Spatial intensity maps are routinely used to summarize point patterns on geographically constrained regions, such as islands, coastlines, watersheds, ecological reserves, and administrative areas with physical barriers. In these settings, the domain is not a nuisance feature– it determines where probability mass may be assigned and which locations should be smoothed together. Standard kernel density estimators can place mass outside the study region and smooth according to Euclidean distance, while diffusion-based estimators respect the geometry but are costly to recompute when the data subset, bandwidth, or evaluation grid changes. Motivated by repeated hotspot mapping of theft and larceny incidents on Oahu, Hawaii, we propose the projected diffusion kernel density estimator (PDKDE). PDKDE expands the Neumann diffusion kernel in a truncated Laplacian eigenbasis, so that the geometry of a fixed domain is computed once and subsequent density estimates are obtained through explicit spectral coefficients. The resulting estimator preserves the boundary-aware and geometry-respecting behavior of diffusion smoothing while making repeated estimation and least-squares cross-validation computationally practical. For the exact projected estimator, we prove MISE consistency and pointwise consistency up to the boundary. Controlled simulations show that PDKDE reduces boundary and barrier artifacts relative to Euclidean kernels and is substantially faster than the geometry-aware comparators considered after the one-time domain computation. In the Oahu application, the method produces coastline-constrained descriptive maps across time windows in seconds, illustrating its intended fixed-domain, changing-data use case.

We consider predictive density estimation under logarithmic score for d-dimensional infinitely divisible location models. Taking the formal Bayes predictive density under the Lebesgue prior as a benchmark, we study the Kullback-Leibler regret of competing Bayes predictive densities. Our main contribution is an exact entropy-energy identity: the integrated regret of a Bayes predictive density p̂^π under prior π relative to the benchmark admits an exact representation as the Dirichlet-form energy of the square-rooted marginal distribution √(M^π) for the symmetric Markov semigroup induced by the benchmark kernel. This converts regret comparisons into a potential-theoretic problem and yields a sharp recurrence/transience characterization of when the benchmark predictive density can or cannot be uniformly improved. We introduce an 𝒜-harmonic class of improper priors – defined through the generator 𝒜 of the induced process – and give explicit tail conditions – an integral test on the induced marginal, equivalent to power-law prior decay in heavy-tailed models – that guarantee admissibility of the resulting Bayes predictive density. We illustrate the theory with new results for several distributions.

We generalize the potential outcome framework to time series with an intervention by defining causal effects on stochastic processes. Interventions in dynamic systems alter not only outcome levels but also evolutionary dynamics – changing persistence and transition laws. Our framework treats potential outcomes as entire trajectories, enabling causal estimands, identification conditions, and estimators to be formulated directly on path space. The resulting Dynamic Average Treatment Effect (DATE) characterizes how causal effects evolve through time and reduces to the classical average treatment effect under one period of time. For observational data, we derive a dynamic inverse-probability weighting estimator that is unbiased under dynamic ignorability and positivity. When treated units are scarce, we show that conditional mean trajectories underlying the DATE admit a linear state-space representation, yielding a dynamic linear model implementation. Simulations demonstrate that modeling time as intrinsic to the causal mechanism exposes dynamic effects that static methods systematically misestimate. An empirical study of COVID-19 lockdowns illustrates the framework's practical value for estimating and decomposing treatment effects.

Single-case experimental design (SCED) is a quantitative research method widely used across diverse fields including behavioral education, special education, medicine, and applied behavior analysis. SCED methodology establishes external validity through a well-established sequential process: demonstrating strong internal validity within individual studies, followed by direct and systematic replication across varied populations, settings, and conditions. While this replication-based approach has proven highly successful as evidenced by robust intervention literature bases such as functional communication training and behavioral skills training the traditional process of accumulating replications across independent research teams can require years or decades. Inspired by the ManyLabs initiative’s success in coordinating replication efforts, we developed a sequential experimental design using dynamic Bayesian models to accelerate coordinated SCED research across laboratories. Applied to simulated and published multiple-baseline design data, our approach demonstrates how posterior distributions from completed cases can serve as priors for subsequent participants, creating cumulative knowledge building across research sites. Findings show that dynamic models effectively estimate intervention effects even when timing and effect sizes vary randomly. Parameter estimates converged to true effects after 15-25 coordinated cases, with credible intervals narrowing systematically as participants were added sequentially. This ManyLabs-inspired framework offers SCED researchers a pathway to accelerate the coordination of replications while maintaining the individualized focus and rigorous internal validity that make single-case designs valuable for evidence-based practice.

Cross-validation (CV) is routinely used across the sciences to select models and tune parameters, and the resulting choices are often interpreted as substantive scientific conclusions (e.g., which variables, mechanisms, or risk factors are “supported by the data”). A key part of the CV procedure – the hold-out size, or equivalently the fold count K – is typically set by convention (e.g., 80/20, K=5) rather than by a principled criterion. Central to the issue is the tradeoff between training and testing: increasing the training sample size improves model accuracy, while sacrificing certainty around the accuracy itself. We formalize the tradeoff by targeting predictive performance and explicitly penalizing evaluation uncertainty, which cannot be identified from the data without additional assumptions. We derive finite-sample expressions of this evaluation uncertainty under symmetric errors and general upper bounds under broader error conditions, yielding a transparent utility-based rule for selecting the hold-out size as a function of an irreducible-noise parameter. Empirical analyses with linear regression and random forests across multiple domains, and a high-dimensional genomics application, show that (i) the choice of K is dependent on the data and model. (ii) the optimal K varies based on the assumption on the irreducible error, and (iii) the implied inferential conclusions can change materially as the irreducible error, and thus K, varies. The resulting framework replaces a one-size-fits-all convention with a context-specific, assumption-explicit choice of K, enabling more reliable model comparisons and downstream scientific inference.

Due to spatial dependence -- often characterized as complex and non-linear -- model misspecification is a prevalent and critical issue in spatial data analysis and prediction. As the data, and thus model performance, is heterogeneous, typical model selection and ensemble methods that assume homogeneity are not suitable. We address the issue of model uncertainty for spatial data by proposing a novel Bayesian ensemble methodology that captures spatially-varying model uncertainty and performance heterogeneity of multiple spatial predictions, and synthesizes them for improved predictions, which we call Bayesian spatial predictive synthesis. Our proposal is defined by specifying a latent factor spatially-varying coefficient model as the synthesis function, which enables spatial characteristics of each model to be learned and ensemble coefficients to vary over regions to achieve flexible predictions. We derive our method from the theoretically best approximation of the data generating process, and show that it provides a finite sample theoretical guarantee for its predictive performance, specifically that the predictions are exact minimax. Two MCMC strategies are implemented for full uncertainty quantification, as well as a variational inference strategy for fast point inference. We also extend the estimation strategy for general responses. Through simulation examples and two real data applications in real estate and ecology, our proposed Bayesian spatial predictive synthesis outperforms standard spatial models and ensemble methods, and advanced machine learning methods, in terms of predictive accuracy and uncertainty quantification, while maintaining interpretability of the prediction mechanism.

The doubly robust estimator, which models both the propensity score and outcomes, is a popular approach to estimate the average treatment effect in the potential outcome setting. The primary appeal of this estimator is its theoretical property, wherein the estimator achieves consistency as long as either the propensity score or outcomes is correctly specified. In most applications, however, both are misspecified, leading to considerable bias that cannot be checked. In this paper, we propose a Bayesian ensemble approach that synthesizes multiple models for both the propensity score and outcomes, which we call doubly robust Bayesian regression synthesis. Our approach applies Bayesian updating to the ensemble model weights that adapt at the unit level, incorporating data heterogeneity, to significantly mitigate misspecification bias. Theoretically, we show that our proposed approach is consistent regarding the estimation of both the propensity score and outcomes, ensuring that the doubly robust estimator is consistent, even if no single model is correctly specified. An efficient algorithm for posterior computation facilitates the characterization of uncertainty regarding the treatment effect. Our proposed approach is compared against standard and state-of-the-art methods through two comprehensive simulation studies, where we find that our approach is superior in all cases. An empirical study on the impact of maternal smoking on birth weight highlights the practical applicability of our proposed method.

We discuss the relation between the statistical question of inadmissibility and the probabilistic question of transience. Brown (1971) proved the mathematical link between the admissibility of the mean of a Gaussian distribution and the recurrence of a Brownian motion, which holds for $\mathbb{R}^{2}$ but not for $\mathbb{R}^{3}$ in Euclidean space. We extend this result to symmetric, non-Gaussian distributions, without assuming the existence of moments. As an application, we prove that the relation between the inadmissibility of the predictive density of a Cauchy distribution under a uniform prior and the transience of the Cauchy process differs from dimensions $\mathbb{R}^{1}$ to $\mathbb{R}^{2}$. We also show that there exists an extreme model that is inadmissible in $\mathbb{R}^{1}$.

We conducted a survey experiment with a sample of university-based start-up researchers to investigate the effects of patents on innovations. The real-world news about the patent waiver of COVID-19 vaccines was exploited in our random information manipulation study. A new ensemble method for causal inference, Bayesian causal synthesis, was employed to estimate heterogeneous treatment effects after our news provision, which provided us with important policy implications regarding patent protection policies.

The estimation of heterogeneous treatment effects in the potential outcome setting is biased when there exists model misspecification or unobserved confounding. As these biases are unobservable, what model to use when remains a critical open question. In this paper, we propose a novel Bayesian methodology to mitigate misspecification and improve estimation via a synthesis of multiple causal estimates, which we call Bayesian causal synthesis. Our development is built upon identifying a synthesis function that correctly specifies the heterogeneous treatment effect under no unobserved confounding, and achieves the irreducible bias under unobserved confounding. We show that our proposed method results in consistent estimates of the heterogeneous treatment effect; either with no bias or with irreducible bias. We provide a computational algorithm for fast posterior sampling. Several benchmark simulations and an empirical study highlight the efficacy of the proposed approach compared to existing methodologies, providing improved point and density estimation of the heterogeneous treatment effect, even under unobserved confounding.

Spatial data are characterized by their spatial dependence, which is often complex, non-linear, and difficult to capture with a single model. Significant levels of model uncertainty -- arising from these characteristics -- cannot be resolved by model selection or simple ensemble methods. We address this issue by proposing a novel methodology that captures spatially varying model uncertainty, which we call Bayesian spatial predictive synthesis. Our proposal is derived by identifying the theoretically best approximate model under reasonable conditions, which is a latent factor spatially varying coefficient model in the Bayesian predictive synthesis framework. We then show that our proposed method produces exact minimax predictive distributions, providing finite sample guarantees. Two MCMC strategies are implemented for full uncertainty quantification, as well as a variational inference strategy for fast point inference. We also extend the estimation strategy for general responses. Through simulation examples and two real data applications, we demonstrate that our proposed spatial Bayesian predictive synthesis outperforms standard spatial models and advanced machine learning methods in terms of predictive accuracy.

We consider learning causal relationships under conditional moment restrictions. Unlike causal inference under unconditional moment restrictions, conditional moment restrictions pose serious challenges for causal inference, especially in high-dimensional settings. To address this issue, we propose a method that transforms conditional moment restrictions to unconditional moment restrictions through importance weighting, using a conditional density ratio estimator. Using this transformation, we successfully estimate nonparametric functions defined under conditional moment restrictions. Our proposed framework is general and can be applied to a wide range of methods, including neural networks. We analyze the estimation error, providing theoretical support for our proposed method. In experiments, we confirm the soundness of our proposed method.

Spatial data are characterized by their spatial dependence, which is often complex, non-linear, and difficult to capture with a single model. Significant levels of model uncertainty– arising from these characteristics– cannot be resolved by model selection or simple ensemble methods, as performances are not homogeneous. We address this issue by proposing a novel methodology that captures spatially-varying model uncertainty, which we call spatial Bayesian predictive synthesis. Our proposal is defined by specifying a latent factor spatially-varying coefficient model as the synthesis function, which enables model coeffi-cients to vary over the region to achieve flexible spatial model ensembling. Two MCMC strategies are implemented for full uncertainty quantification, as well as a variational inference strategy for fast point inference. We also extend the estimation strategy for general responses. A finite sample theoretical guarantee is given for the predictive performance of our methodology, showing that the predictions are exact minimax. Through simulation examples and two real data applications, we demonstrate that our proposed spatial Bayesian predictive synthesis outperforms standard spatial models and advanced machine learning methods, in terms of predictive accuracy, while maintaining interpretability of the prediction mechanism.

We consider the"policy choice"problem -- otherwise known as best arm identification in the bandit literature -- proposed by Kasy and Sautmann (2021) for adaptive experimental design. Theorem 1 of Kasy and Sautmann (2021) provides three asymptotic results that give theoretical guarantees for exploration sampling developed for this setting. We first show that the proof of Theorem 1 (1) has technical issues, and the proof and statement of Theorem 1 (2) are incorrect. We then show, through a counterexample, that Theorem 1 (3) is false. For the former two, we correct the statements and provide rigorous proofs. For Theorem 1 (3), we propose an alternative objective function, which we call posterior weighted policy regret, and derive the asymptotic optimality of exploration sampling.

This paper studies the asymptotic convergence of computed dynamic models when the shock is unbounded. Most dynamic economic models lack a closed-form solution. As such, approximate solutions by numerical methods are utilized. Since the researcher cannot directly evaluate the exact policy function and the associated exact likelihood, it is imperative that the approximate likelihood asymptotically converges – as well as to know the conditions of convergence – to the exact likelihood, in order to justify and validate its usage. In this regard, Fernandez-Villaverde, Rubio-Ramirez, and Santos (2006) show convergence of the likelihood, when the shock has compact support. However, compact support implies that the shock is bounded, which is not an assumption met in most dynamic economic models, e.g., with normally distributed shocks. This paper provides theoretical justification for most dynamic models used in the literature by showing the conditions for convergence of the approximate invariant measure obtained from numerical simulations to the exact invariant measure, thus providing the conditions for convergence of the likelihood.

We consider controlling the false discovery rate for testing many time series with an unknown cross-sectional correlation structure. Given a large number of hypotheses, false and missing discoveries can plague an analysis. While many procedures have been proposed to control false discovery, most of them either assume independent hypotheses or lack statistical power. A problem of particular interest is in financial asset pricing, where the goal is to determine which “factors" lead to excess returns out of a large number of potential factors. Our contribution is two-fold. First, we show the consistency of Fama and French’s prominent method under multiple testing. Second, we propose a novel method for false discovery control using double bootstrapping. We achieve superior statistical power to existing methods and prove that the false discovery rate is controlled. Simulations and a real data application illustrate the efficacy of our method over existing methods.

We analyze the combination of multiple predictive distributions for time series data when all forecasts are misspecified. We show that a specific dynamic form of Bayesian predictive synthesis– a general and coherent Bayesian framework for ensemble methods– produces exact minimax predictive densities with regard to Kullback-Leibler loss, providing theoretical support for finite sample predictive performance over existing ensemble methods. A simulation study that highlights this theoretical result is presented, showing that dynamic Bayesian predictive synthesis is superior to other ensemble methods using multiple metrics.

We address the problem of dynamic variable selection in time series regression with unknown residual variances, where the set of active predictors is allowed to evolve over time. To capture time-varying variable selection uncertainty, we introduce new dynamic shrinkage priors for the time series of regression coefficients. These priors are characterized by two main ingredients: smooth parameter evolutions and intermittent zeroes for modeling predictive breaks. More formally, our proposed Dynamic Spike-and-Slab (DSS) priors are constructed as mixtures of two processes: a spike process for the irrelevant coefficients and a slab autoregressive process for the active coefficients. The mixing weights are themselves time-varying and depend on lagged values of the series. Our DSS priors are probabilistically coherent in the sense that their stationary distribution is fully known and characterized by spike-and-slab marginals. For posterior sampling over dynamic regression coefficients, model selection indicators as well as unknown dynamic residual variances, we propose a Dynamic SSVS algorithm based on forward-filtering and backward-sampling. To scale our method to large data sets, we develop a Dynamic EMVS algorithm for MAP smoothing. We demonstrate, through simulation and a topical macroeconomic dataset, that DSS priors are very effective at separating active and noisy coefficients. Our fast implementation significantly extends the reach of spike-and-slab methods to large time series data.