David M. Blei

Julius von Kügelgen, Michel Besserve, Liang Wendong et al. · eth-zurich

We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional mixtures of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision.

Nino Scherrer, Claudia Shi, Amir Feder et al.

This paper presents a case study on the design, administration, post-processing, and evaluation of surveys on large language models (LLMs). It comprises two components: (1) A statistical method for eliciting beliefs encoded in LLMs. We introduce statistical measures and evaluation metrics that quantify the probability of an LLM "making a choice", the associated uncertainty, and the consistency of that choice. (2) We apply this method to study what moral beliefs are encoded in different LLMs, especially in ambiguous cases where the right choice is not obvious. We design a large-scale survey comprising 680 high-ambiguity moral scenarios (e.g., "Should I tell a white lie?") and 687 low-ambiguity moral scenarios (e.g., "Should I stop for a pedestrian on the road?"). Each scenario includes a description, two possible actions, and auxiliary labels indicating violated rules (e.g., "do not kill"). We administer the survey to 28 open- and closed-source LLMs. We find that (a) in unambiguous scenarios, most models "choose" actions that align with commonsense. In ambiguous cases, most models express uncertainty. (b) Some models are uncertain about choosing the commonsense action because their responses are sensitive to the question-wording. (c) Some models reflect clear preferences in ambiguous scenarios. Specifically, closed-source models tend to agree with each other.

Luhuan Wu, Brian L. Trippe, Christian A. Naesseth et al.

Diffusion models have been successful on a range of conditional generation tasks including molecular design and text-to-image generation. However, these achievements have primarily depended on task-specific conditional training or error-prone heuristic approximations. Ideally, a conditional generation method should provide exact samples for a broad range of conditional distributions without requiring task-specific training. To this end, we introduce the Twisted Diffusion Sampler, or TDS. TDS is a sequential Monte Carlo (SMC) algorithm that targets the conditional distributions of diffusion models through simulating a set of weighted particles. The main idea is to use twisting, an SMC technique that enjoys good computational efficiency, to incorporate heuristic approximations without compromising asymptotic exactness. We first find in simulation and in conditional image generation tasks that TDS provides a computational statistical trade-off, yielding more accurate approximations with many particles but with empirical improvements over heuristics with as few as two particles. We then turn to motif-scaffolding, a core task in protein design, using a TDS extension to Riemannian diffusion models. On benchmark test cases, TDS allows flexible conditioning criteria and often outperforms the state of the art.

David M. Kaplan, David M. Blei

We develop a quantitative method to assess the style of American poems and to visualize a collection of poems in relation to one another. Qualitative poetry criticism helped guide our development of metrics that analyze various orthographic, syntactic, and phonemic features. These features are used to discover comprehensive stylistic information from a poem's multi-layered latent structure, and to compute distances between poems in this space. Visualizations provide ready access to the analytical components. We demonstrate our method on several collections of poetry, showing that it better delineates poetry style than the traditional word-occurrence features that are used in typical text analysis algorithms. Our method has potential applications to academic research of texts, to research of the intuitive personal response to poetry, and to making recommendations to readers based on their favorite poems.

Zhendong Wang, Ruijiang Gao, Mingzhang Yin et al.

This paper proposes probabilistic conformal prediction (PCP), a predictive inference algorithm that estimates a target variable by a discontinuous predictive set. Given inputs, PCP construct the predictive set based on random samples from an estimated generative model. It is efficient and compatible with either explicit or implicit conditional generative models. Theoretically, we show that PCP guarantees correct marginal coverage with finite samples. Empirically, we study PCP on a variety of simulated and real datasets. Compared to existing methods for conformal inference, PCP provides sharper predictive sets.

Charles C. Margossian, David M. Blei

In a probabilistic latent variable model, factorized (or mean-field) variational inference (F-VI) fits a separate parametric distribution for each latent variable. Amortized variational inference (A-VI) instead learns a common inference function, which maps each observation to its corresponding latent variable's approximate posterior. Typically, A-VI is used as a step in the training of variational autoencoders, however it stands to reason that A-VI could also be used as a general alternative to F-VI. In this paper we study when and why A-VI can be used for approximate Bayesian inference. We derive conditions on a latent variable model which are necessary, sufficient, and verifiable under which A-VI can attain F-VI's optimal solution, thereby closing the amortization gap. We prove these conditions are uniquely verified by simple hierarchical models, a broad class that encompasses many models in machine learning. We then show, on a broader class of models, how to expand the domain of AVI's inference function to improve its solution, and we provide examples, e.g. hidden Markov models, where the amortization gap cannot be closed.

Chirag Modi, Charles Margossian, Yuling Yao et al.

Variational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to the target posterior by minimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI based on the principle of score matching, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this, we develop score matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is also a ``black box'' variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We also study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. In all of our studies we find that GSM-VI is faster than BBVI, but without sacrificing accuracy. It requires 10-100x fewer gradient evaluations to obtain a comparable quality of approximation.

Achille Nazaret, David Blei

We introduce the unbounded depth neural network (UDN), an infinitely deep probabilistic model that adapts its complexity to the training data. The UDN contains an infinite sequence of hidden layers and places an unbounded prior on a truncation L, the layer from which it produces its data. Given a dataset of observations, the posterior UDN provides a conditional distribution of both the parameters of the infinite neural network and its truncation. We develop a novel variational inference algorithm to approximate this posterior, optimizing a distribution of the neural network weights and of the truncation depth L, and without any upper limit on L. To this end, the variational family has a special structure: it models neural network weights of arbitrary depth, and it dynamically creates or removes free variational parameters as its distribution of the truncation is optimized. (Unlike heuristic approaches to model search, it is solely through gradient-based optimization that this algorithm explores the space of truncations.) We study the UDN on real and synthetic data. We find that the UDN adapts its posterior depth to the dataset complexity; it outperforms standard neural networks of similar computational complexity; and it outperforms other approaches to infinite-depth neural networks.

Sachit Menon, David Blei, Carl Vondrick

Variational autoencoders (VAEs) suffer from posterior collapse, where the powerful neural networks used for modeling and inference optimize the objective without meaningfully using the latent representation. We introduce inference critics that detect and incentivize against posterior collapse by requiring correspondence between latent variables and the observations. By connecting the critic's objective to the literature in self-supervised contrastive representation learning, we show both theoretically and empirically that optimizing inference critics increases the mutual information between observations and latents, mitigating posterior collapse. This approach is straightforward to implement and requires significantly less training time than prior methods, yet obtains competitive results on three established datasets. Overall, the approach lays the foundation to bridge the previously disconnected frameworks of contrastive learning and probabilistic modeling with variational autoencoders, underscoring the benefits both communities may find at their intersection.

Achille Nazaret, Justin Hong, Elham Azizi et al.

Inferring causal relationships as directed acyclic graphs (DAGs) is an important but challenging problem. Differentiable Causal Discovery (DCD) is a promising approach to this problem, framing the search as a continuous optimization. But existing DCD methods are numerically unstable, with poor performance beyond tens of variables. In this paper, we propose Stable Differentiable Causal Discovery (SDCD), a new method that improves previous DCD methods in two ways: (1) It employs an alternative constraint for acyclicity; this constraint is more stable, both theoretically and empirically, and fast to compute. (2) It uses a training procedure tailored for sparse causal graphs, which are common in real-world scenarios. We first derive SDCD and prove its stability and correctness. We then evaluate it with both observational and interventional data and on both small-scale and large-scale settings. We find that SDCD outperforms existing methods in both convergence speed and accuracy and can scale to thousands of variables. We provide code at https://github.com/azizilab/sdcd.

Keyon Vafa, Susan Athey, David M. Blei

The rise of foundation models marks a paradigm shift in machine learning: instead of training specialized models from scratch, foundation models are first trained on massive datasets before being adapted or fine-tuned to make predictions on smaller datasets. Initially developed for text, foundation models have also excelled at making predictions about social science data. However, while many estimation problems in the social sciences use prediction as an intermediate step, they ultimately require different criteria for success. In this paper, we develop methods for fine-tuning foundation models to perform these estimation problems. We first characterize an omitted variable bias that can arise when a foundation model is only fine-tuned to maximize predictive accuracy. We then provide a novel set of conditions for fine-tuning under which estimates derived from a foundation model are root-n-consistent. Based on this theory, we develop new fine-tuning algorithms that empirically mitigate this omitted variable bias. To demonstrate our ideas, we study gender wage decomposition. This is a statistical estimation problem from econometrics where the goal is to decompose the gender wage gap into components that can and cannot be explained by career histories of workers. Classical methods for decomposing the wage gap employ simple predictive models of wages which condition on coarse summaries of career history that may omit factors that are important for explaining the gap. Instead, we use a custom-built foundation model to decompose the gender wage gap, which captures a richer representation of career history. Using data from the Panel Study of Income Dynamics, we find that career history explains more of the gender wage gap than standard econometric models can measure, and we identify elements of career history that are omitted by standard models but are important for explaining the wage gap.

Yookoon Park, David M. Blei

Assessing the predictive uncertainty of deep neural networks is crucial for safety-related applications of deep learning. Although Bayesian deep learning offers a principled framework for estimating model uncertainty, the common approaches that approximate the parameter posterior often fail to deliver reliable estimates of predictive uncertainty. In this paper, we propose a novel criterion for reliable predictive uncertainty: a model's predictive variance should be grounded in the empirical density of the input. That is, the model should produce higher uncertainty for inputs that are improbable in the training data and lower uncertainty for inputs that are more probable. To operationalize this criterion, we develop the density uncertainty layer, a stochastic neural network architecture that satisfies the density uncertain criterion by design. We study density uncertainty layers on the UCI and CIFAR-10/100 uncertainty benchmarks. Compared to existing approaches, density uncertainty layers provide more reliable uncertainty estimates and robust out-of-distribution detection performance.

Linying Zhang, Lauren R. Richter, Yixin Wang et al.

Healthcare continues to grapple with the persistent issue of treatment disparities, sparking concerns regarding the equitable allocation of treatments in clinical practice. While various fairness metrics have emerged to assess fairness in decision-making processes, a growing focus has been on causality-based fairness concepts due to their capacity to mitigate confounding effects and reason about bias. However, the application of causal fairness notions in evaluating the fairness of clinical decision-making with electronic health record (EHR) data remains an understudied domain. This study aims to address the methodological gap in assessing causal fairness of treatment allocation with electronic health records data. We propose a causal fairness algorithm to assess fairness in clinical decision-making. Our algorithm accounts for the heterogeneity of patient populations and identifies potential unfairness in treatment allocation by conditioning on patients who have the same likelihood to benefit from the treatment. We apply this framework to a patient cohort with coronary artery disease derived from an EHR database to evaluate the fairness of treatment decisions. In addition, we investigate the impact of social determinants of health on the assessment of causal fairness of treatment allocation.

Dhanya Sridhar, Caterina De Bacco, David Blei

We consider the problem of estimating social influence, the effect that a person's behavior has on the future behavior of their peers. The key challenge is that shared behavior between friends could be equally explained by influence or by two other confounding factors: 1) latent traits that caused people to both become friends and engage in the behavior, and 2) latent preferences for the behavior. This paper addresses the challenges of estimating social influence with three contributions. First, we formalize social influence as a causal effect, one which requires inferences about hypothetical interventions. Second, we develop Poisson Influence Factorization (PIF), a method for estimating social influence from observational data. PIF fits probabilistic factor models to networks and behavior data to infer variables that serve as substitutes for the confounding latent traits. Third, we develop assumptions under which PIF recovers estimates of social influence. We empirically study PIF with semi-synthetic and real data from Last.fm, and conduct a sensitivity analysis. We find that PIF estimates social influence most accurately compared to related methods and remains robust under some violations of its assumptions.

Amir Feder, Yoav Wald, Claudia Shi et al.

The reliance of text classifiers on spurious correlations can lead to poor generalization at deployment, raising concerns about their use in safety-critical domains such as healthcare. In this work, we propose to use counterfactual data augmentation, guided by knowledge of the causal structure of the data, to simulate interventions on spurious features and to learn more robust text classifiers. We show that this strategy is appropriate in prediction problems where the label is spuriously correlated with an attribute. Under the assumptions of such problems, we discuss the favorable sample complexity of counterfactual data augmentation, compared to importance re-weighting. Pragmatically, we match examples using auxiliary data, based on diff-in-diff methodology, and use a large language model (LLM) to represent a conditional probability of text. Through extensive experimentation on learning caregiver-invariant predictors of clinical diagnoses from medical narratives and on semi-synthetic data, we demonstrate that our method for simulating interventions improves out-of-distribution (OOD) accuracy compared to baseline invariant learning algorithms.

Yuli Slavutsky, Matthew Shen, Bohan Wu et al.

We consider multi-environment prediction problems. We assume the environments change the distribution of a latent variable, while the mechanisms generating observed covariates and targets remain stable conditional on that variable. For example, hospitals or clinical cohorts may differ in the prevalence of latent patient states, even though the relationships between those states, physiological measurements, and outcomes remain unchanged. Given a dataset from multiple environments, we formulate a Bayesian model for such problems and derive the corresponding variational objective. We show that this objective decomposes into per-environment terms and an additional cross-environment balancing term induced by the model's structure. We use an empirical Bayes method to set the prior and incorporate it into the objective. Based on this objective, we develop an amortized variational algorithm for posterior approximation, and use the resulting learned latent variables to form predictions in new environments.We study our approach through simulations and real-world studies of astronomical source identification, microbiome-based disease detection, and ICU sepsis prediction. Across these settings, our method outperforms previous approaches for prediction in new environments.

Bohan Wu, Julius von Kügelgen, David M. Blei

Causal representation learning (CRL) aims to learn low-dimensional causal latent variables from high-dimensional observations. While identifiability has been extensively studied for CRL, estimation has been less explored. In this paper, we explore the use of empirical Bayes (EB) to estimate causal representations. In particular, we consider the problem of learning from data from multiple domains, where differences between domains are modeled by interventions in a shared underlying causal model. Multi-domain CRL naturally poses a simultaneous inference problem that EB is designed to tackle. Here, we propose an EB $f$-modeling algorithm that improves the quality of learned causal variables by exploiting invariant structure within and across domains. Specifically, we consider a linear measurement model and interventional priors arising from a shared acyclic SCM. When the graph and intervention targets are known, we develop an EM-style algorithm based on causally structured score matching. We further discuss EB $\rmg$-modeling in the context of existing CRL approaches. In experiments on synthetic data, our proposed method achieves more accurate estimation than other methods for CRL.

Yuli Slavutsky, Sebastian Salazar, David M. Blei

Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.

Yuli Slavutsky, David M. Blei

We consider learning from labeled data collected across multiple environments, where the data distribution may vary across these environments. This problem is commonly approached from a causal perspective, seeking invariant representations that retain causal factors while discarding spurious ones. However, this framework assumes that the environment has no direct effect on the target. In contrast, we consider settings in which this assumption fails, but still aim to learn representations that support robust prediction on average across previously unseen environments. To this end, we study representations learned by explicitly modeling variation across environments and then marginalizing that variation out. We analyze the resulting representations and characterize when they are preferable to those learned by causal invariant-representation methods. We propose a concrete method based on generalized random-intercept models, a class of predictors in which such marginalization is possible, and study their generalization properties. Empirically, we show that these models outperform invariant-learning methods across a range of challenging settings.

Claudia Shi, Nicolas Beltran-Velez, Achille Nazaret et al.

Large language models (LLMs) demonstrate surprising capabilities, but we do not understand how they are implemented. One hypothesis suggests that these capabilities are primarily executed by small subnetworks within the LLM, known as circuits. But how can we evaluate this hypothesis? In this paper, we formalize a set of criteria that a circuit is hypothesized to meet and develop a suite of hypothesis tests to evaluate how well circuits satisfy them. The criteria focus on the extent to which the LLM's behavior is preserved, the degree of localization of this behavior, and whether the circuit is minimal. We apply these tests to six circuits described in the research literature. We find that synthetic circuits -- circuits that are hard-coded in the model -- align with the idealized properties. Circuits discovered in Transformer models satisfy the criteria to varying degrees. To facilitate future empirical studies of circuits, we created the \textit{circuitry} package, a wrapper around the \textit{TransformerLens} library, which abstracts away lower-level manipulations of hooks and activations. The software is available at \url{https://github.com/blei-lab/circuitry}.

Achille Nazaret, David Blei

The goal of causal discovery is to learn a directed acyclic graph from data. One of the most well-known methods for this problem is Greedy Equivalence Search (GES). GES searches for the graph by incrementally and greedily adding or removing edges to maximize a model selection criterion. It has strong theoretical guarantees on infinite data but can fail in practice on finite data. In this paper, we first identify some of the causes of GES's failure, finding that it can get blocked in local optima, especially in denser graphs. We then propose eXtremely Greedy Equivalent Search (XGES), which involves a new heuristic to improve the search strategy of GES while retaining its theoretical guarantees. In particular, XGES favors deleting edges early in the search over inserting edges, which reduces the possibility of the search ending in local optima. A further contribution of this work is an efficient algorithmic formulation of XGES (and GES). We benchmark XGES on simulated datasets with known ground truth. We find that XGES consistently outperforms GES in recovering the correct graphs, and it is 10 times faster. XGES implementations in Python and C++ are available at https://github.com/ANazaret/XGES.

Nicolas Beltran-Velez, Alessandro Antonio Grande, Achille Nazaret et al.

Probabilistic prediction aims to compute predictive distributions rather than single point predictions. These distributions enable practitioners to quantify uncertainty, compute risk, and detect outliers. However, most probabilistic methods assume parametric responses, such as Gaussian or Poisson distributions. When these assumptions fail, such models lead to bad predictions and poorly calibrated uncertainty. In this paper, we propose Treeffuser, an easy-to-use method for probabilistic prediction on tabular data. The idea is to learn a conditional diffusion model where the score function is estimated using gradient-boosted trees. The conditional diffusion model makes Treeffuser flexible and non-parametric, while the gradient-boosted trees make it robust and easy to train on CPUs. Treeffuser learns well-calibrated predictive distributions and can handle a wide range of regression tasks -- including those with multivariate, multimodal, and skewed responses. We study Treeffuser on synthetic and real data and show that it outperforms existing methods, providing better calibrated probabilistic predictions. We further demonstrate its versatility with an application to inventory allocation under uncertainty using sales data from Walmart. We implement Treeffuser in https://github.com/blei-lab/treeffuser.

Yookoon Park, Sangho Lee, Gunhee Kim et al.

We present neural activation coding (NAC) as a novel approach for learning deep representations from unlabeled data for downstream applications. We argue that the deep encoder should maximize its nonlinear expressivity on the data for downstream predictors to take full advantage of its representation power. To this end, NAC maximizes the mutual information between activation patterns of the encoder and the data over a noisy communication channel. We show that learning for a noise-robust activation code increases the number of distinct linear regions of ReLU encoders, hence the maximum nonlinear expressivity. More interestingly, NAC learns both continuous and discrete representations of data, which we respectively evaluate on two downstream tasks: (i) linear classification on CIFAR-10 and ImageNet-1K and (ii) nearest neighbor retrieval on CIFAR-10 and FLICKR-25K. Empirical results show that NAC attains better or comparable performance on both tasks over recent baselines including SimCLR and DistillHash. In addition, NAC pretraining provides significant benefits to the training of deep generative models. Our code is available at https://github.com/yookoon/nac.

Luhuan Wu, Andrew Miller, Lauren Anderson et al.

We examine the general problem of inter-domain Gaussian Processes (GPs): problems where the GP realization and the noisy observations of that realization lie on different domains. When the mapping between those domains is linear, such as integration or differentiation, inference is still closed form. However, many of the scaling and approximation techniques that our community has developed do not apply to this setting. In this work, we introduce the hierarchical inducing point GP (HIP-GP), a scalable inter-domain GP inference method that enables us to improve the approximation accuracy by increasing the number of inducing points to the millions. HIP-GP, which relies on inducing points with grid structure and a stationary kernel assumption, is suitable for low-dimensional problems. In developing HIP-GP, we introduce (1) a fast whitening strategy, and (2) a novel preconditioner for conjugate gradients which can be helpful in general GP settings. Our code is available at https: //github.com/cunningham-lab/hipgp.

Wesley Tansey, Christopher Tosh, David M. Blei

Exploratory cancer drug studies test multiple tumor cell lines against multiple candidate drugs. The goal in each paired (cell line, drug) experiment is to map out the dose-response curve of the cell line as the dose level of the drug increases. We propose Bayesian Tensor Filtering (BTF), a hierarchical Bayesian model for dose-response modeling in multi-sample, multi-treatment cancer drug studies. BTF uses low-dimensional embeddings to share statistical strength between similar drugs and similar cell lines. Structured shrinkage priors in BTF encourage smoothness in the dose-response curves while remaining adaptive to sharp jumps when the data call for it. We focus on a pair of cancer drug studies exhibiting a particular pathology in their experimental design, leading us to a non-conjugate monotone mixture-of-Gammas likelihood. To perform posterior inference, we develop a variant of the elliptical slice sampling algorithm for sampling from linearly-constrained multivariate normal priors with non-conjugate likelihoods. In benchmarks, BTF outperforms state-of-the-art methods for covariance regression and dynamic Poisson matrix factorization. On the two cancer drug studies, BTF outperforms the current standard approach in biology and reveals potential new biomarkers of drug sensitivity in cancer. Code is available at https://github.com/tansey/functionalmf.

Claudia Shi, David M. Blei, Victor Veitch

This paper addresses the use of neural networks for the estimation of treatment effects from observational data. Generally, estimation proceeds in two stages. First, we fit models for the expected outcome and the probability of treatment (propensity score) for each unit. Second, we plug these fitted models into a downstream estimator of the effect. Neural networks are a natural choice for the models in the first step. The question we address is: how can we adapt the design and training of the neural networks used in the first step in order to improve the quality of the final estimate of the treatment effect? We propose two adaptations based on insights from the statistical literature on the estimation of treatment effects. The first is a new architecture, the Dragonnet, that exploits the sufficiency of the propensity score for estimation adjustment. The second is a regularization procedure, targeted regularization, that induces a bias towards models that have non-parametrically optimal asymptotic properties `out-of-the-box`. Studies on benchmark datasets for causal inference show these adaptations outperform existing methods. Code is available at github.com/claudiashi57/dragonnet.

Victor Veitch, Dhanya Sridhar, David M. Blei

Does adding a theorem to a paper affect its chance of acceptance? Does labeling a post with the author's gender affect the post popularity? This paper develops a method to estimate such causal effects from observational text data, adjusting for confounding features of the text such as the subject or writing quality. We assume that the text suffices for causal adjustment but that, in practice, it is prohibitively high-dimensional. To address this challenge, we develop causally sufficient embeddings, low-dimensional document representations that preserve sufficient information for causal identification and allow for efficient estimation of causal effects. Causally sufficient embeddings combine two ideas. The first is supervised dimensionality reduction: causal adjustment requires only the aspects of text that are predictive of both the treatment and outcome. The second is efficient language modeling: representations of text are designed to dispose of linguistically irrelevant information, and this information is also causally irrelevant. Our method adapts language models (specifically, word embeddings and topic models) to learn document embeddings that are able to predict both treatment and outcome. We study causally sufficient embeddings with semi-synthetic datasets and find that they improve causal estimation over related embedding methods. We illustrate the methods by answering the two motivating questions---the effect of a theorem on paper acceptance and the effect of a gender label on post popularity. Code and data available at https://github.com/vveitch/causal-text-embeddings-tf2}{github.com/vveitch/causal-text-embeddings-tf2

Victor Veitch, Yixin Wang, David M. Blei

We consider causal inference in the presence of unobserved confounding. We study the case where a proxy is available for the unobserved confounding in the form of a network connecting the units. For example, the link structure of a social network carries information about its members. We show how to effectively use the proxy to do causal inference. The main idea is to reduce the causal estimation problem to a semi-supervised prediction of both the treatments and outcomes. Networks admit high-quality embedding models that can be used for this semi-supervised prediction. We show that the method yields valid inferences under suitable (weak) conditions on the quality of the predictive model. We validate the method with experiments on a semi-synthetic social network dataset. Code is available at github.com/vveitch/causal-network-embeddings.

Victor Veitch, Morgane Austern, Wenda Zhou et al.

Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnkey solver for the risk minimization problem. We establish basic theoretical properties of the procedure. Finally, we demonstrate relational ERM with application to two non-standard problems: one-stage training for semi-supervised node classification, and learning embedding vectors for vertex attributes. Experiments confirm that the turnkey inference procedure is effective in practice, and that the sampling scheme used for model specification has a strong effect on model performance. Code is available at https://github.com/wooden-spoon/relational-ERM.

Alp Kucukelbir, Rajesh Ranganath, Andrew Gelman et al.

Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference algorithm, automatic differentiation variational inference (ADVI). The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and support a broad class of models. The algorithm automatically determines an appropriate variational family and optimizes the variational objective. We implement ADVI in Stan (code available now), a probabilistic programming framework. We compare ADVI to MCMC sampling across hierarchical generalized linear models, nonconjugate matrix factorization, and a mixture model. We train the mixture model on a quarter million images. With ADVI we can use variational inference on any model we write in Stan.

Diana Cai, Chirag Modi, Loucas Pillaud-Vivien et al.

Most leading implementations of black-box variational inference (BBVI) are based on optimizing a stochastic evidence lower bound (ELBO). But such approaches to BBVI often converge slowly due to the high variance of their gradient estimates and their sensitivity to hyperparameters. In this work, we propose batch and match (BaM), an alternative approach to BBVI based on a score-based divergence. Notably, this score-based divergence can be optimized by a closed-form proximal update for Gaussian variational families with full covariance matrices. We analyze the convergence of BaM when the target distribution is Gaussian, and we prove that in the limit of infinite batch size the variational parameter updates converge exponentially quickly to the target mean and covariance. We also evaluate the performance of BaM on Gaussian and non-Gaussian target distributions that arise from posterior inference in hierarchical and deep generative models. In these experiments, we find that BaM typically converges in fewer (and sometimes significantly fewer) gradient evaluations than leading implementations of BBVI based on ELBO maximization.

Diana Cai, Chirag Modi, Charles C. Margossian et al.

We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI). EigenVI constructs its variational approximations from orthogonal function expansions. For distributions over $\mathbb{R}^D$, the lowest order term in these expansions provides a Gaussian variational approximation, while higher-order terms provide a systematic way to model non-Gaussianity. These approximations are flexible enough to model complex distributions (multimodal, asymmetric), but they are simple enough that one can calculate their low-order moments and draw samples from them. EigenVI can also model other types of random variables (e.g., nonnegative, bounded) by constructing variational approximations from different families of orthogonal functions. Within these families, EigenVI computes the variational approximation that best matches the score function of the target distribution by minimizing a stochastic estimate of the Fisher divergence. Notably, this optimization reduces to solving a minimum eigenvalue problem, so that EigenVI effectively sidesteps the iterative gradient-based optimizations that are required for many other BBVI algorithms. (Gradient-based methods can be sensitive to learning rates, termination criteria, and other tunable hyperparameters.) We use EigenVI to approximate a variety of target distributions, including a benchmark suite of Bayesian models from posteriordb. On these distributions, we find that EigenVI is more accurate than existing methods for Gaussian BBVI.

Luhuan Wu, Mingzhang Yin, Yixin Wang et al.

Invariant prediction [Peters et al., 2016] analyzes feature/outcome data from multiple environments to identify invariant features - those with a stable predictive relationship to the outcome. Such features support generalization to new environments and help reveal causal mechanisms. Previous methods have primarily tackled this problem through hypothesis testing or regularized optimization. Here we develop Bayesian Invariant Prediction (BIP), a probabilistic model for invariant prediction. BIP encodes the indices of invariant features as a latent variable and recover them by posterior inference. Under the assumptions of Peters et al. [2016], the BIP posterior targets the true invariant features. We prove that the posterior is consistent and that greater environment heterogeneity leads to faster posterior contraction. To handle many features, we design an efficient variational approximation called VI-BIP. In simulations and real data, we find that BIP and VI-BIP are more accurate and scalable than existing methods for invariant prediction.

Bohan Wu, David Blei

Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $Ξ$-variational inference ($Ξ$-VI). $Ξ$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $Ξ$-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $Ξ$-variational approximation and how it affects computational considerations, providing a rough characterization of the statistical-computational trade-off in $Ξ$-VI. We also investigate the frequentist properties of $Ξ$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for achieving polynomial-time approximate inference using the method. Finally, we demonstrate the practical advantage of $Ξ$-VI over mean-field variational inference on simulated and real data.

Andrew Jesson, Nicolas Beltran-Velez, David Blei

This work is about estimating when a conditional generative model (CGM) can solve an in-context learning (ICL) problem. An in-context learning (ICL) problem comprises a CGM, a dataset, and a prediction task. The CGM could be a multi-modal foundation model; the dataset, a collection of patient histories, test results, and recorded diagnoses; and the prediction task to communicate a diagnosis to a new patient. A Bayesian interpretation of ICL assumes that the CGM computes a posterior predictive distribution over an unknown Bayesian model defining a joint distribution over latent explanations and observable data. From this perspective, Bayesian model criticism is a reasonable approach to assess the suitability of a given CGM for an ICL problem. However, such approaches -- like posterior predictive checks (PPCs) -- often assume that we can sample from the likelihood and posterior defined by the Bayesian model, which are not explicitly given for contemporary CGMs. To address this, we show when ancestral sampling from the predictive distribution of a CGM is equivalent to sampling datasets from the posterior predictive of the assumed Bayesian model. Then we develop the generative predictive $p$-value, which enables PPCs and their cousins for contemporary CGMs. The generative predictive $p$-value can then be used in a statistical decision procedure to determine when the model is appropriate for an ICL problem. Our method only requires generating queries and responses from a CGM and evaluating its response log probability. We empirically evaluate our method on synthetic tabular, imaging, and natural language ICL tasks using large language models.

Eli N. Weinstein, Elizabeth B. Wood, David M. Blei

A central question in human immunology is how a patient's repertoire of T cells impacts disease. Here, we introduce a method to infer the causal effects of T cell receptor (TCR) sequences on patient outcomes using observational TCR repertoire sequencing data and clinical outcomes data. Our approach corrects for unobserved confounders, such as a patient's environment and life history, by using the patient's immature, pre-selection TCR repertoire. The pre-selection repertoire can be estimated from nonproductive TCR data, which is widely available. It is generated by a randomized mutational process, V(D)J recombination, which provides a natural experiment. We show formally how to use the pre-selection repertoire to draw causal inferences, and develop a scalable neural-network estimator for our identification formula. Our method produces an estimate of the effect of interventions that add a specific TCR sequence to patient repertoires. As a demonstration, we use it to analyze the effects of TCRs on COVID-19 severity, uncovering potentially therapeutic TCRs that are (1) observed in patients, (2) bind SARS-CoV-2 antigens in vitro and (3) have strong positive effects on clinical outcomes.

Carolina Zheng, Keyon Vafa, David M. Blei

A recent line of work in natural language processing has aimed to combine language models and topic models. These topic-guided language models augment neural language models with topic models, unsupervised learning methods that can discover document-level patterns of word use. This paper compares the effectiveness of these methods in a standardized setting. We study four topic-guided language models and two baselines, evaluating the held-out predictive performance of each model on four corpora. Surprisingly, we find that none of these methods outperform a standard LSTM language model baseline, and most fail to learn good topics. Further, we train a probe of the neural language model that shows that the baseline's hidden states already encode topic information. We make public all code used for this study.

Diana Cai, Robert M. Gower, David M. Blei et al.

We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be able to sample from their densities. We show how to do this by reformulating these products of experts as latent variable models with auxiliary Dirichlet random variables. These Dirichlet variables emerge from a Feynman identity, originally developed for loop integrals in quantum field theory, that expresses the product of multiple fractions (or in our case, $t$-distributions) as an integral over the simplex. We leverage this simplicial latent space to draw weighted samples from these products of experts -- samples which BBVI then uses to find the PoE that best approximates a target density. Given a collection of experts, we derive an iterative procedure to optimize the exponents that determine their geometric weighting in the PoE. At each iteration, this procedure minimizes a regularized Fisher divergence to match the scores of the variational and target densities at a batch of samples drawn from the current approximation. This minimization reduces to a convex quadratic program, and we prove under general conditions that these updates converge exponentially fast to a near-optimal weighting of experts. We conclude by evaluating this approach on a variety of synthetic and real-world target distributions.

Dominic Sobhani, Amir Feder, David Blei

Probabilistic topic models are a powerful tool for extracting latent themes from large text datasets. In many text datasets, we also observe per-document covariates (e.g., source, style, political affiliation) that act as environments that modulate a "global" (environment-agnostic) topic representation. Accurately learning these representations is important for prediction on new documents in unseen environments and for estimating the causal effect of topics on real-world outcomes. To this end, we introduce the Multi-environment Topic Model (MTM), an unsupervised probabilistic model that separates global and environment-specific terms. Through experimentation on various political content, from ads to tweets and speeches, we show that the MTM produces interpretable global topics with distinct environment-specific words. On multi-environment data, the MTM outperforms strong baselines in and out-of-distribution. It also enables the discovery of accurate causal effects.

Andrew Jesson, Nicolas Beltran-Velez, Quentin Chu et al.

This paper presents a method for estimating the hallucination rate for in-context learning (ICL) with generative AI. In ICL, a conditional generative model (CGM) is prompted with a dataset and a prediction question and asked to generate a response. One interpretation of ICL assumes that the CGM computes the posterior predictive of an unknown Bayesian model, which implicitly defines a joint distribution over observable datasets and latent mechanisms. This joint distribution factorizes into two components: the model prior over mechanisms and the model likelihood of datasets given a mechanism. With this perspective, we define a hallucination as a generated response to the prediction question with low model likelihood given the mechanism. We develop a new method that takes an ICL problem and estimates the probability that a CGM will generate a hallucination. Our method only requires generating prediction questions and responses from the CGM and evaluating its response log probability. We empirically evaluate our method using large language models for synthetic regression and natural language ICL tasks.

Carolina Zheng, Claudia Shi, Keyon Vafa et al.

Controlled generation refers to the problem of creating text that contains stylistic or semantic attributes of interest. Many approaches reduce this problem to training a predictor of the desired attribute. For example, researchers hoping to deploy a large language model to produce non-toxic content may use a toxicity classifier to filter generated text. In practice, the generated text to classify, which is determined by user prompts, may come from a wide range of distributions. In this paper, we show that the performance of controlled generation may be poor if the distributions of text in response to user prompts differ from the distribution the predictor was trained on. To address this problem, we cast controlled generation under distribution shift as an invariant learning problem: the most effective predictor should be invariant across multiple text environments. We then discuss a natural solution that arises from this characterization and propose heuristics for selecting natural environments. We study this characterization and the proposed method empirically using both synthetic and real data. Experiments demonstrate both the challenge of distribution shift in controlled generation and the potential of invariance methods in this setting.

Keyon Vafa, Emil Palikot, Tianyu Du et al.

Labor economists regularly analyze employment data by fitting predictive models to small, carefully constructed longitudinal survey datasets. Although machine learning methods offer promise for such problems, these survey datasets are too small to take advantage of them. In recent years large datasets of online resumes have also become available, providing data about the career trajectories of millions of individuals. However, standard econometric models cannot take advantage of their scale or incorporate them into the analysis of survey data. To this end we develop CAREER, a foundation model for job sequences. CAREER is first fit to large, passively-collected resume data and then fine-tuned to smaller, better-curated datasets for economic inferences. We fit CAREER to a dataset of 24 million job sequences from resumes, and adjust it on small longitudinal survey datasets. We find that CAREER forms accurate predictions of job sequences, outperforming econometric baselines on three widely-used economics datasets. We further find that CAREER can be used to form good predictions of other downstream variables. For example, incorporating CAREER into a wage model provides better predictions than the econometric models currently in use.

Liyi Zhang, David M. Blei, Christian A. Naesseth

Variational inference often minimizes the "reverse" Kullbeck-Leibler (KL) KL(q||p) from the approximate distribution q to the posterior p. Recent work studies the "forward" KL KL(p||q), which unlike reverse KL does not lead to variational approximations that underestimate uncertainty. This paper introduces Transport Score Climbing (TSC), a method that optimizes KL(p||q) by using Hamiltonian Monte Carlo (HMC) and a novel adaptive transport map. The transport map improves the trajectory of HMC by acting as a change of variable between the latent variable space and a warped space. TSC uses HMC samples to dynamically train the transport map while optimizing KL(p||q). TSC leverages synergies, where better transport maps lead to better HMC sampling, which then leads to better transport maps. We demonstrate TSC on synthetic and real data. We find that TSC achieves competitive performance when training variational autoencoders on large-scale data.

Gemma E. Moran, Dhanya Sridhar, Yixin Wang et al.

We develop the sparse VAE for unsupervised representation learning on high-dimensional data. The sparse VAE learns a set of latent factors (representations) which summarize the associations in the observed data features. The underlying model is sparse in that each observed feature (i.e. each dimension of the data) depends on a small subset of the latent factors. As examples, in ratings data each movie is only described by a few genres; in text data each word is only applicable to a few topics; in genomics, each gene is active in only a few biological processes. We prove such sparse deep generative models are identifiable: with infinite data, the true model parameters can be learned. (In contrast, most deep generative models are not identifiable.) We empirically study the sparse VAE with both simulated and real data. We find that it recovers meaningful latent factors and has smaller heldout reconstruction error than related methods.

Mingzhang Yin, Yixin Wang, David M. Blei

This paper presents a new optimization approach to causal estimation. Given data that contains covariates and an outcome, which covariates are causes of the outcome, and what is the strength of the causality? In classical machine learning (ML), the goal of optimization is to maximize predictive accuracy. However, some covariates might exhibit a non-causal association with the outcome. Such spurious associations provide predictive power for classical ML, but they prevent us from causally interpreting the result. This paper proposes CoCo, an optimization algorithm that bridges the gap between pure prediction and causal inference. CoCo leverages the recently-proposed idea of environments, datasets of covariates/response where the causal relationships remain invariant but where the distribution of the covariates changes from environment to environment. Given datasets from multiple environments-and ones that exhibit sufficient heterogeneity-CoCo maximizes an objective for which the only solution is the causal solution. We describe the theoretical foundations of this approach and demonstrate its effectiveness on simulated and real datasets. Compared to classical ML and existing methods, CoCo provides more accurate estimates of the causal model and more accurate predictions under interventions.

Keyon Vafa, Yuntian Deng, David M. Blei et al.

Sequence models are a critical component of modern NLP systems, but their predictions are difficult to explain. We consider model explanations though rationales, subsets of context that can explain individual model predictions. We find sequential rationales by solving a combinatorial optimization: the best rationale is the smallest subset of input tokens that would predict the same output as the full sequence. Enumerating all subsets is intractable, so we propose an efficient greedy algorithm to approximate this objective. The algorithm, which is called greedy rationalization, applies to any model. For this approach to be effective, the model should form compatible conditional distributions when making predictions on incomplete subsets of the context. This condition can be enforced with a short fine-tuning step. We study greedy rationalization on language modeling and machine translation. Compared to existing baselines, greedy rationalization is best at optimizing the combinatorial objective and provides the most faithful rationales. On a new dataset of annotated sequential rationales, greedy rationales are most similar to human rationales.

Antonio Khalil Moretti, Liyi Zhang, Christian A. Naesseth et al.

Bayesian phylogenetic inference is often conducted via local or sequential search over topologies and branch lengths using algorithms such as random-walk Markov chain Monte Carlo (MCMC) or Combinatorial Sequential Monte Carlo (CSMC). However, when MCMC is used for evolutionary parameter learning, convergence requires long runs with inefficient exploration of the state space. We introduce Variational Combinatorial Sequential Monte Carlo (VCSMC), a powerful framework that establishes variational sequential search to learn distributions over intricate combinatorial structures. We then develop nested CSMC, an efficient proposal distribution for CSMC and prove that nested CSMC is an exact approximation to the (intractable) locally optimal proposal. We use nested CSMC to define a second objective, VNCSMC which yields tighter lower bounds than VCSMC. We show that VCSMC and VNCSMC are computationally efficient and explore higher probability spaces than existing methods on a range of tasks.

Claudia Shi, Victor Veitch, David Blei

The defining challenge for causal inference from observational data is the presence of `confounders', covariates that affect both treatment assignment and the outcome. To address this challenge, practitioners collect and adjust for the covariates, hoping that they adequately correct for confounding. However, including every observed covariate in the adjustment runs the risk of including `bad controls', variables that induce bias when they are conditioned on. The problem is that we do not always know which variables in the covariate set are safe to adjust for and which are not. To address this problem, we develop Nearly Invariant Causal Estimation (NICE). NICE uses invariant risk minimization (IRM) [Arj19] to learn a representation of the covariates that, under some assumptions, strips out bad controls but preserves sufficient information to adjust for confounding. Adjusting for the learned representation, rather than the covariates themselves, avoids the induced bias and provides valid causal inferences. We evaluate NICE on both synthetic and semi-synthetic data. When the covariates contain unknown collider variables and other bad controls, NICE performs better than adjusting for all the covariates.

Keyon Vafa, Suresh Naidu, David M. Blei

Ideal point models analyze lawmakers' votes to quantify their political positions, or ideal points. But votes are not the only way to express a political position. Lawmakers also give speeches, release press statements, and post tweets. In this paper, we introduce the text-based ideal point model (TBIP), an unsupervised probabilistic topic model that analyzes texts to quantify the political positions of its authors. We demonstrate the TBIP with two types of politicized text data: U.S. Senate speeches and senator tweets. Though the model does not analyze their votes or political affiliations, the TBIP separates lawmakers by party, learns interpretable politicized topics, and infers ideal points close to the classical vote-based ideal points. One benefit of analyzing texts, as opposed to votes, is that the TBIP can estimate ideal points of anyone who authors political texts, including non-voting actors. To this end, we use it to study tweets from the 2020 Democratic presidential candidates. Using only the texts of their tweets, it identifies them along an interpretable progressive-to-moderate spectrum.

Christian A. Naesseth, Fredrik Lindsten, David Blei

Modern variational inference (VI) uses stochastic gradients to avoid intractable expectations, enabling large-scale probabilistic inference in complex models. VI posits a family of approximating distributions q and then finds the member of that family that is closest to the exact posterior p. Traditionally, VI algorithms minimize the "exclusive Kullback-Leibler (KL)" KL(q || p), often for computational convenience. Recent research, however, has also focused on the "inclusive KL" KL(p || q), which has good statistical properties that makes it more appropriate for certain inference problems. This paper develops a simple algorithm for reliably minimizing the inclusive KL using stochastic gradients with vanishing bias. This method, which we call Markovian score climbing (MSC), converges to a local optimum of the inclusive KL. It does not suffer from the systematic errors inherent in existing methods, such as Reweighted Wake-Sleep and Neural Adaptive Sequential Monte Carlo, which lead to bias in their final estimates. We illustrate convergence on a toy model and demonstrate the utility of MSC on Bayesian probit regression for classification as well as a stochastic volatility model for financial data.