This paper presents a Bayesian estimation method for sequential direction finding. The proposed method estimates the number of directions of arrivals (DOAs) and their DOAs performing operations on the factor graph. The graph represents a statistical model for sequential beamforming. At each time step, belief propagation predicts the number of DOAs and their DOAs using posterior probability density functions (pdfs) from the previous time and a different Bernoulli-von Mises state transition model. Variational Bayesian inference then updates the number of DOAs and their DOAs. The method promotes sparse solutions through a Bernoulli-Gaussian amplitude model, is gridless, and provides marginal posterior pdfs from which DOA estimates and their uncertainties can be extracted. Compared to nonsequential approaches, the method can reduce DOA estimation errors in scenarios involving multiple time steps and time-varying DOAs. Simulation results demonstrate performance improvements compared to state-of-the-art methods. The proposed method is evaluated using ocean acoustic experimental data.
Animal learning has interested ecologists and psychologists for over a century. Mathematical models that explain how animals store and recall information have gained attention recently. Central to this work is statistical decision theory (SDT), which relates information uptake in animals to Bayesian inference. SDT effectively explains many learning tasks in animals, but extending this theory to predict how animals will learn in changing environments still poses a challenge for ecologists. We addressed this shortcoming with a novel implementation of Bayesian Markov Chain Monte Carlo (MCMC) sampling to simulate how animals sample environmental information and learn as a result. We applied our framework to an individual-based model simulating complex foraging tasks encountered by wild animals. Simulated ``animals" learned behavioral strategies that optimized foraging returns simply by following the principles of an MCMC sampler. In these simulations, behavioral plasticity was most conducive to efficient foraging in unpredictable and uncertain environments. Our model suggests that animals prioritize highly concentrated resources even when these resources are less available overall, in line with existing knowledge on optimal foraging and ideal free distribution theory. Our innovative computational modelling framework can be applied more widely to simulate the learning of many other tasks in animals and humans.
Wasserstein-Fisher-Rao (WFR) distance is a family of metrics to gauge the discrepancy of two Radon measures, which takes into account both transportation and weight change. Spherical WFR distance is a projected version of WFR distance for probability measures so that the space of Radon measures equipped with WFR can be viewed as metric cone over the space of probability measures with spherical WFR. Compared to the case for Wasserstein distance, the understanding of geodesics under the spherical WFR is less clear and still an ongoing research focus. In this paper, we develop a deep learning framework to compute the geodesics under the spherical WFR metric, and the learned geodesics can be adopted to generate weighted samples. Our approach is based on a Benamou-Brenier type dynamic formulation for spherical WFR. To overcome the difficulty in enforcing the boundary constraint brought by the weight change, a Kullback-Leibler (KL) divergence term based on the inverse map is introduced into the cost function. Moreover, a new regularization term using the particle velocity is introduced as a substitute for the Hamilton-Jacobi equation for the potential in dynamic formula. When used for sample generation, our framework can be beneficial for applications with given weighted samples, especially in the Bayesian inference, compared to sample generation with previous flow models.
Mapping people dynamics is a crucial skill for robots, because it enables them to coexist in human-inhabited environments. However, learning a model of people dynamics is a time consuming process which requires observation of large amount of people moving in an environment. Moreover, approaches for mapping dynamics are unable to transfer the learned models across environments: each model is only able to describe the dynamics of the environment it has been built in. However, the impact of architectural geometry on people's movement can be used to anticipate their patterns of dynamics, and recent work has looked into learning maps of dynamics from occupancy. So far however, approaches based on trajectories and those based on geometry have not been combined. In this work we propose a novel Bayesian approach to learn people dynamics able to combine knowledge about the environment geometry with observations from human trajectories. An occupancy-based deep prior is used to build an initial transition model without requiring any observations of pedestrian; the model is then updated when observations become available using Bayesian inference. We demonstrate the ability of our model to increase data efficiency and to generalize across real large-scale environments, which is unprecedented for maps of dynamics.
Current methods for regularization in machine learning require quite specific model assumptions (e.g. a kernel shape) that are not derived from prior knowledge about the application, but must be imposed merely to make the method work. We show in this paper that regularization can indeed be achieved by assuming nothing but invariance principles (w.r.t. scaling, translation, and rotation of input and output space) and the degree of differentiability of the true function. Concretely, we derive a novel (non-Gaussian) stochastic process from the above minimal assumptions, and we present a corresponding Bayesian inference method for regression. The mean posterior turns out to be a polyharmonic spline, and the posterior process is a mixture of t-processes. Compared with Gaussian process regression, the proposed method shows equal performance and has the advantages of being (i) less arbitrary (no choice of kernel) (ii) potentially faster (no kernel parameter optimization), and (iii) having better extrapolation behavior. We believe that the proposed theory has central importance for the conceptual foundations of regularization and machine learning and also has great potential to enable practical advances in ML areas beyond regression.
We introduce a probabilistic model with implicit norm regularization for learning nonnegative matrix factorization (NMF) that is commonly used for predicting missing values and finding hidden patterns in the data, in which the matrix factors are latent variables associated with each data dimension. The nonnegativity constraint for the latent factors is handled by choosing priors with support on the nonnegative subspace, e.g., exponential density or distribution based on exponential function. Bayesian inference procedure based on Gibbs sampling is employed. We evaluate the model on several real-world datasets including Genomics of Drug Sensitivity in Cancer (GDSC $IC_{50}$) and Gene body methylation with different sizes and dimensions, and show that the proposed Bayesian NMF GL$_2^2$ and GL$_\infty$ models lead to robust predictions for different data values and avoid overfitting compared with competitive Bayesian NMF approaches.
When sampling for Bayesian inference, one popular approach is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS) which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density, and can prove slow in practice. We propose Hamiltonian neural networks (HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, HNNs do not require numerical gradients of the target density during sampling. Moreover, they satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose an HNN extension called latent HNNs (L-HNNs), which are capable of predicting latent variable outputs. Compared to HNNs, L-HNNs offer improved expressivity and reduced integration errors. Finally, we employ L-HNNs in NUTS with an online error monitoring scheme to prevent sample degeneracy in regions of low probability density. We demonstrate L-HNNs in NUTS with online error monitoring on several examples involving complex, heavy-tailed, and high-local-curvature probability densities. Overall, L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. Compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1--2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient by an order of magnitude.
This work proposes a novel resource allocation strategy for anti-jamming in Cognitive Radio using Active Inference ($\textit{AIn}$), and a cognitive-UAV is employed as a case study. An Active Generalized Dynamic Bayesian Network (Active-GDBN) is proposed to represent the external environment that jointly encodes the physical signal dynamics and the dynamic interaction between UAV and jammer in the spectrum. We cast the action and planning as a Bayesian inference problem that can be solved by avoiding surprising states (minimizing abnormality) during online learning. Simulation results verify the effectiveness of the proposed $\textit{AIn}$ approach in minimizing abnormalities (maximizing rewards) and has a high convergence speed by comparing it with the conventional Frequency Hopping and Q-learning.
Study of neural networks with infinite width is important for better understanding of the neural network in practical application. In this work, we derive the equivalence of the deep, infinite-width maxout network and the Gaussian process (GP) and characterize the maxout kernel with a compositional structure. Moreover, we build up the connection between our deep maxout network kernel and deep neural network kernels. We also give an efficient numerical implementation of our kernel which can be adapted to any maxout rank. Numerical results show that doing Bayesian inference based on the deep maxout network kernel can lead to competitive results compared with their finite-width counterparts and deep neural network kernels. This enlightens us that the maxout activation may also be incorporated into other infinite-width neural network structures such as the convolutional neural network (CNN).
Divide-and-conquer strategies for Monte Carlo algorithms are an increasingly popular approach to making Bayesian inference scalable to large data sets. In its simplest form, the data are partitioned across multiple computing cores and a separate Markov chain Monte Carlo algorithm on each core targets the associated partial posterior distribution, which we refer to as a sub-posterior, that is the posterior given only the data from the segment of the partition associated with that core. Divide-and-conquer techniques reduce computational, memory and disk bottle-necks, but make it difficult to recombine the sub-posterior samples. We propose SwISS: Sub-posteriors with Inflation, Scaling and Shifting; a new approach for recombining the sub-posterior samples which is simple to apply, scales to high-dimensional parameter spaces and accurately approximates the original posterior distribution through affine transformations of the sub-posterior samples. We prove that our transformation is asymptotically optimal across a natural set of affine transformations and illustrate the efficacy of SwISS against competing algorithms on synthetic and real-world data sets.
We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of $Y$ given $X$ into a target reproducing kernel Hilbert space $\mathcal{H}_Y$. The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between $\mathcal{H}_X$ and $L_2$, to $\mathcal{H}_Y$. This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal $O(\log n / n)$ rates without assuming $\mathcal{H}_Y$ to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.
The recent introduction of gradient-based MCMC for discrete spaces holds great promise, and comes with the tantalising possibility of new discrete counterparts to celebrated continuous methods such as MALA and HMC. Towards this goal, we introduce several discrete Metropolis-Hastings samplers that are conceptually-inspired by MALA, and demonstrate their strong empirical performance across a range of challenging sampling problems in Bayesian inference and energy-based modelling. Methodologically, we identify why discrete analogues to preconditioned MALA are generally intractable, motivating us to introduce a new kind of preconditioning based on auxiliary variables and the `Gaussian integral trick'.
Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges via a neural network that approximates the posterior distribution not only for one instance of data, but a distribution of data pertaining to a specific inverse problem. During inference, the neural network -- in our case a conditional normalizing flow -- provides posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems due to the Earth's heterogeneity. In addition, the network is prone to errors if evaluated over out-of-distribution data. As such, we propose to increase the resilience of amortized variational inference in the presence of moderate data distribution shifts. We achieve this via a correction to the latent distribution that improves the posterior distribution approximation for the data at hand. The correction involves relaxing the standard Gaussian assumption on the latent distribution and parameterizing it via a Gaussian distribution with an unknown mean and (diagonal) covariance. These unknowns are then estimated by minimizing the Kullback-Leibler divergence between the corrected and the (physics-based) true posterior distributions. While generic and applicable to other inverse problems, by means of a linearized seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in the number of seismic sources, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty at approximately the same cost as five reverse-time migrations.
Graph neural networks (GNNs) have shown superiority in many prediction tasks over graphs due to their impressive capability of capturing nonlinear relations in graph-structured data. However, for node classification tasks, often, only marginal improvement of GNNs over their linear counterparts has been observed. Previous works provide very few understandings of this phenomenon. In this work, we resort to Bayesian learning to deeply investigate the functions of non-linearity in GNNs for node classification tasks. Given a graph generated from the statistical model CSBM, we observe that the max-a-posterior estimation of a node label given its own and neighbors' attributes consists of two types of non-linearity, a possibly non-linear transformation of node attributes and a ReLU-activated feature aggregation from neighbors. The latter surprisingly matches the type of non-linearity used in many GNN models. By further imposing Gaussian assumption on node attributes, we prove that the superiority of those ReLU activations is only significant when the node attributes are far more informative than the graph structure, which nicely matches many previous empirical observations. A similar argument can be achieved when there is a distribution shift of node attributes between the training and testing datasets. Finally, we verify our theory on both synthetic and real-world networks.
Variational inference has recently emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference. The core idea is to trade statistical accuracy for computational efficiency. In this work, we study these statistical and computational trade-offs in variational inference via a case study in inferential model selection. Focusing on Gaussian inferential models (or variational approximating families) with diagonal plus low-rank precision matrices, we initiate a theoretical study of the trade-offs in two aspects, Bayesian posterior inference error and frequentist uncertainty quantification error. From the Bayesian posterior inference perspective, we characterize the error of the variational posterior relative to the exact posterior. We prove that, given a fixed computation budget, a lower-rank inferential model produces variational posteriors with a higher statistical approximation error, but a lower computational error; it reduces variance in stochastic optimization and, in turn, accelerates convergence. From the frequentist uncertainty quantification perspective, we consider the precision matrix of the variational posterior as an uncertainty estimate, which involves an additional statistical error originating from the sampling uncertainty of the data. As a consequence, for small datasets, the inferential model need not be full-rank to achieve optimal estimation error (even with unlimited computation budget).
Multiband fusion enhances WiFi sensing by jointly utilizing signals from multiple non-contiguous frequency bands. However, in the multi-band WiFi sensing signal model, there are many local optimums in the associated likelihood function due to the existence of high frequency component and phase distortion factors, posing challenges for high-accuracy parameter estimation. To address this, we propose a two-stage scheme equipped with different signal models derived from the original model, where the first-stage coarse estimation is performed using a weighted root MUSIC algorithm to narrow down the search range for the subsequent stage, and the second-stage refined estimation utilizes a Bayesian approach to avoid convergence to bad suboptimal solutions. Specifically, we apply the block stochastic successive convex approximation (SSCA) approach to derive a novel stochastic particle-based variational Bayesian inference (SPVBI) algorithm in the refined stage. Unlike conventional particle-based VBI (PVBI) that optimizes only particle probability and incurs exponential per-iteration complexity with particle count, our more flexible SPVBI algorithm optimizes both the position and probability of each particle. Additionally, it utilizes block SSCA to significantly improve sampling efficiency by averaging over iterations, making it suitable for high-dimensional problems. Extensive simulations demonstrate the superiority of our proposed algorithm over various baseline methods.
We propose a novel framework for multitask reinforcement learning based on the minimum description length (MDL) principle. In this approach, which we term MDL-control (MDL-C), the agent learns the common structure among the tasks with which it is faced and then distills it into a simpler representation which facilitates faster convergence and generalization to new tasks. In doing so, MDL-C naturally balances adaptation to each task with epistemic uncertainty about the task distribution. We motivate MDL-C via formal connections between the MDL principle and Bayesian inference, derive theoretical performance guarantees, and demonstrate MDL-C's empirical effectiveness on both discrete and high-dimensional continuous control tasks.
Variational inference, such as the mean-field (MF) approximation, requires certain conjugacy structures for efficient computation. These can impose unnecessary restrictions on the viable prior distribution family and further constraints on the variational approximation family. In this work, we introduce a general computational framework to implement MF variational inference for Bayesian models, with or without latent variables, using the Wasserstein gradient flow (WGF), a modern mathematical technique for realizing a gradient flow over the space of probability measures. Theoretically, we analyze the algorithmic convergence of the proposed approaches, providing an explicit expression for the contraction factor. We also strengthen existing results on MF variational posterior concentration from a polynomial to an exponential contraction, by utilizing the fixed point equation of the time-discretized WGF. Computationally, we propose a new constraint-free function approximation method using neural networks to numerically realize our algorithm. This method is shown to be more precise and efficient than traditional particle approximation methods based on Langevin dynamics.
We present a new probabilistic method for detecting, deblending, and cataloging astronomical sources called the Bayesian Light Source Separator (BLISS). BLISS is based on deep generative models, which embed neural networks within a Bayesian model. For posterior inference, BLISS uses a new form of variational inference known as Forward Amortized Variational Inference. The BLISS inference routine is fast, requiring a single forward pass of the encoder networks on a GPU once the encoder networks are trained. BLISS can perform fully Bayesian inference on megapixel images in seconds, and produces highly accurate catalogs. BLISS is highly extensible, and has the potential to directly answer downstream scientific questions in addition to producing probabilistic catalogs.
Simulation-Based Inference (SBI) is a promising Bayesian inference framework that alleviates the need for analytic likelihoods to estimate posterior distributions. Recent advances using neural density estimators in SBI algorithms have demonstrated the ability to achieve high-fidelity posteriors, at the expense of a large number of simulations ; which makes their application potentially very time-consuming when using complex physical simulations. In this work we focus on boosting the sample-efficiency of posterior density estimation using the gradients of the simulator. We present a new method to perform Neural Posterior Estimation (NPE) with a differentiable simulator. We demonstrate how gradient information helps constrain the shape of the posterior and improves sample-efficiency.