Beyond estimating parameters of interest from data, one of the key goals of statistical inference is to properly quantify uncertainty in these estimates. In Bayesian inference, this uncertainty is provided by the posterior distribution, the computation of which typically involves an intractable high-dimensional integral. Among available approximation methods, sampling-based approaches come with strong theoretical guarantees but scale poorly to large problems, while variational approaches scale well but offer few theoretical guarantees. In particular, variational methods are known to produce overconfident estimates of posterior uncertainty and are typically non-identifiable, with many latent variable configurations generating equivalent predictions. Here, we address these challenges by showing how diffusion-based models (DBMs), which have recently produced state-of-the-art performance in generative modeling tasks, can be repurposed for performing calibrated, identifiable Bayesian inference. By exploiting a previously established connection between the stochastic and probability flow ordinary differential equations (pfODEs) underlying DBMs, we derive a class of models, inflationary flows, that uniquely and deterministically map high-dimensional data to a lower-dimensional Gaussian distribution via ODE integration. This map is both invertible and neighborhood-preserving, with controllable numerical error, with the result that uncertainties in the data are correctly propagated to the latent space. We demonstrate how such maps can be learned via standard DBM training using a novel noise schedule and are effective at both preserving and reducing intrinsic data dimensionality. The result is a class of highly expressive generative models, uniquely defined on a low-dimensional latent space, that afford principled Bayesian inference.
A new variational inference method, SPH-ParVI, based on smoothed particle hydrodynamics (SPH), is proposed for sampling partially known densities (e.g. up to a constant) or sampling using gradients. SPH-ParVI simulates the flow of a fluid under external effects driven by the target density; transient or steady state of the fluid approximates the target density. The continuum fluid is modelled as an interacting particle system (IPS) via SPH, where each particle carries smoothed properties, interacts and evolves as per the Navier-Stokes equations. This mesh-free, Lagrangian simulation method offers fast, flexible, scalable and deterministic sampling and inference for a class of probabilistic models such as those encountered in Bayesian inference and generative modelling.
Federated representation learning (FRL) aims to learn personalized federated models with effective feature extraction from local data. FRL algorithms that share the majority of the model parameters face significant challenges with huge communication overhead. This overhead stems from the millions of neural network parameters and slow aggregation progress of the averaging heuristic. To reduce the overhead, we propose to share sufficient data summaries instead of raw model parameters. The data summaries encode minimal sufficient statistics of an exponential family, and Bayesian inference is utilized for global aggregation. It helps to reduce message sizes and communication frequency. To further ensure formal privacy guarantee, we extend it with differential privacy framework. Empirical results demonstrate high learning accuracy with low communication overhead of our method.
Orthogonal delay-Doppler (DD) division multiplexing (ODDM) has been recently proposed as a promising multicarrier modulation scheme to tackle Doppler spread in high-mobility environments. Accurate channel estimation is of paramount importance to guarantee reliable communication for the ODDM, especially when the delays and Dopplers of the propagation paths are off-grid. In this paper, we propose a novel grid refinement and adjustment-based sparse Bayesian inference (GRASBI) scheme for DD domain channel estimation. The GRASBI involves first formulating the channel estimation problem as a sparse signal recovery through the introduction of a virtual DD grid. Then, an iterative process is proposed that involves (i) sparse Bayesian learning to estimate the channel parameters and (ii) a novel grid refinement and adjustment process to adjust the virtual grid points. The grid adjustment in GRASBI relies on the maximum likelihood principle to attain the adjustment and utilizes refined grids that have much higher resolution than the virtual grid. Moreover, a low-complexity grid refinement and adjustment-based channel estimation scheme is proposed, that can provides a good tradeoff between the estimation accuracy and the complexity. Finally, numerical results are provided to demonstrate the accuracy and efficiency of the proposed channel estimation schemes.
Recent years have witnessed a growing call for renewed emphasis on neuroscience-inspired approaches in artificial intelligence research, under the banner of NeuroAI. A prime example of this is predictive coding networks (PCNs), based on the neuroscientific framework of predictive coding. This framework views the brain as a hierarchical Bayesian inference model that minimizes prediction errors through feedback connections. Unlike traditional neural networks trained with backpropagation (BP), PCNs utilize inference learning (IL), a more biologically plausible algorithm that explains patterns of neural activity that BP cannot. Historically, IL has been more computationally intensive, but recent advancements have demonstrated that it can achieve higher efficiency than BP with sufficient parallelization. Furthermore, PCNs can be mathematically considered a superset of traditional feedforward neural networks (FNNs), significantly extending the range of trainable architectures. As inherently probabilistic (graphical) latent variable models, PCNs provide a versatile framework for both supervised learning and unsupervised (generative) modeling that goes beyond traditional artificial neural networks. This work provides a comprehensive review and detailed formal specification of PCNs, particularly situating them within the context of modern ML methods. Additionally, we introduce a Python library (PRECO) for practical implementation. This positions PC as a promising framework for future ML innovations.
There are several approaches to modeling and forecasting time series as applied to prices of commodities and financial assets. One of the approaches is to model the price as a non-stationary time series process with heteroscedastic volatility (variance of price). The goal of the research is to generate probabilistic forecasts of day-ahead electricity prices in a spot marker employing stochastic volatility models. A typical stochastic volatility model - that treats the volatility as a latent stochastic process in discrete time - is explored first. Then the research focuses on enriching the baseline model by introducing several exogenous regressors. A better fitting model - as compared to the baseline model - is derived as a result of the research. Out-of-sample forecasts confirm the applicability and robustness of the enriched model. This model may be used in financial derivative instruments for hedging the risk associated with electricity trading. Keywords: Electricity spot prices forecasting, Stochastic volatility, Exogenous regressors, Autoregression, Bayesian inference, Stan
A new particle-based sampling and approximate inference method, based on electrostatics and Newton mechanics principles, is introduced with theoretical ground, algorithm design and experimental validation. This method simulates an interacting particle system (IPS) where particles, i.e. the freely-moving negative charges and spatially-fixed positive charges with magnitudes proportional to the target distribution, interact with each other via attraction and repulsion induced by the resulting electric fields described by Poisson's equation. The IPS evolves towards a steady-state where the distribution of negative charges conforms to the target distribution. This physics-inspired method offers deterministic, gradient-free sampling and inference, achieving comparable performance as other particle-based and MCMC methods in benchmark tasks of inferring complex densities, Bayesian logistic regression and dynamical system identification. A discrete-time, discrete-space algorithmic design, readily extendable to continuous time and space, is provided for usage in more general inference problems occurring in probabilistic machine learning scenarios such as Bayesian inference, generative modelling, and beyond.
This paper proposes a joint channel and data estimation (JCDE) algorithm for uplink multiuser extremely large-scale multiple-input-multiple-output (XL-MIMO) systems. The initial channel estimation is formulated as a sparse reconstruction problem based on the angle and distance sparsity under the near-field propagation condition. This problem is solved using non-orthogonal pilots through an efficient low complexity two-stage compressed sensing algorithm. Furthermore, the initial channel estimates are refined by employing a JCDE framework driven by both non-orthogonal pilots and estimated data. The JCDE problem is solved by sequential expectation propagation (EP) algorithms, where the channel and data are alternately updated in an iterative manner. In the channel estimation phase, integrating Bayesian inference with a model-based deterministic approach provides precise estimations to effectively exploit the near-field characteristics in the beam-domain. In the data estimation phase, a linear minimum mean square error (LMMSE)-based filter is designed at each sub-array to address the correlation due to energy leakage in the beam-domain arising from the near-field effects. Numerical simulations reveal that the proposed initial channel estimation and JCDE algorithm outperforms the state-ofthe-art approaches in terms of channel estimation, data detection, and computational complexity.
Bayesian inference has predominantly relied on the Markov chain Monte Carlo (MCMC) algorithm for many years. However, MCMC is computationally laborious, especially for complex phylogenetic models of time trees. This bottleneck has led to the search for alternatives, such as variational Bayes, which can scale better to large datasets. In this paper, we introduce torchtree, a framework written in Python that allows developers to easily implement rich phylogenetic models and algorithms using a fixed tree topology. One can either use automatic differentiation, or leverage torchtree's plug-in system to compute gradients analytically for model components for which automatic differentiation is slow. We demonstrate that the torchtree variational inference framework performs similarly to BEAST in terms of speed and approximation accuracy. Furthermore, we explore the use of the forward KL divergence as an optimizing criterion for variational inference, which can handle discontinuous and non-differentiable models. Our experiments show that inference using the forward KL divergence tends to be faster per iteration compared to the evidence lower bound (ELBO) criterion, although the ELBO-based inference may converge faster in some cases. Overall, torchtree provides a flexible and efficient framework for phylogenetic model development and inference using PyTorch.
In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible. While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii). The proposed methodology results in an efficient derivative-free sampler flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.
In this paper, we develop a decision support system for the hierarchical text classification. We consider text collections with a fixed hierarchical structure of topics given by experts in the form of a tree. The system sorts the topics by relevance to a given document. The experts choose one of the most relevant topics to finish the classification. We propose a weighted hierarchical similarity function to calculate topic relevance. The function calculates the similarity of a document and a tree branch. The weights in this function determine word importance. We use the entropy of words to estimate the weights. The proposed hierarchical similarity function formulates a joint hierarchical thematic classification probability model of the document topics, parameters, and hyperparameters. The variational Bayesian inference gives a closed-form EM algorithm. The EM algorithm estimates the parameters and calculates the probability of a topic for a given document. Compared to hierarchical multiclass SVM, hierarchical PLSA with adaptive regularization, and hierarchical naive Bayes, the weighted hierarchical similarity function has better improvement in ranking accuracy in an abstract collection of a major conference EURO and a website collection of industrial companies.
Using both single-index measures and stochastic dominance concepts, we show how Bayesian inference can be used to make multivariate welfare comparisons. A four-dimensional distribution for the well-being attributes income, mental health, education, and happiness are estimated via Bayesian Markov chain Monte Carlo using unit-record data taken from the Household, Income and Labour Dynamics in Australia survey. Marginal distributions of beta and gamma mixtures and discrete ordinal distributions are combined using a copula. Improvements in both well-being generally and poverty magnitude are assessed using posterior means of single-index measures and posterior probabilities of stochastic dominance. The conditions for stochastic dominance depend on the class of utility functions that is assumed to define a social welfare function and the number of attributes in the utility function. Three classes of utility functions are considered, and posterior probabilities of dominance are computed for one, two, and four-attribute utility functions for three time intervals within the period 2001 to 2019.
The immense production of the chemical industry requires an improved predictive risk assessment that can handle constantly evolving challenges while reducing the dependency of risk assessment on animal testing. Integrating 'omics data into mechanistic models offers a promising solution by linking cellular processes triggered after chemical exposure with observed effects in the organism. With the emerging availability of time-resolved RNA data, the goal of integrating gene expression data into mechanistic models can be approached. We propose a biologically anchored TKTD model, which describes key processes that link the gene expression level of the stress regulator $nrf2$ to detoxification and lethality by associating toxicodynamic damage with $nrf2$ expression. Fitting such a model to complex datasets consisting of multiple endpoints required the combination of methods from molecular biology, mechanistic dynamic systems modeling and Bayesian inference. In this study we successfully integrate time-resolved gene expression data into TKTD models, and thus provide a method for assessing the influence of molecular markers on survival. This novel method was used to test whether, $nrf2$, can be applied to predict lethality in zebrafish embryos. With the presented approach we outline a method to successively approach the goal of a predictive risk assessment based on molecular data.
Bayesian bandit algorithms with approximate Bayesian inference have been widely used in real-world applications. Despite the superior practical performance, their theoretical justification is less investigated in the literature, especially for contextual bandit problems. To fill this gap, we propose a theoretical framework to analyze the impact of approximate inference in stochastic linear bandits and conduct regret analysis on two Bayesian bandit algorithms, Linear Thompson sampling (LinTS) and the extension of Bayesian Upper Confidence Bound, namely Linear Bayesian Upper Confidence Bound (LinBUCB). We demonstrate that when applied in the presence of approximate inference, LinTS and LinBUCB can preserve their original rates of regret upper bound but with a sacrifice of larger constant terms. These results hold for general Bayesian inference approaches, assuming the inference error measured by two different $\alpha$-divergences is bounded. Additionally, by introducing a new definition of well-behaved distributions, we show that LinBUCB expedites the regret rate of LinTS from $\tilde{O}(d^{3/2}\sqrt{T})$ to $\tilde{O}(d\sqrt{T})$, matching the minimax optimal rate. To our knowledge, this work provides the first regret bounds in the setting of stochastic linear bandits with bounded approximate inference errors.
We propose a framework to perform Bayesian inference using conditional score-based diffusion models to solve a class of inverse problems in mechanics involving the inference of a specimen's spatially varying material properties from noisy measurements of its mechanical response to loading. Conditional score-based diffusion models are generative models that learn to approximate the score function of a conditional distribution using samples from the joint distribution. More specifically, the score functions corresponding to multiple realizations of the measurement are approximated using a single neural network, the so-called score network, which is subsequently used to sample the posterior distribution using an appropriate Markov chain Monte Carlo scheme based on Langevin dynamics. Training the score network only requires simulating the forward model. Hence, the proposed approach can accommodate black-box forward models and complex measurement noise. Moreover, once the score network has been trained, it can be re-used to solve the inverse problem for different realizations of the measurements. We demonstrate the efficacy of the proposed approach on a suite of high-dimensional inverse problems in mechanics that involve inferring heterogeneous material properties from noisy measurements. Some examples we consider involve synthetic data, while others include data collected from actual elastography experiments. Further, our applications demonstrate that the proposed approach can handle different measurement modalities, complex patterns in the inferred quantities, non-Gaussian and non-additive noise models, and nonlinear black-box forward models. The results show that the proposed framework can solve large-scale physics-based inverse problems efficiently.
Traditional regression and prediction tasks often only provide deterministic point estimates. To estimate the distribution or uncertainty of the response variable, traditional methods either assume that the posterior distribution of samples follows a Gaussian process or require thousands of forward passes for sample generation. We propose a novel approach called DistPred for regression and forecasting tasks, which overcomes the limitations of existing methods while remaining simple and powerful. Specifically, we transform proper scoring rules that measure the discrepancy between the predicted distribution and the target distribution into a differentiable discrete form and use it as a loss function to train the model end-to-end. This allows the model to sample numerous samples in a single forward pass to estimate the potential distribution of the response variable. We have compared our method with several existing approaches on multiple datasets and achieved state-of-the-art performance. Additionally, our method significantly improves computational efficiency. For example, compared to state-of-the-art models, DistPred has a 180x faster inference speed Experimental results can be reproduced through https://github.com/Anoise/DistPred.
Domain generalization aims to develop a model that can perform well on unseen target domains by learning from multiple source domains. However, recent-proposed domain generalization models usually rely on domain labels, which may not be available in many real-world scenarios. To address this challenge, we propose a Discriminant Risk Minimization (DRM) theory and the corresponding algorithm to capture the invariant features without domain labels. In DRM theory, we prove that reducing the discrepancy of prediction distribution between overall source domain and any subset of it can contribute to obtaining invariant features. To apply the DRM theory, we develop an algorithm which is composed of Bayesian inference and a new penalty termed as Categorical Discriminant Risk (CDR). In Bayesian inference, we transform the output of the model into a probability distribution to align with our theoretical assumptions. We adopt sliding update approach to approximate the overall prediction distribution of the model, which enables us to obtain CDR penalty. We also indicate the effectiveness of these components in finding invariant features. We evaluate our algorithm against various domain generalization methods on multiple real-world datasets, providing empirical support for our theory.
Faced with complex problems, the human brain demonstrates a remarkable capacity to transcend sensory input and form latent understandings of perceived world patterns. However, this cognitive capacity is not explicitly considered or encoded in current large language models (LLMs). As a result, LLMs often struggle to capture latent structures and model uncertainty in complex compositional reasoning tasks. This work introduces a novel Bayesian prompting approach that facilitates training-free Bayesian inference with LLMs by using a verbalized Probabilistic Graphical Model (PGM). While traditional Bayesian approaches typically depend on extensive data and predetermined mathematical structures for learning latent factors and dependencies, our approach efficiently reasons latent variables and their probabilistic dependencies by prompting LLMs to adhere to Bayesian principles. We evaluated our model on several compositional reasoning tasks, both close-ended and open-ended. Our results indicate that the model effectively enhances confidence elicitation and text generation quality, demonstrating its potential to improve AI language understanding systems, especially in modeling uncertainty.
To obtain high-resolution images of subsurface structures from seismic data, seismic imaging techniques such as Full Waveform Inversion (FWI) serve as crucial tools. However, FWI involves solving a nonlinear and often non-unique inverse problem, presenting challenges such as local minima trapping and inadequate handling of inherent uncertainties. In addressing these challenges, we propose leveraging deep generative models as the prior distribution of geophysical parameters for stochastic Bayesian inversion. This approach integrates the adjoint state gradient for efficient back-propagation from the numerical solution of partial differential equations. Additionally, we introduce explicit and implicit variational Bayesian inference methods. The explicit method computes variational distribution density using a normalizing flow-based neural network, enabling computation of the Bayesian posterior of parameters. Conversely, the implicit method employs an inference network attached to a pretrained generative model to estimate density, incorporating an entropy estimator. Furthermore, we also experimented with the Stein Variational Gradient Descent (SVGD) method as another variational inference technique, using particles. We compare these variational Bayesian inference methods with conventional Markov chain Monte Carlo (McMC) sampling. Each method is able to quantify uncertainties and to generate seismic data-conditioned realizations of subsurface geophysical parameters. This framework provides insights into subsurface structures while accounting for inherent uncertainties.
Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. Despite its empirical success, the theoretical properties of VI have only received attention recently, and mostly when the parametric family is the one of Gaussians. This work aims to contribute to the theoretical study of VI in the non-Gaussian case by investigating the setting of Mixture of Gaussians with fixed covariance and constant weights. In this view, VI over this specific family can be casted as the minimization of a Mollified relative entropy, i.e. the KL between the convolution (with respect to a Gaussian kernel) of an atomic measure supported on Diracs, and the target distribution. The support of the atomic measure corresponds to the localization of the Gaussian components. Hence, solving variational inference becomes equivalent to optimizing the positions of the Diracs (the particles), which can be done through gradient descent and takes the form of an interacting particle system. We study two sources of error of variational inference in this context when optimizing the mollified relative entropy. The first one is an optimization result, that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second one is an approximation error, that upper bounds the objective between an optimal finite mixture and the target distribution.