arXiv subjects

David J. Nott

Publications and source records attributed to David J. Nott.

At least 19 recordsLinked to original sources

arXiv2026

Flexible Transformations for Bayesian Score Calibration

Modern statistical models are growing increasingly complex in an effort to realistically capture system dynamics. Using standard simulation-based inference, these models may be computationally prohibitive, necessitating the use of model calibration methods. Bayesian score calibration is a computationally efficient framework for model calibration with strong theoretical guarantees. This framework learns an appropriate correction for an approximate model using a small number of simulations from the data-generating process. Currently, only a location-scale transformation has been explored, which may lack the flexibility to correct the complex error introduced by some approximate models. In this paper, we develop two flexible transformations for use in the Bayesian score calibration framework. The first is a polynomial extension, which can appropriately adjust approximate models with location-varying error. The second is a sequential application of Bayesian score calibration, which can accommodate approximate models with posteriors that have low support for the true parameter values. We also discuss an additional diagnostic for use with this framework. We demonstrate the increased flexibility these two approaches provide over Bayesian score calibration in two illustrative simulation studies.

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arXiv2026

Deep adaptive design with an evidential bias criterion

Bayesian optimal experimental design (BOED) aims to collect informative data by optimizing an expected utility reflecting the goals of an experiment. However, this optimization is computationally challenging for common utilities and complex models. This is especially so for sequential or adaptive designs, where design and data collection alternate, so that feedback from already observed data must be taken into account. Most existing BOED research employs information gain as the utility, leading to the expected information gain (EIG) criterion. While EIG is widely useful, it may not always adequately reflect experimental goals. EIG can be viewed as rewarding experiments that produce large positive evidence for the truth on average, but it does not directly control the risk of an experiment producing misleading evidence. Here we consider an alternative criterion, which we call bias against (BA), that prioritizes such control. To address computational challenges when applying this criterion for adaptive design, we consider a policy-based deep adaptive design framework, which has previously been used for the EIG criterion. Minimizing a tractable upper bound on the BA objective is equivalent to maximizing a variance-penalized EIG criterion, and we optimize the latter by approximating it by Monte Carlo and learning design policies using stochastic gradient methods. The differences between BA and EIG designs are demonstrated in several examples including the adaptive design of a complex discrete choice experiment.

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arXiv2026

Logistic Gaussian process density regression: a generalized Bayesian approach

Density regression extends conventional parametric regression by allowing the entire distribution of the response to vary flexibly with covariates rather than just low-order moments. In the Bayesian setting, logistic Gaussian process (GP) priors have been widely used for density estimation and extend naturally to density regression. The prior can be centred on a base density model, with the nonparametric component providing an interpretable correction that is useful for model criticism. However, logistic GP density regression models have seen limited use, since they require computation of a normalizing constant for every observation, typically via numerical integration. We address this difficulty by proposing a generalized Bayesian approach using a loss function based on the Hyvarinen score. The Hyvarinen score depends only on derivatives of the log density with respect to the response, eliminating the need to compute normalizing constants. Since GP computations remain expensive, we also employ sparse inducing point approximations and variational inference to develop a scalable approach. We demonstrate the method on one simulated and two real datasets, including a German weather dataset with more than 150,000 observations.

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arXiv2026

Learning the distance for ABC and localized neural posterior estimation

Likelihood-free inference methods can perform Bayesian inference when evaluating the likelihood is impractical but simulating synthetic data from the model is feasible. Approximate Bayesian computation (ABC) is a well-established likelihood-free approach that constructs particle posterior approximations by evaluating the similarity between simulated and observed data using a distance function, which is used in rejection or weighting steps. Here we extend previous work on adaptive distance learning for ABC to misspecified time series, while also exploring applications in neural posterior estimation using prior-data fitted networks (NPE-PFN) with localization. The adaptation of the distance that we consider optimizes out-of-sample predictive performance using a scoring rule. We also establish a connection between linear pooling for forecast combination and our posterior estimation methods with randomized distances, showing that empirical estimation of pooling weights can be interpreted as another form of adaptive distance learning. For both ABC algorithms and NPE-PFN methods with localization, adaptive distance learning improves forecasting performance in simulated and real examples.

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arXiv2026

Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

Copula models of multivariate data are popular because they allow separate specification of marginal distributions and the copula function. These components can be treated as inter-related modules in a modified Bayesian inference approach called ''cutting feedback'' that is robust to their misspecification. Recent work uses a two module approach, where all $d$ marginals form a single module, to robustify inference for the marginals against copula function misspecification, or vice versa. However, marginals can exhibit differing levels of misspecification, making it attractive to assign each its own module with an individual influence parameter controlling its contribution to a joint semi-modular inference (SMI) posterior. This generalizes existing two module SMI methods, which interpolate between cut and conventional posteriors using a single influence parameter. We develop a novel copula SMI method and select the influence parameters using Bayesian optimization. It provides an efficient continuous relaxation of the discrete optimization problem over $2^d$ cut/uncut configurations. We establish theoretical properties of the resulting semi-modular posterior and demonstrate the approach on simulated and real data. The real data application uses a skew-normal copula model of asymmetric dependence between equity volatility and bond yields, where robustifying copula estimation against marginal misspecification is strongly motivated.

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arXiv2026

Predictive variational inference for flexible regression models

A conventional Bayesian approach to prediction uses the posterior distribution to integrate out parameters in a density for unobserved data conditional on the observed data and parameters. When the true posterior is intractable, it is replaced by an approximation; here we focus on variational approximations. Recent work has explored methods that learn posteriors optimized for predictive accuracy under a chosen scoring rule, while regularizing toward the prior or conventional posterior. Our work builds on an existing predictive variational inference (PVI) framework that improves prediction, but also diagnoses model deficiencies through implicit model expansion. In models where the sampling density depends on the parameters through a linear predictor, we improve the interpretability of existing PVI methods as a diagnostic tool. This is achieved by adopting PVI posteriors of Gaussian mixture form (GM-PVI) and establishing connections with plug-in prediction for mixture-of-experts models. We make three main contributions. First, we show that GM-PVI prediction is equivalent to plug-in prediction for certain mixture-of-experts models with covariate-independent weights in generalized linear models and hierarchical extensions of them. Second, we extend standard PVI by allowing GM-PVI posteriors to vary with the prediction covariate and in this case an equivalence to plug-in prediction for mixtures of experts with covariate-dependent weights is established. Third, we demonstrate the diagnostic value of this approach across several examples, including generalized linear models, linear mixed models, and latent Gaussian process models, demonstrating how the parameters of the original model must vary across the covariate space to achieve improvements in prediction.

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arXiv2025

Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies

Evidence synthesis models combine multiple data sources to estimate latent quantities of interest, enabling reliable inference on parameters that are difficult to measure directly. However, shared parameters across data sources can induce conflicts both among the data and with the assumed model structure. Detecting and quantifying such conflicts remains a challenge in model criticism. Here we propose a general framework for conflict detection in evidence synthesis models based on score discrepancies, extending prior-data conflict diagnostics to more general conflict checks in the latent space of hierarchical models. Simulation studies in an exchangeable model demonstrate that the proposed approach effectively detects between-data inconsistencies. Application to an influenza severity model illustrates its use, complementary to traditional deviance-based diagnostics, in complex real-world hierarchical settings. The proposed framework thus provides a flexible and broadly applicable tool for consistency assessment in Bayesian evidence synthesis.

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arXiv2025

Weighted Fisher divergence for high-dimensional Gaussian variational inference

Bayesian inference has many advantages for complex models, but standard Monte Carlo methods for summarizing the posterior can be computationally demanding, and it is attractive to consider optimization-based variational methods. Our work considers Gaussian approximations with sparse precision matrices which are tractable to optimize in high-dimensions. The optimal Gaussian approximation is usually defined as being closest to the posterior in Kullback-Leibler divergence, but it is useful to consider other divergences when the Gaussian assumption is crude, to capture important posterior features for given applications. Our work studies the weighted Fisher divergence, which focuses on gradient differences between the target posterior and its approximation, with the Fisher and score-based divergences as special cases. We make three main contributions. First, we compare approximations for weighted Fisher divergences under mean-field assumptions for Gaussian and non-Gaussian targets with Kullback-Leibler approximations. Second, we go beyond mean-field and consider approximations with sparse precision matrices reflecting posterior conditional independence structure for hierarchical models. Using stochastic gradient descent to enforce sparsity, we develop two approaches to minimize the Fisher and score-based divergences, based on the reparametrization trick and a batch approximation of the objective. Finally, we study the performances of our methods using logistic regression, generalized linear mixed models and stochastic volatility models.

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arXiv2025

Optimization-centric cutting feedback for semiparametric models

Complex statistical models are often built by combining multiple submodels, called modules. Here we consider modular inference where the modules contain both parametric and nonparametric components. In such cases, standard Bayesian inference can be highly sensitive to misspecification in any module, and influential prior specifications for the nonparametric components can compromise inference for the parametric components, and vice versa. We propose a novel "optimization-centric" approach to cutting feedback for semiparametric modular inference, which can address misspecification and prior-data conflicts. The proposed cut posteriors are defined via a variational optimization problem like other generalized posteriors, but regularization is based on R\'enyi divergence, instead of Kullback-Leibler divergence (KLD). We show empirically that defining the cut posterior using R\'enyi divergence delivers more robust inference than KLD, and R\'enyi divergence reduces the tendency to underestimate uncertainty when the variational approximations impose strong parametric or independence assumptions. Novel posterior concentration results that accommodate the R\'enyi divergence and allow for semiparametric components are derived, extending existing results for cut posteriors that only apply to KLD and parametric models. These new methods are demonstrated in a benchmark example and two real examples: Gaussian process adjustments for confounding in causal inference and misspecified copula models with nonparametric marginals.

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arXiv2025

Pooling information in likelihood-free inference

Likelihood-free inference (LFI) methods, such as approximate Bayesian computation, have become commonplace for conducting inference in complex models. Many approaches are based on summary statistics or discrepancies derived from synthetic data. However, determining which summary statistics or discrepancies to use for constructing the posterior remains a challenging question, both practically and theoretically. Instead of relying on a single vector of summaries for inference, we propose a new pooled posterior that optimally combines inferences from multiple LFI posteriors. This pooled approach eliminates the need to select a single vector of summaries or even a specific LFI algorithm. Our approach is straightforward to implement and avoids performing a high-dimensional LFI analysis involving all summary statistics. We give theoretical guarantees for the improved performance of the pooled posterior mean in terms of asymptotic frequentist risk and demonstrate the effectiveness of the approach in a number of benchmark examples.

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arXiv2025

Positional Encoder Graph Quantile Neural Networks for Geographic Data

Positional Encoder Graph Neural Networks (PE-GNNs) are among the most effective models for learning from continuous spatial data. However, their predictive distributions are often poorly calibrated, limiting their utility in applications that require reliable uncertainty quantification. We propose the Positional Encoder Graph Quantile Neural Network (PE-GQNN), a novel framework that combines PE-GNNs with Quantile Neural Networks, partially monotonic neural blocks, and post-hoc recalibration techniques. The PE-GQNN enables flexible and robust conditional density estimation with minimal assumptions about the target distribution, and it extends naturally to tasks beyond spatial data. Empirical results on benchmark datasets show that the PE-GQNN outperforms existing methods in both predictive accuracy and uncertainty quantification, without incurring additional computational cost. We also provide theoretical insights and identify important special cases arising from our formulation, including the PE-GNN.

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arXiv2025

Robust Bayesian methods using amortized simulation-based inference

Bayesian simulation-based inference (SBI) methods are used in statistical models where simulation is feasible but the likelihood is intractable. Standard SBI methods can perform poorly in cases of model misspecification, and there has been much recent work on modified SBI approaches which are robust to misspecified likelihoods. However, less attention has been given to the issue of inappropriate prior specification, which is the focus of this work. In conventional Bayesian modelling, there will often be a wide range of prior distributions consistent with limited prior knowledge expressed by an expert. Choosing a single prior can lead to an inappropriate choice, possibly conflicting with the likelihood information. Robust Bayesian methods, where a class of priors is considered instead of a single prior, can address this issue. For each density in the prior class, a posterior can be computed, and the range of the resulting inferences is informative about posterior sensitivity to the prior imprecision. We consider density ratio classes for the prior and implement robust Bayesian SBI using amortized neural methods developed recently in the literature. We also discuss methods for checking for conflict between a density ratio class of priors and the likelihood, and sequential updating methods for examining conflict between different groups of summary statistics. The methods are illustrated for several simulated and real examples.

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arXiv2025

Variational inference for hierarchical models with conditional scale and skewness corrections

Gaussian variational approximations are widely used for summarizing posterior distributions in Bayesian models, especially in high-dimensional settings. However, a drawback of such approximations is the inability to capture skewness or more complex features of the posterior. Recent work suggests applying skewness corrections to existing Gaussian or other symmetric approximations to address this limitation. We propose to incorporate the skewness correction into the definition of an approximating variational family. We consider approximating the posterior for hierarchical models, in which there are ``global'' and ``local'' parameters. A baseline variational approximation is defined as the product of a Gaussian marginal posterior for global parameters and a Gaussian conditional posterior for local parameters given the global ones. Skewness corrections are then considered. The adjustment of the conditional posterior term for local variables is adaptive to the global parameter value. Optimization of baseline variational parameters is performed jointly with the skewness correction. Our approach allows the location, scale and skewness to be captured separately, without using additional parameters for skewness adjustments. The proposed method substantially improves accuracy for only a modest increase in computational cost compared to state-of-the-art Gaussian approximations. Good performance is demonstrated in generalized linear mixed models and multinomial logit discrete choice models.

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arXiv2025

Simulation-based Bayesian inference under model misspecification

Simulation-based Bayesian inference (SBI) methods are widely used for parameter estimation in complex models where evaluating the likelihood is challenging but generating simulations is relatively straightforward. However, these methods commonly assume that the simulation model accurately reflects the true data-generating process, an assumption that is frequently violated in realistic scenarios. In this paper, we focus on the challenges faced by SBI methods under model misspecification. We consolidate recent research aimed at mitigating the effects of misspecification, highlighting three key strategies: i) robust summary statistics, ii) generalised Bayesian inference, and iii) error modelling and adjustment parameters. To illustrate both the vulnerabilities of popular SBI methods and the effectiveness of misspecification-robust alternatives, we present empirical results on an illustrative example.

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arXiv2024

Posterior risk of modular and semi-modular Bayesian inference

Modular Bayesian methods perform inference in models that are specified through a collection of coupled sub-models, known as modules. These modules often arise from modelling different data sources or from combining domain knowledge from different disciplines. ``Cutting feedback'' is a Bayesian inference method that ensures misspecification of one module does not affect inferences for parameters in other modules, and produces what is known as the cut posterior. However, choosing between the cut posterior and the standard Bayesian posterior is challenging. When misspecification is not severe, cutting feedback can greatly increase posterior uncertainty without a large reduction of estimation bias, leading to a bias-variance trade-off. This trade-off motivates semi-modular posteriors, which interpolate between standard and cut posteriors based on a tuning parameter. In this work, we provide the first precise formulation of the bias-variance trade-off that is present in cutting feedback, and we propose a new semi-modular posterior that takes advantage of it. Under general regularity conditions, we prove that this semi-modular posterior is more accurate than the cut posterior according to a notion of posterior risk. An important implication of this result is that point inferences made under the cut posterior are inadmissable. The new method is demonstrated in a number of examples.

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arXiv2024

Calibrated Multivariate Regression with Localized PIT Mappings

Calibration ensures that predicted uncertainties align with observed uncertainties. While there is an extensive literature on recalibration methods for univariate probabilistic forecasts, work on calibration for multivariate forecasts is much more limited. This paper introduces a novel post-hoc recalibration approach that addresses multivariate calibration for potentially misspecified models. Our method involves constructing local mappings between vectors of marginal probability integral transform values and the space of observations, providing a flexible and model free solution applicable to continuous, discrete, and mixed responses. We present two versions of our approach: one uses K-nearest neighbors, and the other uses normalizing flows. Each method has its own strengths in different situations. We demonstrate the effectiveness of our approach on two real data applications: recalibrating a deep neural network's currency exchange rate forecast and improving a regression model for childhood malnutrition in India for which the multivariate response has both discrete and continuous components.

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arXiv2024

Dropout Regularization in Extended Generalized Linear Models based on Double Exponential Families

Even though dropout is a popular regularization technique, its theoretical properties are not fully understood. In this paper we study dropout regularization in extended generalized linear models based on double exponential families, for which the dispersion parameter can vary with the features. A theoretical analysis shows that dropout regularization prefers rare but important features in both the mean and dispersion, generalizing an earlier result for conventional generalized linear models. To illustrate, we apply dropout to adaptive smoothing with B-splines, where both the mean and dispersion parameters are modeled flexibly. The important B-spline basis functions can be thought of as rare features, and we confirm in experiments that dropout is an effective form of regularization for mean and dispersion parameters that improves on a penalized maximum likelihood approach with an explicit smoothness penalty. An application to traffic detection data from Berlin further illustrates the benefits of our method.

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arXiv2024

Cutting Feedback in Misspecified Copula Models

In copula models the marginal distributions and copula function are specified separately. We treat these as two modules in a modular Bayesian inference framework, and propose conducting modified Bayesian inference by "cutting feedback". Cutting feedback limits the influence of potentially misspecified modules in posterior inference. We consider two types of cuts. The first limits the influence of a misspecified copula on inference for the marginals, which is a Bayesian analogue of the popular Inference for Margins (IFM) estimator. The second limits the influence of misspecified marginals on inference for the copula parameters by using a pseudo likelihood of the ranks to define the cut model. We establish that if only one of the modules is misspecified, then the appropriate cut posterior gives accurate uncertainty quantification asymptotically for the parameters in the other module. Computation of the cut posteriors is difficult, and new variational inference methods to do so are proposed. The efficacy of the new methodology is demonstrated using both simulated data and a substantive multivariate time series application from macroeconomic forecasting. In the latter, cutting feedback from misspecified marginals to a 1096 dimension copula improves posterior inference and predictive accuracy greatly, compared to conventional Bayesian inference.

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