Tamara Broderick

Brian L. Trippe, Jason Yim, Doug Tischer et al.

Construction of a scaffold structure that supports a desired motif, conferring protein function, shows promise for the design of vaccines and enzymes. But a general solution to this motif-scaffolding problem remains open. Current machine-learning techniques for scaffold design are either limited to unrealistically small scaffolds (up to length 20) or struggle to produce multiple diverse scaffolds. We propose to learn a distribution over diverse and longer protein backbone structures via an E(3)-equivariant graph neural network. We develop SMCDiff to efficiently sample scaffolds from this distribution conditioned on a given motif; our algorithm is the first to theoretically guarantee conditional samples from a diffusion model in the large-compute limit. We evaluate our designed backbones by how well they align with AlphaFold2-predicted structures. We show that our method can (1) sample scaffolds up to 80 residues and (2) achieve structurally diverse scaffolds for a fixed motif.

Vijay Gadepally, Gregory Angelides, Andrei Barbu et al.

Through a series of federal initiatives and orders, the U.S. Government has been making a concerted effort to ensure American leadership in AI. These broad strategy documents have influenced organizations such as the United States Department of the Air Force (DAF). The DAF-MIT AI Accelerator is an initiative between the DAF and MIT to bridge the gap between AI researchers and DAF mission requirements. Several projects supported by the DAF-MIT AI Accelerator are developing public challenge problems that address numerous Federal AI research priorities. These challenges target priorities by making large, AI-ready datasets publicly available, incentivizing open-source solutions, and creating a demand signal for dual use technologies that can stimulate further research. In this article, we describe these public challenges being developed and how their application contributes to scientific advances.

Ryan Giordano, Martin Ingram, Tamara Broderick

Automatic differentiation variational inference (ADVI) offers fast and easy-to-use posterior approximation in multiple modern probabilistic programming languages. However, its stochastic optimizer lacks clear convergence criteria and requires tuning parameters. Moreover, ADVI inherits the poor posterior uncertainty estimates of mean-field variational Bayes (MFVB). We introduce "deterministic ADVI" (DADVI) to address these issues. DADVI replaces the intractable MFVB objective with a fixed Monte Carlo approximation, a technique known in the stochastic optimization literature as the "sample average approximation" (SAA). By optimizing an approximate but deterministic objective, DADVI can use off-the-shelf second-order optimization, and, unlike standard mean-field ADVI, is amenable to more accurate posterior covariances via linear response (LR). In contrast to existing worst-case theory, we show that, on certain classes of common statistical problems, DADVI and the SAA can perform well with relatively few samples even in very high dimensions, though we also show that such favorable results cannot extend to variational approximations that are too expressive relative to mean-field ADVI. We show on a variety of real-world problems that DADVI reliably finds good solutions with default settings (unlike ADVI) and, together with LR covariances, is typically faster and more accurate than standard ADVI.

Renato Berlinghieri, Brian L. Trippe, David R. Burt et al.

Given sparse observations of buoy velocities, oceanographers are interested in reconstructing ocean currents away from the buoys and identifying divergences in a current vector field. As a first and modular step, we focus on the time-stationary case - for instance, by restricting to short time periods. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current reconstruction and divergence identification, due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method with theory and experiments on synthetic and real ocean data.

Sameer K. Deshpande, Soumya Ghosh, Tin D. Nguyen et al.

Test log-likelihood is commonly used to compare different models of the same data or different approximate inference algorithms for fitting the same probabilistic model. We present simple examples demonstrating how comparisons based on test log-likelihood can contradict comparisons according to other objectives. Specifically, our examples show that (i) approximate Bayesian inference algorithms that attain higher test log-likelihoods need not also yield more accurate posterior approximations and (ii) conclusions about forecast accuracy based on test log-likelihood comparisons may not agree with conclusions based on root mean squared error.

Yunyi Shen, Renato Berlinghieri, Tamara Broderick

Practitioners often aim to infer an unobserved population trajectory using sample snapshots at multiple time points. E.g., given single-cell sequencing data, scientists would like to learn how gene expression changes over a cell's life cycle. But sequencing any cell destroys that cell. So we can access data for any particular cell only at a single time point, but we have data across many cells. The deep learning community has recently explored using Schrödinger bridges (SBs) and their extensions in similar settings. However, existing methods either (1) interpolate between just two time points or (2) require a single fixed reference dynamic (often set to Brownian motion within SBs). But learning piecewise from adjacent time points can fail to capture long-term dependencies. And practitioners are typically able to specify a model family for the reference dynamic but not the exact values of the parameters within it. So we propose a new method that (1) learns the unobserved trajectories from sample snapshots across multiple time points and (2) requires specification only of a family of reference dynamics, not a single fixed one. We demonstrate the advantages of our method on simulated and real data.

Tamara Broderick, Ali Jadbabaie, Vanessa Lin et al.

Surety bonds are financial agreements between a contractor (principal) and obligee (project owner) to complete a project. However, most large-scale projects involve multiple contractors, creating a network and introducing the possibility of incomplete obligations to propagate and result in project failures. Typical models for risk assessment assume independent failure probabilities within each contractor. However, we take a network approach, modeling the contractor network as a directed graph where nodes represent contractors and project owners and edges represent contractual obligations with associated financial records. To understand risk propagation throughout the contractor network, we extend the celebrated Friedkin-Johnsen model and introduce a stochastic process to simulate principal failures across the network. From a theoretical perspective, we show that under natural monotonicity conditions on the contractor network, incorporating network effects leads to increases in the average risk for the surety organization. We further use data from a partnering insurance company to validate our findings, estimating an approximately 2% higher exposure when accounting for network effects.

Andre Ye, Jenny Y. Huang, Alicia Guo et al.

When language models answer open-ended problems, they implicitly make hidden decisions that shape their outputs, leaving users with uncontextualized answers rather than a working map of the problem; drawing on multiverse analysis from statistics, we build and evaluate the conceptual multiverse, an interactive system that represents conceptual decisions such as how to frame a question or what to value as a space users can transparently inspect, intervenably change, and check against principled domain reasoning; for this structure to be worth navigating rather than misleading, it must be rigorous and checkable against domain reasoning norms, so we develop a general verification framework that enforces properties of good decision structures like unambiguity and completeness calibrated by expert-level reasoning; across three domains, the conceptual multiverse helped participants develop a working map of the problem, with philosophy students rewriting essays with sharper framings and reversed theses, alignment annotators moving from surface preferences to reasoning about user intent and harm, and poets identifying compositional patterns that clarified their taste.

Renato Berlinghieri, David R. Burt, Paolo Giani et al.

Wildfire frequency is increasing as the climate changes, and the resulting air pollution poses health risks. Just as people routinely use hourly weather forecasts to plan their day's activities around precipitation, reliable hourly air quality forecasts could help individuals reduce their exposure to air pollution. In the present work, we evaluate six existing forecasts of ground-level fine particulate matter (PM2.5) within the continental United States during the 2023 fire season. We include forecasts using physical simulation, ensembling, and artificial intelligence. We focus our evaluation on individual decisions, such as (1) whether to go outside on a day with potentially high PM2.5 or (2) when to go outside for the lowest PM2.5 exposure. Our evaluation consists of both visualizations of hourly PM2.5 forecasts in particular locations as well as metrics summarizing forecast skill for the two tasks above. As part of our analysis, we introduce a new evaluation metric for the task of deciding when to go outside. We find meaningful room for improvement in PM2.5 forecasting, which might be realized by improving physical models, incorporating more data sources, and using artificial intelligence tools.

David R. Burt, Yunyi Shen, Tamara Broderick

Spatial prediction tasks are key to weather forecasting, studying air pollution impacts, and other scientific endeavors. Determining how much to trust predictions made by statistical or physical methods is essential for the credibility of scientific conclusions. Unfortunately, classical approaches for validation fail to handle mismatch between locations available for validation and (test) locations where we want to make predictions. This mismatch is often not an instance of covariate shift (as commonly formalized) because the validation and test locations are fixed (e.g., on a grid or at select points) rather than i.i.d. from two distributions. In the present work, we formalize a check on validation methods: that they become arbitrarily accurate as validation data becomes arbitrarily dense. We show that classical and covariate-shift methods can fail this check. We propose a method that builds from existing ideas in the covariate-shift literature, but adapts them to the validation data at hand. We prove that our proposal passes our check. And we demonstrate its advantages empirically on simulated and real data.

Renato Berlinghieri, Yunyi Shen, Jialong Jiang et al.

Scientists often want to make predictions beyond the observed time horizon of "snapshot" data following latent stochastic dynamics. For example, in time course single-cell mRNA profiling, scientists have access to cellular transcriptional state measurements (snapshots) from different biological replicates at different time points, but they cannot access the trajectory of any one cell because measurement destroys the cell. Researchers want to forecast (e.g.) differentiation outcomes from early state measurements of stem cells. Recent Schrödinger-bridge (SB) methods are natural for interpolating between snapshots. But past SB papers have not addressed forecasting -- likely since existing methods either (1) reduce to following pre-set reference dynamics (chosen before seeing data) or (2) require the user to choose a fixed, state-independent volatility since they minimize a Kullback-Leibler divergence. Either case can lead to poor forecasting quality. In the present work, we propose a new framework, SnapMMD, that learns dynamics by directly fitting the joint distribution of both state measurements and observation time with a maximum mean discrepancy (MMD) loss. Unlike past work, our method allows us to infer unknown and state-dependent volatilities from the observed data. We show in a variety of real and synthetic experiments that our method delivers accurate forecasts. Moreover, our approach allows us to learn in the presence of incomplete state measurements and yields an $R^2$-style statistic that diagnoses fit. We also find that our method's performance at interpolation (and general velocity-field reconstruction) is at least as good as (and often better than) state-of-the-art in almost all of our experiments.

David R. Burt, Renato Berlinghieri, Stephen Bates et al.

Estimating associations between spatial covariates and responses - rather than merely predicting responses - is central to environmental science, epidemiology, and economics. For instance, public health officials might be interested in whether air pollution has a strictly positive association with a health outcome, and the magnitude of any effect. Standard machine learning methods often provide accurate predictions but offer limited insight into covariate-response relationships. And we show that existing methods for constructing confidence (or credible) intervals for associations can fail to provide nominal coverage in the face of model misspecification and nonrandom locations - despite both being essentially always present in spatial problems. We introduce a method that constructs valid frequentist confidence intervals for associations in spatial settings. Our method requires minimal assumptions beyond a form of spatial smoothness and a homoskedastic Gaussian error assumption. In particular, we do not require model correctness or covariate overlap between training and target locations. Our approach is the first to guarantee nominal coverage in this setting and outperforms existing techniques in both real and simulated experiments. Our confidence intervals are valid in finite samples when the noise of the Gaussian error is known, and we provide an asymptotically consistent estimation procedure for this noise variance when it is unknown.

Manuel Quintero, Advik Shreekumar, William T. Stephenson et al.

Scientists often want to explain why an outcome is different in two groups. For instance, differences in patient mortality rates across two hospitals could be due to differences in the patients themselves (covariates) or differences in medical care (outcomes given covariates). The Oaxaca--Blinder decomposition (OBD) is a standard tool to tease apart these factors. It is well known that the OBD requires choosing one of the groups as a reference, and the numerical answer can vary with the reference. To the best of our knowledge, there has not been a systematic investigation into whether the choice of OBD reference can yield different substantive conclusions and how common this issue is. In the present paper, we give existence proofs in real and simulated data that the OBD references can yield substantively different conclusions and that these differences are not entirely driven by model misspecification or small data. We prove that substantively different conclusions occur in up to half of the parameter space, but find these discrepancies rare in the real-data analyses we study. We explain this empirical rarity by examining how realistic data-generating processes can be biased towards parameters that do not change conclusions under the OBD.

David R. Burt, Renato Berlinghieri, Tamara Broderick

Scientists are often interested in estimating an association between a covariate and a binary- or count-valued response. For instance, public health officials are interested in how much disease presence (a binary response per individual) varies as temperature or pollution (covariates) increases. Many existing methods can be used to estimate associations, and corresponding uncertainty intervals, but make unrealistic assumptions in the spatial domain. For instance, they incorrectly assume models are well-specified. Or they assume the training and target locations are i.i.d. -- whereas in practice, these locations are often not even randomly sampled. Some recent work avoids these assumptions but works only for continuous responses with spatially constant noise. In the present work, we provide the first confidence intervals with guaranteed asymptotic nominal coverage for spatial associations given discrete responses, even under simultaneous model misspecification and nonrandom sampling of spatial locations. To do so, we demonstrate how to handle spatially varying noise, provide a novel proof of consistency for our proposed estimator, and use a delta method argument with a Lyapunov central limit theorem. We show empirically that standard approaches can produce unreliable confidence intervals and can even get the sign of an association wrong, while our method reliably provides correct coverage.

Jenny Y. Huang, Yunyi Shen, Dennis Wei et al.

We propose a method for evaluating the robustness of a widely used LLM ranking system -- the Bradley--Terry ranking system -- to dropping a worst-case very small fraction of evaluation data. Our approach is computationally fast and easy to adopt. When we apply our method to matchups from two popular human-preference platforms, Chatbot Arena and MT-Bench, we find that the Bradley--Terry rankings of top-performing models are remarkably sensitive to the removal of a small fraction of evaluations. Our framework also identifies the specific evaluations most responsible for such ranking flips, allowing for inspections of these influential preferences. We observe that the rankings derived from MT-Bench preferences are notably more robust than those from Chatbot Arena, likely due to MT-bench's use of expert annotators and carefully constructed prompts. Finally, we find that rankings based on crowdsourced human-evaluated systems are just as sensitive as those based on LLM-as-a-judge evaluations, where in both, dropping as little as 0.02% of the total evaluations in the dataset can change the top-ranked model.

Manuel Quintero, William T. Stephenson, Advik Shreekumar et al.

In science and social science, we often wish to explain why an outcome is different in two populations. For instance, if a jobs program benefits members of one city more than another, is that due to differences in program participants (particular covariates) or the local labor markets (outcomes given covariates)? The Kitagawa-Oaxaca-Blinder (KOB) decomposition is a standard tool in econometrics that explains the difference in the mean outcome across two populations. However, the KOB decomposition assumes a linear relationship between covariates and outcomes, while the true relationship may be meaningfully nonlinear. Modern machine learning boasts a variety of nonlinear functional decompositions for the relationship between outcomes and covariates in one population. It seems natural to extend the KOB decomposition using these functional decompositions. We observe that a successful extension should not attribute the differences to covariates -- or, respectively, to outcomes given covariates -- if those are the same in the two populations. Unfortunately, we demonstrate that, even in simple examples, two common decompositions -- functional ANOVA and Accumulated Local Effects -- can attribute differences to outcomes given covariates, even when they are identical in two populations. We provide a characterization of when functional ANOVA misattributes, as well as a general property that any discrete decomposition must satisfy to avoid misattribution. We show that if the decomposition is independent of its input distribution, it does not misattribute. We further conjecture that misattribution arises in any reasonable additive decomposition that depends on the distribution of the covariates.

Tin D. Nguyen, Brian L. Trippe, Tamara Broderick

Markov chain Monte Carlo (MCMC) methods are often used in clustering since they guarantee asymptotically exact expectations in the infinite-time limit. In finite time, though, slow mixing often leads to poor performance. Modern computing environments offer massive parallelism, but naive implementations of parallel MCMC can exhibit substantial bias. In MCMC samplers of continuous random variables, Markov chain couplings can overcome bias. But these approaches depend crucially on paired chains meetings after a small number of transitions. We show that straightforward applications of existing coupling ideas to discrete clustering variables fail to meet quickly. This failure arises from the "label-switching problem": semantically equivalent cluster relabelings impede fast meeting of coupled chains. We instead consider chains as exploring the space of partitions rather than partitions' (arbitrary) labelings. Using a metric on the partition space, we formulate a practical algorithm using optimal transport couplings. Our theory confirms our method is accurate and efficient. In experiments ranging from clustering of genes or seeds to graph colorings, we show the benefits of our coupling in the highly parallel, time-limited regime.

Tamara Broderick, Andrew Gelman, Rachael Meager et al.

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.

William T. Stephenson, Zachary Frangella, Madeleine Udell et al.

Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees. Cross-validation (CV) is widely used for hyperparameter tuning in these models, but do practical optimization methods minimize the true out-of-sample loss? A recent line of research promises to show that the optimum of the CV loss matches the optimum of the out-of-sample loss (possibly after simple corrections). It remains to show how tractable it is to minimize the CV loss. In the present paper, we show that, in the case of ridge regression, the CV loss may fail to be quasiconvex and thus may have multiple local optima. We can guarantee that the CV loss is quasiconvex in at least one case: when the spectrum of the covariate matrix is nearly flat and the noise in the observed responses is not too high. More generally, we show that quasiconvexity status is independent of many properties of the observed data (response norm, covariate-matrix right singular vectors and singular-value scaling) and has a complex dependence on the few that remain. We empirically confirm our theory using simulated experiments.

Brian L. Trippe, Hilary K. Finucane, Tamara Broderick

Hierarchical Bayesian methods enable information sharing across multiple related regression problems. While standard practice is to model regression parameters (effects) as (1) exchangeable across datasets and (2) correlated to differing degrees across covariates, we show that this approach exhibits poor statistical performance when the number of covariates exceeds the number of datasets. For instance, in statistical genetics, we might regress dozens of traits (defining datasets) for thousands of individuals (responses) on up to millions of genetic variants (covariates). When an analyst has more covariates than datasets, we argue that it is often more natural to instead model effects as (1) exchangeable across covariates and (2) correlated to differing degrees across datasets. To this end, we propose a hierarchical model expressing our alternative perspective. We devise an empirical Bayes estimator for learning the degree of correlation between datasets. We develop theory that demonstrates that our method outperforms the classic approach when the number of covariates dominates the number of datasets, and corroborate this result empirically on several high-dimensional multiple regression and classification problems.

Ryan Giordano, Runjing Liu, Michael I. Jordan et al.

Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. Prior specification is, however, relatively difficult for such models, given that their flexibility implies that the consequences of prior choices are often relatively opaque. Moreover, these choices can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis.

Raj Agrawal, Tamara Broderick

Many scientific problems require identifying a small set of covariates that are associated with a target response and estimating their effects. Often, these effects are nonlinear and include interactions, so linear and additive methods can lead to poor estimation and variable selection. Unfortunately, methods that simultaneously express sparsity, nonlinearity, and interactions are computationally intractable -- with runtime at least quadratic in the number of covariates, and often worse. In the present work, we solve this computational bottleneck. We show that suitable interaction models have a kernel representation, namely there exists a "kernel trick" to perform variable selection and estimation in $O$(# covariates) time. Our resulting fit corresponds to a sparse orthogonal decomposition of the regression function in a Hilbert space (i.e., a functional ANOVA decomposition), where interaction effects represent all variation that cannot be explained by lower-order effects. On a variety of synthetic and real data sets, our approach outperforms existing methods used for large, high-dimensional data sets while remaining competitive (or being orders of magnitude faster) in runtime.

William T. Stephenson, Soumya Ghosh, Tin D. Nguyen et al.

Gaussian processes (GPs) are used to make medical and scientific decisions, including in cardiac care and monitoring of atmospheric carbon dioxide levels. Notably, the choice of GP kernel is often somewhat arbitrary. In particular, uncountably many kernels typically align with qualitative prior knowledge (e.g.\ function smoothness or stationarity). But in practice, data analysts choose among a handful of convenient standard kernels (e.g.\ squared exponential). In the present work, we ask: Would decisions made with a GP differ under other, qualitatively interchangeable kernels? We show how to answer this question by solving a constrained optimization problem over a finite-dimensional space. We can then use standard optimizers to identify substantive changes in relevant decisions made with a GP. We demonstrate in both synthetic and real-world examples that decisions made with a GP can exhibit non-robustness to kernel choice, even when prior draws are qualitatively interchangeable to a user.

Tin D. Nguyen, Jonathan Huggins, Lorenzo Masoero et al.

Completely random measures (CRMs) and their normalizations (NCRMs) offer flexible models in Bayesian nonparametrics. But their infinite dimensionality presents challenges for inference. Two popular finite approximations are truncated finite approximations (TFAs) and independent finite approximations (IFAs). While the former have been well-studied, IFAs lack similarly general bounds on approximation error, and there has been no systematic comparison between the two options. In the present work, we propose a general recipe to construct practical finite-dimensional approximations for homogeneous CRMs and NCRMs, in the presence or absence of power laws. We call our construction the automated independent finite approximation (AIFA). Relative to TFAs, we show that AIFAs facilitate more straightforward derivations and use of parallel computing in approximate inference. We upper bound the approximation error of AIFAs for a wide class of common CRMs and NCRMs -- and thereby develop guidelines for choosing the approximation level. Our lower bounds in key cases suggest that our upper bounds are tight. We prove that, for worst-case choices of observation likelihoods, TFAs are more efficient than AIFAs. Conversely, we find that in real-data experiments with standard likelihoods, AIFAs and TFAs perform similarly. Moreover, we demonstrate that AIFAs can be used for hyperparameter estimation even when other potential IFA options struggle or do not apply.

William T. Stephenson, Madeleine Udell, Tamara Broderick

Many recent advances in machine learning are driven by a challenging trifecta: large data size $N$; high dimensions; and expensive algorithms. In this setting, cross-validation (CV) serves as an important tool for model assessment. Recent advances in approximate cross validation (ACV) provide accurate approximations to CV with only a single model fit, avoiding traditional CV's requirement for repeated runs of expensive algorithms. Unfortunately, these ACV methods can lose both speed and accuracy in high dimensions -- unless sparsity structure is present in the data. Fortunately, there is an alternative type of simplifying structure that is present in most data: approximate low rank (ALR). Guided by this observation, we develop a new algorithm for ACV that is fast and accurate in the presence of ALR data. Our first key insight is that the Hessian matrix -- whose inverse forms the computational bottleneck of existing ACV methods -- is ALR. We show that, despite our use of the \emph{inverse} Hessian, a low-rank approximation using the largest (rather than the smallest) matrix eigenvalues enables fast, reliable ACV. Our second key insight is that, in the presence of ALR data, error in existing ACV methods roughly grows with the (approximate, low) rank rather than with the (full, high) dimension. These insights allow us to prove theoretical guarantees on the quality of our proposed algorithm -- along with fast-to-compute upper bounds on its error. We demonstrate the speed and accuracy of our method, as well as the usefulness of our bounds, on a range of real and simulated data sets.

Diana Cai, Trevor Campbell, Tamara Broderick

Scientists and engineers are often interested in learning the number of subpopulations (or components) present in a data set. A common suggestion is to use a finite mixture model (FMM) with a prior on the number of components. Past work has shown the resulting FMM component-count posterior is consistent; that is, the posterior concentrates on the true, generating number of components. But consistency requires the assumption that the component likelihoods are perfectly specified, which is unrealistic in practice. In this paper, we add rigor to data-analysis folk wisdom by proving that under even the slightest model misspecification, the FMM component-count posterior diverges: the posterior probability of any particular finite number of components converges to 0 in the limit of infinite data. Contrary to intuition, posterior-density consistency is not sufficient to establish this result. We develop novel sufficient conditions that are more realistic and easily checkable than those common in the asymptotics literature. We illustrate practical consequences of our theory on simulated and real data.

Soumya Ghosh, William T. Stephenson, Tin D. Nguyen et al.

Many modern data analyses benefit from explicitly modeling dependence structure in data -- such as measurements across time or space, ordered words in a sentence, or genes in a genome. A gold standard evaluation technique is structured cross-validation (CV), which leaves out some data subset (such as data within a time interval or data in a geographic region) in each fold. But CV here can be prohibitively slow due to the need to re-run already-expensive learning algorithms many times. Previous work has shown approximate cross-validation (ACV) methods provide a fast and provably accurate alternative in the setting of empirical risk minimization. But this existing ACV work is restricted to simpler models by the assumptions that (i) data across CV folds are independent and (ii) an exact initial model fit is available. In structured data analyses, both these assumptions are often untrue. In the present work, we address (i) by extending ACV to CV schemes with dependence structure between the folds. To address (ii), we verify -- both theoretically and empirically -- that ACV quality deteriorates smoothly with noise in the initial fit. We demonstrate the accuracy and computational benefits of our proposed methods on a diverse set of real-world applications.

Jonathan H. Huggins, Mikołaj Kasprzak, Trevor Campbell et al.

Variational inference has become an increasingly attractive fast alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of post-hoc accuracy measures that are both theoretically justified and computationally efficient. In this paper, we provide rigorous bounds on the error of posterior mean and uncertainty estimates that arise from full-distribution approximations, as in variational inference. Our bounds are widely applicable, as they require only that the approximating and exact posteriors have polynomial moments. Our bounds are also computationally efficient for variational inference because they require only standard values from variational objectives, straightforward analytic calculations, and simple Monte Carlo estimates. We show that our analysis naturally leads to a new and improved workflow for validated variational inference. Finally, we demonstrate the utility of our proposed workflow and error bounds on a robust regression problem and on a real-data example with a widely used multilevel hierarchical model.

Ryan Giordano, Michael I. Jordan, Tamara Broderick

Cross validation (CV) and the bootstrap are ubiquitous model-agnostic tools for assessing the error or variability of machine learning and statistical estimators. However, these methods require repeatedly re-fitting the model with different weighted versions of the original dataset, which can be prohibitively time-consuming. For sufficiently regular optimization problems the optimum depends smoothly on the data weights, and so the process of repeatedly re-fitting can be approximated with a Taylor series that can be often evaluated relatively quickly. The first-order approximation is known as the "infinitesimal jackknife" in the statistics literature and has been the subject of recent interest in machine learning for approximate CV. In this work, we consider high-order approximations, which we call the "higher-order infinitesimal jackknife" (HOIJ). Under mild regularity conditions, we provide a simple recursive procedure to compute approximations of all orders with finite-sample accuracy bounds. Additionally, we show that the HOIJ can be efficiently computed even in high dimensions using forward-mode automatic differentiation. We show that a linear approximation with bootstrap weights approximation is equivalent to those provided by asymptotic normal approximations. Consequently, the HOIJ opens up the possibility of enjoying higher-order accuracy properties of the bootstrap using local approximations. Consistency of the HOIJ for leave-one-out CV under different asymptotic regimes follows as corollaries from our finite-sample bounds under additional regularity assumptions. The generality of the computation and bounds motivate the name "higher-order Swiss Army infinitesimal jackknife."

William T. Stephenson, Tamara Broderick

Leave-one-out cross-validation (LOOCV) can be particularly accurate among cross-validation (CV) variants for machine learning assessment tasks -- e.g., assessing methods' error or variability. But it is expensive to re-fit a model $N$ times for a dataset of size $N$. Previous work has shown that approximations to LOOCV can be both fast and accurate -- when the unknown parameter is of small, fixed dimension. But these approximations incur a running time roughly cubic in dimension -- and we show that, besides computational issues, their accuracy dramatically deteriorates in high dimensions. Authors have suggested many potential and seemingly intuitive solutions, but these methods have not yet been systematically evaluated or compared. We find that all but one perform so poorly as to be unusable for approximating LOOCV. Crucially, though, we are able to show, both empirically and theoretically, that one approximation can perform well in high dimensions -- in cases where the high-dimensional parameter exhibits sparsity. Under interpretable assumptions, our theory demonstrates that the problem can be reduced to working within an empirically recovered (small) support. This procedure is straightforward to implement, and we prove that its running time and error depend on the (small) support size even when the full parameter dimension is large.

Brian L. Trippe, Jonathan H. Huggins, Raj Agrawal et al.

Due to the ease of modern data collection, applied statisticians often have access to a large set of covariates that they wish to relate to some observed outcome. Generalized linear models (GLMs) offer a particularly interpretable framework for such an analysis. In these high-dimensional problems, the number of covariates is often large relative to the number of observations, so we face non-trivial inferential uncertainty; a Bayesian approach allows coherent quantification of this uncertainty. Unfortunately, existing methods for Bayesian inference in GLMs require running times roughly cubic in parameter dimension, and so are limited to settings with at most tens of thousand parameters. We propose to reduce time and memory costs with a low-rank approximation of the data in an approach we call LR-GLM. When used with the Laplace approximation or Markov chain Monte Carlo, LR-GLM provides a full Bayesian posterior approximation and admits running times reduced by a full factor of the parameter dimension. We rigorously establish the quality of our approximation and show how the choice of rank allows a tunable computational-statistical trade-off. Experiments support our theory and demonstrate the efficacy of LR-GLM on real large-scale datasets.

Raj Agrawal, Jonathan H. Huggins, Brian Trippe et al.

Discovering interaction effects on a response of interest is a fundamental problem faced in biology, medicine, economics, and many other scientific disciplines. In theory, Bayesian methods for discovering pairwise interactions enjoy many benefits such as coherent uncertainty quantification, the ability to incorporate background knowledge, and desirable shrinkage properties. In practice, however, Bayesian methods are often computationally intractable for even moderate-dimensional problems. Our key insight is that many hierarchical models of practical interest admit a particular Gaussian process (GP) representation; the GP allows us to capture the posterior with a vector of O(p) kernel hyper-parameters rather than O(p^2) interactions and main effects. With the implicit representation, we can run Markov chain Monte Carlo (MCMC) over model hyper-parameters in time and memory linear in p per iteration. We focus on sparsity-inducing models and show on datasets with a variety of covariate behaviors that our method: (1) reduces runtime by orders of magnitude over naive applications of MCMC, (2) provides lower Type I and Type II error relative to state-of-the-art LASSO-based approaches, and (3) offers improved computational scaling in high dimensions relative to existing Bayesian and LASSO-based approaches.

Miriam Shiffman, William T. Stephenson, Geoffrey Schiebinger et al.

Until recently, transcriptomics was limited to bulk RNA sequencing, obscuring the underlying expression patterns of individual cells in favor of a global average. Thanks to technological advances, we can now profile gene expression across thousands or millions of individual cells in parallel. This new type of data has led to the intriguing discovery that individual cell profiles can reflect the imprint of time or dynamic processes. However, synthesizing this information to reconstruct dynamic biological phenomena from data that are noisy, heterogenous, and sparse---and from processes that may unfold asynchronously---poses a complex computational and statistical challenge. Here, we develop a full generative model for probabilistically reconstructing trees of cellular differentiation from single-cell RNA-seq data. Specifically, we extend the framework of the classical Dirichlet diffusion tree to simultaneously infer branch topology and latent cell states along continuous trajectories over the full tree. In tandem, we construct a novel Markov chain Monte Carlo sampler that interleaves Metropolis-Hastings and message passing to leverage model structure for efficient inference. Finally, we demonstrate that these techniques can recover latent trajectories from simulated single-cell transcriptomes. While this work is motivated by cellular differentiation, we derive a tractable model that provides flexible densities for any data (coupled with an appropriate noise model) that arise from continuous evolution along a latent nonparametric tree.

Raj Agrawal, Trevor Campbell, Jonathan H. Huggins et al.

Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large-data setting. Random feature maps (RFMs) and the Nystrom method both consider low-rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features J+, and the latter may be prohibitively expensive for high-dimensional problems. We propose to combine the simplicity and generality of RFMs with a data-dependent feature selection scheme to achieve desirable theoretical approximation properties of Nystrom with just O(log J+) features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a data-dependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experiments--including on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than state-of-the-art methods.

Jonathan H. Huggins, Trevor Campbell, Mikołaj Kasprzak et al.

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, remain the gold standard for approximate Bayesian inference because they have a robust finite-sample theory and reliable convergence diagnostics. However, alternative methods, which are more scalable or apply to problems where Markov Chain Monte Carlo cannot be used, lack the same finite-data approximation theory and tools for evaluating their accuracy. In this work, we develop a flexible new approach to bounding the error of mean and uncertainty estimates of scalable inference algorithms. Our strategy is to control the estimation errors in terms of Wasserstein distance, then bound the Wasserstein distance via a generalized notion of Fisher distance. Unlike computing the Wasserstein distance, which requires access to the normalized posterior distribution, the Fisher distance is tractable to compute because it requires access only to the gradient of the log posterior density. We demonstrate the usefulness of our Fisher distance approach by deriving bounds on the Wasserstein error of the Laplace approximation and Hilbert coresets. We anticipate that our approach will be applicable to many other approximate inference methods such as the integrated Laplace approximation, variational inference, and approximate Bayesian computation

Jonathan H. Huggins, Trevor Campbell, Mikołaj Kasprzak et al.

Gaussian processes (GPs) offer a flexible class of priors for nonparametric Bayesian regression, but popular GP posterior inference methods are typically prohibitively slow or lack desirable finite-data guarantees on quality. We develop an approach to scalable approximate GP regression with finite-data guarantees on the accuracy of pointwise posterior mean and variance estimates. Our main contribution is a novel objective for approximate inference in the nonparametric setting: the preconditioned Fisher (pF) divergence. We show that unlike the Kullback--Leibler divergence (used in variational inference), the pF divergence bounds the 2-Wasserstein distance, which in turn provides tight bounds the pointwise difference of the mean and variance functions. We demonstrate that, for sparse GP likelihood approximations, we can minimize the pF divergence efficiently. Our experiments show that optimizing the pF divergence has the same computational requirements as variational sparse GPs while providing comparable empirical performance--in addition to our novel finite-data quality guarantees.

Raj Agrawal, Tamara Broderick, Caroline Uhler

Learning a Bayesian network (BN) from data can be useful for decision-making or discovering causal relationships. However, traditional methods often fail in modern applications, which exhibit a larger number of observed variables than data points. The resulting uncertainty about the underlying network as well as the desire to incorporate prior information recommend a Bayesian approach to learning the BN, but the highly combinatorial structure of BNs poses a striking challenge for inference. The current state-of-the-art methods such as order MCMC are faster than previous methods but prevent the use of many natural structural priors and still have running time exponential in the maximum indegree of the true directed acyclic graph (DAG) of the BN. We here propose an alternative posterior approximation based on the observation that, if we incorporate empirical conditional independence tests, we can focus on a high-probability DAG associated with each order of the vertices. We show that our method allows the desired flexibility in prior specification, removes timing dependence on the maximum indegree and yields provably good posterior approximations; in addition, we show that it achieves superior accuracy, scalability, and sampler mixing on several datasets.

Trevor Campbell, Tamara Broderick

Coherent uncertainty quantification is a key strength of Bayesian methods. But modern algorithms for approximate Bayesian posterior inference often sacrifice accurate posterior uncertainty estimation in the pursuit of scalability. This work shows that previous Bayesian coreset construction algorithms---which build a small, weighted subset of the data that approximates the full dataset---are no exception. We demonstrate that these algorithms scale the coreset log-likelihood suboptimally, resulting in underestimated posterior uncertainty. To address this shortcoming, we develop greedy iterative geodesic ascent (GIGA), a novel algorithm for Bayesian coreset construction that scales the coreset log-likelihood optimally. GIGA provides geometric decay in posterior approximation error as a function of coreset size, and maintains the fast running time of its predecessors. The paper concludes with validation of GIGA on both synthetic and real datasets, demonstrating that it reduces posterior approximation error by orders of magnitude compared with previous coreset constructions.

Trevor Campbell, Tamara Broderick

The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern datasets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to shrink the dataset itself as a preprocessing step, providing fully-automated, scalable Bayesian inference with theoretical guarantees. We begin with an intuitive reformulation of Bayesian coreset construction as sparse vector sum approximation, and demonstrate that its automation and performance-based shortcomings arise from the use of the supremum norm. To address these shortcomings we develop Hilbert coresets, i.e., Bayesian coresets constructed under a norm induced by an inner-product on the log-likelihood function space. We propose two Hilbert coreset construction algorithms---one based on importance sampling, and one based on the Frank-Wolfe algorithm---along with theoretical guarantees on approximation quality as a function of coreset size. Since the exact computation of the proposed inner-products is model-specific, we automate the construction with a random finite-dimensional projection of the log-likelihood functions. The resulting automated coreset construction algorithm is simple to implement, and experiments on a variety of models with real and synthetic datasets show that it provides high-quality posterior approximations and a significant reduction in the computational cost of inference.

Jonathan H. Huggins, Ryan P. Adams, Tamara Broderick

Generalized linear models (GLMs) -- such as logistic regression, Poisson regression, and robust regression -- provide interpretable models for diverse data types. Probabilistic approaches, particularly Bayesian ones, allow coherent estimates of uncertainty, incorporation of prior information, and sharing of power across experiments via hierarchical models. In practice, however, the approximate Bayesian methods necessary for inference have either failed to scale to large data sets or failed to provide theoretical guarantees on the quality of inference. We propose a new approach based on constructing polynomial approximate sufficient statistics for GLMs (PASS-GLM). We demonstrate that our method admits a simple algorithm as well as trivial streaming and distributed extensions that do not compound error across computations. We provide theoretical guarantees on the quality of point (MAP) estimates, the approximate posterior, and posterior mean and uncertainty estimates. We validate our approach empirically in the case of logistic regression using a quadratic approximation and show competitive performance with stochastic gradient descent, MCMC, and the Laplace approximation in terms of speed and multiple measures of accuracy -- including on an advertising data set with 40 million data points and 20,000 covariates.

Diana Cai, Trevor Campbell, Tamara Broderick

Many popular network models rely on the assumption of (vertex) exchangeability, in which the distribution of the graph is invariant to relabelings of the vertices. However, the Aldous-Hoover theorem guarantees that these graphs are dense or empty with probability one, whereas many real-world graphs are sparse. We present an alternative notion of exchangeability for random graphs, which we call edge exchangeability, in which the distribution of a graph sequence is invariant to the order of the edges. We demonstrate that edge-exchangeable models, unlike models that are traditionally vertex exchangeable, can exhibit sparsity. To do so, we outline a general framework for graph generative models; by contrast to the pioneering work of Caron and Fox (2015), models within our framework are stationary across steps of the graph sequence. In particular, our model grows the graph by instantiating more latent atoms of a single random measure as the dataset size increases, rather than adding new atoms to the measure.

Fangjian Guo, Xiangyu Wang, Kai Fan et al.

Variational inference (VI) provides fast approximations of a Bayesian posterior in part because it formulates posterior approximation as an optimization problem: to find the closest distribution to the exact posterior over some family of distributions. For practical reasons, the family of distributions in VI is usually constrained so that it does not include the exact posterior, even as a limit point. Thus, no matter how long VI is run, the resulting approximation will not approach the exact posterior. We propose to instead consider a more flexible approximating family consisting of all possible finite mixtures of a parametric base distribution (e.g., Gaussian). For efficient inference, we borrow ideas from gradient boosting to develop an algorithm we call boosting variational inference (BVI). BVI iteratively improves the current approximation by mixing it with a new component from the base distribution family and thereby yields progressively more accurate posterior approximations as more computing time is spent. Unlike a number of common VI variants including mean-field VI, BVI is able to capture multimodality, general posterior covariance, and nonstandard posterior shapes.

Ryan Giordano, Tamara Broderick, Rachael Meager et al.

Bayesian hierarchical models are increasing popular in economics. When using hierarchical models, it is useful not only to calculate posterior expectations, but also to measure the robustness of these expectations to reasonable alternative prior choices. We use variational Bayes and linear response methods to provide fast, accurate posterior means and robustness measures with an application to measuring the effectiveness of microcredit in the developing world.

Jonathan H. Huggins, Trevor Campbell, Tamara Broderick

The use of Bayesian methods in large-scale data settings is attractive because of the rich hierarchical models, uncertainty quantification, and prior specification they provide. Standard Bayesian inference algorithms are computationally expensive, however, making their direct application to large datasets difficult or infeasible. Recent work on scaling Bayesian inference has focused on modifying the underlying algorithms to, for example, use only a random data subsample at each iteration. We leverage the insight that data is often redundant to instead obtain a weighted subset of the data (called a coreset) that is much smaller than the original dataset. We can then use this small coreset in any number of existing posterior inference algorithms without modification. In this paper, we develop an efficient coreset construction algorithm for Bayesian logistic regression models. We provide theoretical guarantees on the size and approximation quality of the coreset -- both for fixed, known datasets, and in expectation for a wide class of data generative models. Crucially, the proposed approach also permits efficient construction of the coreset in both streaming and parallel settings, with minimal additional effort. We demonstrate the efficacy of our approach on a number of synthetic and real-world datasets, and find that, in practice, the size of the coreset is independent of the original dataset size. Furthermore, constructing the coreset takes a negligible amount of time compared to that required to run MCMC on it.

Diana Cai, Tamara Broderick

Network data appear in a number of applications, such as online social networks and biological networks, and there is growing interest in both developing models for networks as well as studying the properties of such data. Since individual network datasets continue to grow in size, it is necessary to develop models that accurately represent the real-life scaling properties of networks. One behavior of interest is having a power law in the degree distribution. However, other types of power laws that have been observed empirically and considered for applications such as clustering and feature allocation models have not been studied as frequently in models for graph data. In this paper, we enumerate desirable asymptotic behavior that may be of interest for modeling graph data, including sparsity and several types of power laws. We outline a general framework for graph generative models using completely random measures; by contrast to the pioneering work of Caron and Fox (2015), we consider instantiating more of the existing atoms of the random measure as the dataset size increases rather than adding new atoms to the measure. We see that these two models can be complementary; they respectively yield interpretations as (1) time passing among existing members of a network and (2) new individuals joining a network. We detail a particular instance of this framework and show simulated results that suggest this model exhibits some desirable asymptotic power-law behavior.

Tamara Broderick, Diana Cai

A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs --- which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous-Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.

Ryan Giordano, Tamara Broderick, Michael Jordan

Mean field variational Bayes (MFVB) is a popular posterior approximation method due to its fast runtime on large-scale data sets. However, it is well known that a major failing of MFVB is that it underestimates the uncertainty of model variables (sometimes severely) and provides no information about model variable covariance. We generalize linear response methods from statistical physics to deliver accurate uncertainty estimates for model variables---both for individual variables and coherently across variables. We call our method linear response variational Bayes (LRVB). When the MFVB posterior approximation is in the exponential family, LRVB has a simple, analytic form, even for non-conjugate models. Indeed, we make no assumptions about the form of the true posterior. We demonstrate the accuracy and scalability of our method on a range of models for both simulated and real data.

Ryan Giordano, Tamara Broderick

Mean field variational Bayes (MFVB) is a popular posterior approximation method due to its fast runtime on large-scale data sets. However, it is well known that a major failing of MFVB is that it underestimates the uncertainty of model variables (sometimes severely) and provides no information about model variable covariance. We develop a fast, general methodology for exponential families that augments MFVB to deliver accurate uncertainty estimates for model variables -- both for individual variables and coherently across variables. MFVB for exponential families defines a fixed-point equation in the means of the approximating posterior, and our approach yields a covariance estimate by perturbing this fixed point. Inspired by linear response theory, we call our method linear response variational Bayes (LRVB). We also show how LRVB can be used to quickly calculate a measure of the influence of individual data points on parameter point estimates. We demonstrate the accuracy and scalability of our method by learning Gaussian mixture models for both simulated and real data.

Ryan Giordano, Tamara Broderick

Mean Field Variational Bayes (MFVB) is a popular posterior approximation method due to its fast runtime on large-scale data sets. However, it is well known that a major failing of MFVB is its (sometimes severe) underestimates of the uncertainty of model variables and lack of information about model variable covariance. We develop a fast, general methodology for exponential families that augments MFVB to deliver accurate uncertainty estimates for model variables -- both for individual variables and coherently across variables. MFVB for exponential families defines a fixed-point equation in the means of the approximating posterior, and our approach yields a covariance estimate by perturbing this fixed point. Inspired by linear response theory, we call our method linear response variational Bayes (LRVB). We demonstrate the accuracy of our method on simulated data sets.

Tamara Broderick, Rebecca C. Steorts

Bayesian entity resolution merges together multiple, noisy databases and returns the minimal collection of unique individuals represented, together with their true, latent record values. Bayesian methods allow flexible generative models that share power across databases as well as principled quantification of uncertainty for queries of the final, resolved database. However, existing Bayesian methods for entity resolution use Markov monte Carlo method (MCMC) approximations and are too slow to run on modern databases containing millions or billions of records. Instead, we propose applying variational approximations to allow scalable Bayesian inference in these models. We derive a coordinate-ascent approximation for mean-field variational Bayes, qualitatively compare our algorithm to existing methods, note unique challenges for inference that arise from the expected distribution of cluster sizes in entity resolution, and discuss directions for future work in this domain.