federated learning where the goal is to find a global variational posterior that well-approximates the true posterior. This naturally motivates an expectation propagation approach to federated learning (FedEP), where approximations to the global posterior are iteratively refined through probabilistic message-passing between the central server and the clients. We conduct an extensive empirical study across various algorithmic considerations and describe practical strategies for scaling up expectation propagation to the modern federated setting. We apply FedEP on standard federated learning benchmarks and find that it outperforms strong baselines in terms of both convergence speed and accuracy.

cs.LG↗

arXiv2022

Prediction scoring of data-driven discoveries for reproducible research

Predictive modeling uncovers knowledge and insights regarding a hypothesized data generating mechanism (DGM). Results from different studies on a complex DGM, derived from different data sets, and using complicated models and algorithms, are hard to quantitatively compare due to random noise and statistical uncertainty in model results. This has been one of the main contributors to the replication crisis in the behavioral sciences. The contribution of this paper is to apply prediction scoring to the problem of comparing two studies, such as can arise when evaluating replications or competing evidence. We examine the role of predictive models in quantitatively assessing agreement between two datasets that are assumed to come from two distinct DGMs. We formalize a distance between the DGMs that is estimated using cross validation. We argue that the resulting prediction scores depend on the predictive models created by cross validation. In this sense, the prediction scores measure the distance between DGMs, along the dimension of the particular predictive model. Using human behavior data from experimental economics, we demonstrate that prediction scores can be used to evaluate preregistered hypotheses and provide insights comparing data from different populations and settings. We examine the asymptotic behavior of the prediction scores using simulated experimental data and demonstrate that leveraging competing predictive models can reveal important differences between underlying DGMs. Our proposed cross-validated prediction scores are capable of quantifying differences between unobserved data generating mechanisms and allow for the validation and assessment of results from complex models.

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arXiv2022

How Democracies Polarize: A Multilevel Perspective

Democracies employ elections at various scales to select officials at the corresponding levels of administration. The geographical distribution of political opinion, the policy issues delegated to each level, and the multilevel interactions between elections can all greatly impact the makeup of these representative bodies. This perspective is not new: the adoption of federal systems has been motivated by the idea that they possess desirable traits not provided by democracies on a single scale. Yet most existing models of polarization do not capture how nested local and national elections interact with heterogeneous political geographies. We begin by developing a framework to describe the multilevel distribution of opinions and analyze the flow of variance among geographic scales, applying it to historical data in the United States from 1912 to 2020. We describe how unstable elections can arise due to the spatial distribution of opinions and how tradeoffs occur between national and local elections. We also examine multi-dimensional spaces of political opinion, for which we show that a decrease in local salience can constrain the dimensions along which elections occur, preventing a federal system from serving as an effective safeguard against polarization. These analyses, based on the interactions between elections and opinion distributions at various scales, offer insights into how democracies can be strengthened to mitigate polarization and increase electoral representation.

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arXiv2022

Using leave-one-out cross-validation (LOO) in a multilevel regression and poststratification (MRP) workflow: A cautionary tale

In recent decades, multilevel regression and poststratification (MRP) has surged in popularity for population inference. However, the validity of the estimates can depend on details of the model, and there is currently little research on validation. We explore how leave-one-out cross-validation (LOO) can be used to compare Bayesian models for MRP. We investigate two approximate calculations of LOO, the Pareto smoothed importance sampling (PSIS-LOO) and a survey-weighted alternative (WTD-PSIS-LOO). Using two simulation designs, we examine how accurately these two criteria recover the correct ordering of model goodness at predicting population and small area level estimands. Focusing first on variable selection, we find that neither PSIS-LOO nor WTD-PSIS-LOO correctly recovers the models' order for an MRP population estimand (although both criteria correctly identify the best and worst model). When considering small-area estimation, the best model differs for different small areas, highlighting the complexity of MRP validation. When considering different priors, the models' order seems slightly better at smaller area levels. These findings suggest that while not terrible, PSIS-LOO-based ranking techniques may not be suitable to evaluate MRP as a method. We suggest this is due to the aggregation stage of MRP, where individual-level prediction errors average out. These results show that in practice, PSIS-LOO-based model validation tools need to be used with caution and might not convey the full story when validating MRP as a method.

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arXiv2022

The worst of both worlds: A comparative analysis of errors in learning from data in psychology and machine learning

Recent arguments that machine learning (ML) is facing a reproducibility and replication crisis suggest that some published claims in ML research cannot be taken at face value. These concerns inspire analogies to the replication crisis affecting the social and medical sciences. They also inspire calls for the integration of statistical approaches to causal inference and predictive modeling. A deeper understanding of what reproducibility concerns in supervised ML research have in common with the replication crisis in experimental science puts the new concerns in perspective, and helps researchers avoid "the worst of both worlds," where ML researchers begin borrowing methodologies from explanatory modeling without understanding their limitations and vice versa. We contribute a comparative analysis of concerns about inductive learning that arise in causal attribution as exemplified in psychology versus predictive modeling as exemplified in ML. We identify themes that re-occur in reform discussions, like overreliance on asymptotic theory and non-credible beliefs about real-world data generating processes. We argue that in both fields, claims from learning are implied to generalize outside the specific environment studied (e.g., the input dataset or subject sample, modeling implementation, etc.) but are often impossible to refute due to undisclosed sources of variance in the learning pipeline. In particular, errors being acknowledged in ML expose cracks in long-held beliefs that optimizing predictive accuracy using huge datasets absolves one from having to consider a true data generating process or formally represent uncertainty in performance claims. We conclude by discussing risks that arise when sources of errors are misdiagnosed and the need to acknowledge the role of human inductive biases in learning and reform.

cs.LG↗

arXiv2022

Pathfinder: Parallel quasi-Newton variational inference

We propose Pathfinder, a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the inverse Hessian estimates produced by the optimizer. Pathfinder returns draws from the approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior. We evaluate Pathfinder on a wide range of posterior distributions, demonstrating that its approximate draws are better than those from automatic differentiation variational inference (ADVI) and comparable to those produced by short chains of dynamic Hamiltonian Monte Carlo (HMC), as measured by 1-Wasserstein distance. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors. Importance resampling over multiple runs of Pathfinder improves the diversity of approximate draws, reducing 1-Wasserstein distance further and providing a measure of robustness to optimization failures on plateaus, saddle points, or in minor modes. The Monte Carlo KL divergence estimates are embarrassingly parallelizable in the core Pathfinder algorithm, as are multiple runs in the resampling version, further increasing Pathfinder's speed advantage with multiple cores.

stat.ML↗

arXiv2022

Delivering data differently

Human-computer interaction relies on mouse/touchpad, keyboard, and screen, but tools have recently been developed that engage sound, smell, touch, muscular resistance, voice dialogue, balance, and multiple senses at once. How might these improvements impact upon the practice of statistics and data science? People with low vision may be better able to grasp and explore data. More generally, methods developed to enable this have the potential to allow sighted people to use more senses and become better analysts. We would like to adapt some of the wide range of available computer and sensory input/output technologies to transform data science workflows. Here is a vision of what this synthesis might accomplish.

cs.HC↗

arXiv2022

Using sex and gender in survey adjustment

Accounting for sex and gender is a challenge in social science research. While other methodology papers consider issues surrounding appropriate measurement, we consider the problem of adjustment for survey nonresponse and generalization from samples to populations in the context of the recent push toward measuring sex or gender as a non-binary construct. This is challenging not only in that response categories differ between sex and gender measurement, but also in that both these attributes are potentially multidimensional. We reflect on similarities to measuring race/ethnicity before considering the ethical and statistical implications of the options available to us. We present a simulation study to understand the statistical implications under a variety of scenarios, and demonstrate the application of the decision process with the New York City Poverty Tracker. Overall, we conclude not with a single best recommendation for all surveys but rather with an awareness of the complexity of the problem and the benefits and weaknesses of different approaches.

stat.AP↗

arXiv2022

Beyond Vaccination Rates: A Synthetic Random Proxy Metric of Total SARS-CoV-2 Immunity Seroprevalence in the Community

Explicit knowledge of total community-level immune seroprevalence is critical to developing policies to mitigate the social and clinical impact of SARS-CoV-2. Publicly available vaccination data are frequently cited as a proxy for population immunity, but this metric ignores the effects of naturally-acquired immunity, which varies broadly throughout the country and world. Without broad or random sampling of the population, accurate measurement of persistent immunity post natural infection is generally unavailable. To enable tracking of both naturally-acquired and vaccine-induced immunity, we set up a synthetic random proxy based on routine hospital testing for estimating total Immunoglobulin G (IgG) prevalence in the sampled community. Our approach analyzes viral IgG testing data of asymptomatic patients who present for elective procedures within a hospital system. We apply multilevel regression and poststratification to adjust for demographic and geographic discrepancies between the sample and the community population. We then apply state-based vaccination data to categorize immune status as driven by natural infection or by vaccine. We have validated the model using verified clinical metrics of viral and symptomatic disease incidence to show the expected biological correlation of these entities with the timing, rate, and magnitude of seroprevalence. In mid-July 2021, the estimated immunity level was 74% with the administered vaccination rate of 45% in the two counties. The metric improves real-time understanding of immunity to COVID-19 as it evolves and the coordination of policy responses to the disease, toward an inexpensive and easily operational surveillance system that transcends the limits of vaccination datasets alone.

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arXiv2021

Toward a Taxonomy of Trust for Probabilistic Machine Learning

Probabilistic machine learning increasingly informs critical decisions in medicine, economics, politics, and beyond. We need evidence to support that the resulting decisions are well-founded. To aid development of trust in these decisions, we develop a taxonomy delineating where trust in an analysis can break down: (1) in the translation of real-world goals to goals on a particular set of available training data, (2) in the translation of abstract goals on the training data to a concrete mathematical problem, (3) in the use of an algorithm to solve the stated mathematical problem, and (4) in the use of a particular code implementation of the chosen algorithm. We detail how trust can fail at each step and illustrate our taxonomy with two case studies: an analysis of the efficacy of microcredit and The Economist's predictions of the 2020 US presidential election. Finally, we describe a wide variety of methods that can be used to increase trust at each step of our taxonomy. The use of our taxonomy highlights steps where existing research work on trust tends to concentrate and also steps where establishing trust is particularly challenging.

cs.LG↗

arXiv2021

Stacking for Non-mixing Bayesian Computations: The Curse and Blessing of Multimodal Posteriors

When working with multimodal Bayesian posterior distributions, Markov chain Monte Carlo (MCMC) algorithms have difficulty moving between modes, and default variational or mode-based approximate inferences will understate posterior uncertainty. And, even if the most important modes can be found, it is difficult to evaluate their relative weights in the posterior. Here we propose an approach using parallel runs of MCMC, variational, or mode-based inference to hit as many modes or separated regions as possible and then combine these using Bayesian stacking, a scalable method for constructing a weighted average of distributions. The result from stacking efficiently samples from multimodal posterior distribution, minimizes cross validation prediction error, and represents the posterior uncertainty better than variational inference, but it is not necessarily equivalent, even asymptotically, to fully Bayesian inference. We present theoretical consistency with an example where the stacked inference approximates the true data generating process from the misspecified model and a non-mixing sampler, from which the predictive performance is better than full Bayesian inference, hence the multimodality can be considered a blessing rather than a curse under model misspecification. We demonstrate practical implementation in several model families: latent Dirichlet allocation, Gaussian process regression, hierarchical regression, horseshoe variable selection, and neural networks.

stat.ME↗

arXiv2021

Fast methods for posterior inference of two-group normal-normal models

We describe a class of algorithms for evaluating posterior moments of certain Bayesian linear regression models with a normal likelihood and a normal prior on the regression coefficients. The proposed methods can be used for hierarchical mixed effects models with partial pooling over one group of predictors, as well as random effects models with partial pooling over two groups of predictors. We demonstrate the performance of the methods on two applications, one involving U.S. opinion polls and one involving the modeling of COVID-19 outbreaks in Israel using survey data. The algorithms involve analytical marginalization of regression coefficients followed by numerical integration of the remaining low-dimensional density. The dominant cost of the algorithms is an eigendecomposition computed once for each value of the outside parameter of integration. Our approach drastically reduces run times compared to state-of-the-art Markov chain Monte Carlo (MCMC) algorithms. The latter, in addition to being computationally expensive, can also be difficult to tune when applied to hierarchical models.

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