Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of \emph{entropic} optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems.
Background: Mathematical models based on ordinary differential equations (ODEs) are essential tools across various scientific disciplines, including biology, ecology, and healthcare informatics. They are used to simulate complex dynamic systems and inform decision-making. In this paper, we introduce BayesianFitForecast, an R toolbox specifically developed to streamline Bayesian parameter estimation and forecasting in ODE models, making it particularly relevant to health informatics and public health decision-making. The toolbox is available at https://github.com/gchowell/BayesianFitForecast/. Results: This toolbox enables automatic generation of Stan files, allowing users to configure models, define priors, and analyze results with minimal programming expertise. To demonstrate the versatility and robustness of BayesianFitForecast, we apply it to the analysis of the 1918 influenza pandemic in San Francisco, comparing Poisson and negative binomial error structures within the SEIR model. We also test it by fitting multiple time series of state variables using simulated data. BayesianFitForecast provides robust tools for evaluating model performance, including convergence diagnostics, posterior distributions, credible intervals, and performance metrics. Conclusion: By improving the accessibility of advanced Bayesian methods, this toolbox significantly broadens the application of Bayesian inference methods to dynamical systems critical for healthcare and epidemiological forecasting. A tutorial video demonstrating the toolbox's functionality is available at https://youtu.be/jnxMjz3V3n8.
For the retrieval of large-scale vegetation biophysical parameters, the inversion of radiative transfer models (RTMs) is the most commonly used approach. In recent years, Artificial Neural Network (ANN)-based methods have become the mainstream for inverting RTMs due to their high accuracy and computational efficiency. It has been widely used in the retrieval of biophysical variables (BV). However, due to the lack of the Bayesian inversion theory interpretation, it faces challenges in quantifying the retrieval uncertainty, a crucial metric for product quality validation and downstream applications such as data assimilation or ecosystem carbon cycling modeling. This study proved that the ANN trained with squared loss outputs the posterior mean, providing a rigorous foundation for its uncertainty quantification, regularization, and incorporation of prior information. A Bayesian theoretical framework was subsequently proposed for ANN-based methods. Using this framework, we derived a new algorithm called Uncertainty Prediction Neural Network (UpNet), which enables the simultaneous training of two ANNs to retrieve BV and provide retrieval uncertainty. To validate our method, we compared UpNet with the standard Bayesian inference method, i.e., Markov Chain Monte Carlo (MCMC), in the inversion of a widely used RTM called ProSAIL for retrieving BVs and estimating uncertainty. The results demonstrated that the BVs retrieved and the uncertainties estimated by UpNet were highly consistent with those from MCMC, achieving over a million-fold acceleration. These results indicated that UpNet has significant potential for fast retrieval and uncertainty quantification of BVs or other parameters with medium and high-resolution remote sensing data. Our Python implementation is available at: https://github.com/Dash-RSer/UpNet.
In this paper, we propose a novel model called Recurrent Explicit Duration Switching Linear Dynamical Systems (REDSLDS) that incorporates recurrent explicit duration variables into the rSLDS model. We also propose an inference and learning scheme that involves the use of P\'olya-gamma augmentation. We demonstrate the improved segmentation capabilities of our model on three benchmark datasets, including two quantitative datasets and one qualitative dataset.
Recent advances in large language models (LLMs) show the potential of using LLMs as evaluators for assessing the quality of text generations from LLMs. However, applying LLM evaluators naively to compare or judge between different systems can lead to unreliable results due to the intrinsic win rate estimation bias of LLM evaluators. In order to mitigate this problem, we propose two calibration methods, Bayesian Win Rate Sampling (BWRS) and Bayesian Dawid-Skene, both of which leverage Bayesian inference to more accurately infer the true win rate of generative language models. We empirically validate our methods on six datasets covering story generation, summarization, and instruction following tasks. We show that both our methods are effective in improving the accuracy of win rate estimation using LLMs as evaluators, offering a promising direction for reliable automatic text quality evaluation.
Bayesian bandit algorithms with approximate Bayesian inference have been widely used in real-world applications. However, there is a large discrepancy between the superior practical performance of these approaches and their theoretical justification. Previous research only indicates a negative theoretical result: Thompson sampling could have a worst-case linear regret $\Omega(T)$ with a constant threshold on the inference error measured by one $\alpha$-divergence. To bridge this gap, we propose an Enhanced Bayesian Upper Confidence Bound (EBUCB) framework that can efficiently accommodate bandit problems in the presence of approximate inference. Our theoretical analysis demonstrates that for Bernoulli multi-armed bandits, EBUCB can achieve the optimal regret order $O(\log T)$ if the inference error measured by two different $\alpha$-divergences is less than a constant, regardless of how large this constant is. To our best knowledge, our study provides the first theoretical regret bound that is better than $o(T)$ in the setting of constant approximate inference error. Furthermore, in concordance with the negative results in previous studies, we show that only one bounded $\alpha$-divergence is insufficient to guarantee a sub-linear regret.
Alzheimer's disease (AD) is a complex neurodegenerative disorder characterized by the progressive accumulation of misfolded proteins, leading to cognitive decline. This study presents a novel stochastic modelling approach to simulate the propagation of these proteins within the brain. We employ a network diffusion model utilizing the Laplacian matrix derived from MRI data provided by the Human Connectome Project (https://braingraph.org/cms/). The deterministic model is extended by incorporating stochastic differential equations (SDEs) to account for inherent uncertainties in disease progression. Introducing stochastic components into the model allows for a more realistic simulation of the disease due to the multi-factorial nature of AD. By simulation, the model captures the variability in misfolded protein concentration across brain regions over time. Bayesian inference is a statistical method that uses prior beliefs and given data to model a posterior distribution for relevant parameter values. This allows us to better understand the impact of noise and external factors on AD progression. Deterministic results suggest that AD progresses at different speeds within each lobe of the brain, moreover, the frontal takes the longest to reach a perfect disease state. We find that in the presence of noise, the model never reaches a perfect disease state and the later years of AD are more unpredictable than earlier on in the disease. These results highlight the importance of integrating stochastic elements into deterministic models to achieve more realistic simulations, providing valuable insights for future studies on the dynamics of neurodegenerative diseases.
Amortized simulation-based inference (SBI) methods train neural networks on simulated data to perform Bayesian inference. While this approach avoids the need for tractable likelihoods, it often requires a large number of simulations and has been challenging to scale to time-series data. Scientific simulators frequently emulate real-world dynamics through thousands of single-state transitions over time. We propose an SBI framework that can exploit such Markovian simulators by locally identifying parameters consistent with individual state transitions. We then compose these local results to obtain a posterior over parameters that align with the entire time series observation. We focus on applying this approach to neural posterior score estimation but also show how it can be applied, e.g., to neural likelihood (ratio) estimation. We demonstrate that our approach is more simulation-efficient than directly estimating the global posterior on several synthetic benchmark tasks and simulators used in ecology and epidemiology. Finally, we validate scalability and simulation efficiency of our approach by applying it to a high-dimensional Kolmogorov flow simulator with around one million dimensions in the data domain.
This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications.
Deep neural network ensembles are powerful tools for uncertainty quantification, which have recently been re-interpreted from a Bayesian perspective. However, current methods inadequately leverage second-order information of the loss landscape, despite the recent availability of efficient Hessian approximations. We propose a novel approximate Bayesian inference method that modifies deep ensembles to incorporate Stein Variational Newton updates. Our approach uniquely integrates scalable modern Hessian approximations, achieving faster convergence and more accurate posterior distribution approximations. We validate the effectiveness of our method on diverse regression and classification tasks, demonstrating superior performance with a significantly reduced number of training epochs compared to existing ensemble-based methods, while enhancing uncertainty quantification and robustness against overfitting.
Real-world offline datasets are often subject to data corruptions (such as noise or adversarial attacks) due to sensor failures or malicious attacks. Despite advances in robust offline reinforcement learning (RL), existing methods struggle to learn robust agents under high uncertainty caused by the diverse corrupted data (i.e., corrupted states, actions, rewards, and dynamics), leading to performance degradation in clean environments. To tackle this problem, we propose a novel robust variational Bayesian inference for offline RL (TRACER). It introduces Bayesian inference for the first time to capture the uncertainty via offline data for robustness against all types of data corruptions. Specifically, TRACER first models all corruptions as the uncertainty in the action-value function. Then, to capture such uncertainty, it uses all offline data as the observations to approximate the posterior distribution of the action-value function under a Bayesian inference framework. An appealing feature of TRACER is that it can distinguish corrupted data from clean data using an entropy-based uncertainty measure, since corrupted data often induces higher uncertainty and entropy. Based on the aforementioned measure, TRACER can regulate the loss associated with corrupted data to reduce its influence, thereby enhancing robustness and performance in clean environments. Experiments demonstrate that TRACER significantly outperforms several state-of-the-art approaches across both individual and simultaneous data corruptions.
This paper discusses the theory and algorithms for interacting large language model agents (LLMAs) using methods from statistical signal processing and microeconomics. While both fields are mature, their application to decision-making involving interacting LLMAs remains unexplored. Motivated by Bayesian sentiment analysis on online platforms, we construct interpretable models and algorithms that enable LLMAs to interact and perform Bayesian inference. Because interacting LLMAs learn from both prior decisions and external inputs, they can exhibit bias and herding behavior. Thus, developing interpretable models and stochastic control algorithms is essential to understand and mitigate these behaviors. This paper has three main results. First, we show using Bayesian revealed preferences from microeconomics that an individual LLMA satisfies the necessary and sufficient conditions for rationally inattentive (bounded rationality) Bayesian utility maximization and, given an observation, the LLMA chooses an action that maximizes a regularized utility. Second, we utilize Bayesian social learning to construct interpretable models for LLMAs that interact sequentially with each other and the environment while performing Bayesian inference. Our proposed models capture the herding behavior exhibited by interacting LLMAs. Third, we propose a stochastic control framework to delay herding and improve state estimation accuracy under 2 settings: (a) centrally controlled LLMAs (b) autonomous LLMAs with incentives. We demonstrate the effectiveness of our methods on real datasets for hate speech classification and product quality assessment, using open-source models like LLaMA and closed-source models like ChatGPT. The main takeaway of this paper, based on empirical analysis and mathematical formalism, is that LLMAs act as rationally bounded Bayesian agents that exhibit social learning when interacting.
When interacting with each other, humans adjust their behavior based on perceived trust. However, to achieve similar adaptability, robots must accurately estimate human trust at sufficiently granular timescales during the human-robot collaboration task. A beta reputation is a popular way to formalize a mathematical estimation of human trust. However, it relies on binary performance, which updates trust estimations only after each task concludes. Additionally, manually crafting a reward function is the usual method of building a performance indicator, which is labor-intensive and time-consuming. These limitations prevent efficiently capturing continuous changes in trust at more granular timescales throughout the collaboration task. Therefore, this paper presents a new framework for the estimation of human trust using a beta reputation at fine-grained timescales. To achieve granularity in beta reputation, we utilize continuous reward values to update trust estimations at each timestep of a task. We construct a continuous reward function using maximum entropy optimization to eliminate the need for the laborious specification of a performance indicator. The proposed framework improves trust estimations by increasing accuracy, eliminating the need for manually crafting a reward function, and advancing toward developing more intelligent robots. The source code is publicly available. https://github.com/resuldagdanov/robot-learning-human-trust
Parkinson's disease (PD) belongs to the class of neurodegenerative disorders that affect the central nervous system. It is usually defined as the gradual loss of dopaminergic neurons in the substantia nigra pars compacta, which causes both motor and non-motor symptoms. Understanding the neuronal processes that underlie PD is critical for creating successful therapies. This study combines machine learning (ML), stochastic modelling, and Bayesian inference with connectomic data to analyze the brain networks involved in PD. We use modern computational methods to study large-scale neural networks to identify neuronal activity patterns related to PD development. We aim to define the subtle structural and functional connection changes in PD brains by combining connectomic with stochastic noises. Stochastic modelling approaches reflect brain dynamics' intrinsic variability and unpredictability, shedding light on the origin and spread of pathogenic events in PD. We employ a novel hybrid model to assess how stochastic noise impacts the cortex-basal ganglia-thalamus (CBGTH) network, using data from the Human Connectome Project (HCP). Bayesian inference allows us to quantify uncertainty in model parameters, improving the accuracy of our predictions. Our findings reveal that stochastic disturbances increase thalamus activity, even under deep brain stimulation (DBS). Bayesian analysis suggests that reducing these disturbances could enhance healthy brain states, providing insights for potential therapeutic interventions. This approach offers a deeper understanding of PD dynamics and paves the way for personalized treatment strategies. This is an extended version of our work presented at the ICCS-2024 conference.
This paper presents a novel framework for full-waveform seismic source inversion using simulation-based inference (SBI). Traditional probabilistic approaches often rely on simplifying assumptions about data errors, which we show can lead to inaccurate uncertainty quantification. SBI addresses this limitation by building an empirical probabilistic model of the data errors using machine learning models, known as neural density estimators, which can then be integrated into the Bayesian inference framework. We apply the SBI framework to point-source moment tensor inversions as well as joint moment tensor and time-location inversions. We construct a range of synthetic examples to explore the quality of the SBI solutions, as well as to compare the SBI results with standard Gaussian likelihood-based Bayesian inversions. We then demonstrate that under real seismic noise, common Gaussian likelihood assumptions for treating full-waveform data yield overconfident posterior distributions that underestimate the moment tensor component uncertainties by up to a factor of 3. We contrast this with SBI, which produces well-calibrated posteriors that generally agree with the true seismic source parameters, and offers an order-of-magnitude reduction in the number of simulations required to perform inference compared to standard Monte Carlo techniques. Finally, we apply our methodology to a pair of moderate magnitude earthquakes in the North Atlantic. We utilise seismic waveforms recorded by the recent UPFLOW ocean bottom seismometer array as well as by regional land stations in the Azores, comparing full moment tensor and source-time location posteriors between SBI and a Gaussian likelihood approach. We find that our adaptation of SBI can be directly applied to real earthquake sources to efficiently produce high quality posterior distributions that significantly improve upon Gaussian likelihood approaches.
Stein variational gradient descent (SVGD) [Liu and Wang, 2016] performs approximate Bayesian inference by representing the posterior with a set of particles. However, SVGD suffers from variance collapse, i.e. poor predictions due to underestimating uncertainty [Ba et al., 2021], even for moderately-dimensional models such as small Bayesian neural networks (BNNs). To address this issue, we generalize SVGD by letting each particle parameterize a component distribution in a mixture model. Our method, Stein Mixture Inference (SMI), optimizes a lower bound to the evidence (ELBO) and introduces user-specified guides parameterized by particles. SMI extends the Nonlinear SVGD framework [Wang and Liu, 2019] to the case of variational Bayes. SMI effectively avoids variance collapse, judging by a previously described test developed for this purpose, and performs well on standard data sets. In addition, SMI requires considerably fewer particles than SVGD to accurately estimate uncertainty for small BNNs. The synergistic combination of NSVGD, ELBO optimization and user-specified guides establishes a promising approach towards variational Bayesian inference in the case of tall and wide data.
Diverse computing paradigms have emerged to meet the growing needs for intelligent energy-efficient systems. The Margin Propagation (MP) framework, being one such initiative in the analog computing domain, stands out due to its scalability across biasing conditions, temperatures, and diminishing process technology nodes. However, the lack of digital-like automation tools for designing analog systems (including that of MP analog) hinders their adoption for designing large systems. The inherent scalability and modularity of MP systems present a unique opportunity in this regard. This paper introduces KALAM (toolKit for Automating high-Level synthesis of Analog computing systeMs), which leverages factor graphs as the foundational paradigm for synthesizing MP-based analog computing systems. Factor graphs are the basis of various signal processing tasks and, when coupled with MP, can be used to design scalable and energy-efficient analog signal processors. Using Python scripting language, the KALAM automation flow translates an input factor graph to its equivalent SPICE-compatible circuit netlist that can be used to validate the intended functionality. KALAM also allows the integration of design optimization strategies such as precision tuning, variable elimination, and mathematical simplification. We demonstrate KALAM's versatility for tasks such as Bayesian inference, Low-Density Parity Check (LDPC) decoding, and Artificial Neural Networks (ANN). Simulation results of the netlists align closely with software implementations, affirming the efficacy of our proposed automation tool.
Bayesian inference for doubly intractable distributions is challenging because they include intractable terms, which are functions of parameters of interest. Although several alternatives have been developed for such models, they are computationally intensive due to repeated auxiliary variable simulations. We propose a novel Monte Carlo Stein variational gradient descent (MC-SVGD) approach for inference for doubly intractable distributions. Through an efficient gradient approximation, our MC-SVGD approach rapidly transforms an arbitrary reference distribution to approximate the posterior distribution of interest, without necessitating any predefined variational distribution class for the posterior. Such a transport map is obtained by minimizing Kullback-Leibler divergence between the transformed and posterior distributions in a reproducing kernel Hilbert space (RKHS). We also investigate the convergence rate of the proposed method. We illustrate the application of the method to challenging examples, including a Potts model, an exponential random graph model, and a Conway--Maxwell--Poisson regression model. The proposed method achieves substantial computational gains over existing algorithms, while providing comparable inferential performance for the posterior distributions.
The analysis of neural power spectra plays a crucial role in understanding brain function and dysfunction. While recent efforts have led to the development of methods for decomposing spectral data, challenges remain in performing statistical analysis and group-level comparisons. Here, we introduce Bayesian Spectral Decomposition (BSD), a Bayesian framework for analysing neural spectral power. BSD allows for the specification, inversion, comparison, and analysis of parametric models of neural spectra, addressing limitations of existing methods. We first establish the face validity of BSD on simulated data and show how it outperforms an established method (\fooof{}) for peak detection on artificial spectral data. We then demonstrate the efficacy of BSD on a group-level study of EEG spectra in 204 healthy subjects from the LEMON dataset. Our results not only highlight the effectiveness of BSD in model selection and parameter estimation, but also illustrate how BSD enables straightforward group-level regression of the effect of continuous covariates such as age. By using Bayesian inference techniques, BSD provides a robust framework for studying neural spectral data and their relationship to brain function and dysfunction.
Gradient-based dimension reduction decreases the cost of Bayesian inference and probabilistic modeling by identifying maximally informative (and informed) low-dimensional projections of the data and parameters, allowing high-dimensional problems to be reformulated as cheaper low-dimensional problems. A broad family of such techniques identify these projections and provide error bounds on the resulting posterior approximations, via eigendecompositions of certain diagnostic matrices. Yet these matrices require gradients or even Hessians of the log-likelihood, excluding the purely data-driven setting and many problems of simulation-based inference. We propose a framework, derived from score-matching, to extend gradient-based dimension reduction to problems where gradients are unavailable. Specifically, we formulate an objective function to directly learn the score ratio function needed to compute the diagnostic matrices, propose a tailored parameterization for the score ratio network, and introduce regularization methods that capitalize on the hypothesized low-dimensional structure. We also introduce a novel algorithm to iteratively identify the low-dimensional reduced basis vectors more accurately with limited data based on eigenvalue deflation methods. We show that our approach outperforms standard score-matching for problems with low-dimensional structure, and demonstrate its effectiveness for PDE-constrained Bayesian inverse problems and conditional generative modeling.