Suppose we observe a random vector $X$ from some distribution $P$ in a known family with unknown parameters. We ask the following question: when is it possible to split $X$ into two parts $f(X)$ and $g(X)$ such that neither part is sufficient to reconstruct $X$ by itself, but both together can recover $X$ fully, and the joint distribution of $(f(X),g(X))$ is tractable? As one example, if $X=(X_1,\dots,X_n)$ and $P$ is a product distribution, then for any $m<n$, we can split the sample to define $f(X)=(X_1,\dots,X_m)$ and $g(X)=(X_{m+1},\dots,X_n)$. Rasines and Young (2022) offers an alternative approach that uses additive Gaussian noise -- this enables post-selection inference in finite samples for Gaussian distributed data and asymptotically when errors are non-Gaussian. In this paper, we offer a more general methodology for achieving such a split in finite samples by borrowing ideas from Bayesian inference to yield a (frequentist) solution that can be viewed as a continuous analog of data splitting. We call our method data fission, as an alternative to data splitting, data carving and p-value masking. We exemplify the method on a few prototypical applications, such as post-selection inference for trend filtering and other regression problems.
Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a challenge, a shortcoming that we address by introducing a surrogate likelihood that can be learned jointly with other variational parameters. We argue theoretically that the resulting algorithm permits the user to make an intuitive trade-off between inference fidelity and computational cost. In an extensive empirical comparison we show that our method performs well in practice and that it is well-suited for black-box inference in probabilistic programming frameworks.
Bayesian models of behavior have provided computational level explanations in a range of psychophysical tasks. One fundamental experimental paradigm is the production or reproduction task, in which subjects are instructed to generate an action that either reproduces a previously sensed stimulus magnitude or achieves a target response. This type of task therefore distinguishes itself from other psychophysical tasks in that the responses are on a continuum and effort plays an important role with increasing response magnitude. Based on Bayesian decision theory we present an inference method to recover perceptual uncertainty, response variability, and the cost function underlying human responses. Crucially, the cost function is parameterized such that effort is explicitly included. We present a hybrid inference method employing MCMC sampling utilizing appropriate proposal distributions and an inner loop utilizing amortized inference with a neural network that approximates the mode of the optimal response distribution. We show how this model can be utilized to avoid unidentifiability of experimental designs and that parameters can be recovered through validation on synthetic and application to experimental data. Our approach will enable behavioral scientists to perform Bayesian inference of decision making parameters in production and reproduction tasks.
A central concept in active inference is that the internal states of a physical system parametrise probability measures over states of the external world. These can be seen as an agent's beliefs, expressed as a Bayesian prior or posterior. Here we begin the development of a general theory that would tell us when it is appropriate to interpret states as representing beliefs in this way. We focus on the case in which a system can be interpreted as performing either Bayesian filtering or Bayesian inference. We provide formal definitions of what it means for such an interpretation to exist, using techniques from category theory.
We introduce Reactive Message Passing (RMP) as a framework for executing schedule-free, robust and scalable message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style that only describes how nodes in a factor graph react to changes in connected nodes. The absence of a fixed message passing schedule improves robustness, scalability and execution time of the inference procedure. We also present ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-defined specification of local form and factorization constraints on the variational posterior distribution, ReactiveMP.jl executes hybrid message passing algorithms including belief propagation, variational message passing, expectation propagation, and expectation maximisation update rules. Experimental results demonstrate the improved performance of ReactiveMP-based RMP in comparison to other Julia packages for Bayesian inference across a range of probabilistic models. In particular, we show that the RMP framework is able to run Bayesian inference for large-scale probabilistic state space models with hundreds of thousands of random variables on a standard laptop computer.
Currently, it is hard to reap the benefits of deep learning for Bayesian methods, which allow the explicit specification of prior knowledge and accurately capture model uncertainty. We present Prior-Data Fitted Networks (PFNs). PFNs leverage in-context learning in large-scale machine learning techniques to approximate a large set of posteriors. The only requirement for PFNs to work is the ability to sample from a prior distribution over supervised learning tasks (or functions). Our method restates the objective of posterior approximation as a supervised classification problem with a set-valued input: it repeatedly draws a task (or function) from the prior, draws a set of data points and their labels from it, masks one of the labels and learns to make probabilistic predictions for it based on the set-valued input of the rest of the data points. Presented with a set of samples from a new supervised learning task as input, PFNs make probabilistic predictions for arbitrary other data points in a single forward propagation, having learned to approximate Bayesian inference. We demonstrate that PFNs can near-perfectly mimic Gaussian processes and also enable efficient Bayesian inference for intractable problems, with over 200-fold speedups in multiple setups compared to current methods. We obtain strong results in very diverse areas such as Gaussian process regression, Bayesian neural networks, classification for small tabular data sets, and few-shot image classification, demonstrating the generality of PFNs. Code and trained PFNs are released at https://github.com/automl/TransformersCanDoBayesianInference.
Neural density estimators have proven remarkably powerful in performing efficient simulation-based Bayesian inference in various research domains. In particular, the BayesFlow framework uses a two-step approach to enable amortized parameter estimation in settings where the likelihood function is implicitly defined by a simulation program. But how faithful is such inference when simulations are poor representations of reality? In this paper, we conceptualize the types of model misspecification arising in simulation-based inference and systematically investigate the performance of the BayesFlow framework under these misspecifications. We propose an augmented optimization objective which imposes a probabilistic structure on the latent data space and utilize maximum mean discrepancy (MMD) to detect potentially catastrophic misspecifications during inference undermining the validity of the obtained results. We verify our detection criterion on a number of artificial and realistic misspecifications, ranging from toy conjugate models to complex models of decision making and disease outbreak dynamics applied to real data. Further, we show that posterior inference errors increase as a function of the distance between the true data-generating distribution and the typical set of simulations in the latent summary space. Thus, we demonstrate the dual utility of MMD as a method for detecting model misspecification and as a proxy for verifying the faithfulness of amortized Bayesian inference.
Autonomous Land Vehicles (ALV) shall efficiently recognize the ground in unknown environments. A novel $\mathcal{GP}$-based method is proposed for the ground segmentation task in rough driving scenarios. A non-stationary covariance function is utilized as the kernel for the $\mathcal{GP}$. The ground surface behavior is assumed to only demonstrate local-smoothness. Thus, point estimates of the kernel's length-scales are obtained. Thus, two Gaussian processes are introduced to separately model the observation and local characteristics of the data. While, the \textit{observation process} is used to model the ground, the \textit{latent process} is put on length-scale values to estimate point values of length-scales at each input location. Input locations for this latent process are chosen in a physically-motivated procedure to represent an intuition about ground condition. Furthermore, an intuitive guess of length-scale value is represented by assuming the existence of hypothetical surfaces in the environment that every bunch of data points may be assumed to be resulted from measurements from this surfaces. Bayesian inference is implemented using \textit{maximum a Posteriori} criterion. The log-marginal likelihood function is assumed to be a multi-task objective function, to represent a whole-frame unbiased view of the ground at each frame. Simulation results shows the effectiveness of the proposed method even in an uneven, rough scene which outperforms similar Gaussian process based ground segmentation methods. While adjacent segments do not have similar ground structure in an uneven scene, the proposed method gives an efficient ground estimation based on a whole-frame viewpoint instead of just estimating segment-wise probable ground surfaces.
Understanding capabilities and limitations of different network architectures is of fundamental importance to machine learning. Bayesian inference on Gaussian processes has proven to be a viable approach for studying recurrent and deep networks in the limit of infinite layer width, $n\to\infty$. Here we present a unified and systematic derivation of the mean-field theory for both architectures that starts from first principles by employing established methods from statistical physics of disordered systems. The theory elucidates that while the mean-field equations are different with regard to their temporal structure, they yet yield identical Gaussian kernels when readouts are taken at a single time point or layer, respectively. Bayesian inference applied to classification then predicts identical performance and capabilities for the two architectures. Numerically, we find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks and the convergence speed depends non-trivially on the parameters of the weight prior as well as the depth or number of time steps, respectively. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$ and we compute next-to-leading-order corrections which turn out to be architecture-specific. The formalism thus paves the way to investigate the fundamental differences between recurrent and deep architectures at finite widths $n$.
Local volatility is a versatile option pricing model due to its state dependent diffusion coefficient. Calibration is, however, non-trivial as it involves both proposing a hypothesis model of the latent function and a method for fitting it to data. In this paper we present novel Bayesian inference with Gaussian process priors. We obtain a rich representation of the local volatility function with a probabilistic notion of uncertainty attached to the calibrate. We propose an inference algorithm and apply our approach to S&P 500 market data.
Multiple techniques for producing calibrated predictive probabilities using deep neural networks in supervised learning settings have emerged that leverage approaches to ensemble diverse solutions discovered during cyclic training or training from multiple random starting points (deep ensembles). However, only a limited amount of work has investigated the utility of exploring the local region around each diverse solution (posterior mode). Using three well-known deep architectures on the CIFAR-10 dataset, we evaluate several simple methods for exploring local regions of the weight space with respect to Brier score, accuracy, and expected calibration error. We consider both Bayesian inference techniques (variational inference and Hamiltonian Monte Carlo applied to the softmax output layer) as well as utilizing the stochastic gradient descent trajectory near optima. While adding separate modes to the ensemble uniformly improves performance, we show that the simple mode exploration methods considered here produce little to no improvement over ensembles without mode exploration.
Approximate Bayesian deep learning methods hold significant promise for addressing several issues that occur when deploying deep learning components in intelligent systems, including mitigating the occurrence of over-confident errors and providing enhanced robustness to out of distribution examples. However, the computational requirements of existing approximate Bayesian inference methods can make them ill-suited for deployment in intelligent IoT systems that include lower-powered edge devices. In this paper, we present a range of approximate Bayesian inference methods for supervised deep learning and highlight the challenges and opportunities when applying these methods on current edge hardware. We highlight several potential solutions to decreasing model storage requirements and improving computational scalability, including model pruning and distillation methods.
Active inference is a mathematical framework which originated in computational neuroscience as a theory of how the brain implements action, perception and learning. Recently, it has been shown to be a promising approach to the problems of state-estimation and control under uncertainty, as well as a foundation for the construction of goal-driven behaviours in robotics and artificial agents in general. Here, we review the state-of-the-art theory and implementations of active inference for state-estimation, control, planning and learning; describing current achievements with a particular focus on robotics. We showcase relevant experiments that illustrate its potential in terms of adaptation, generalization and robustness. Furthermore, we connect this approach with other frameworks and discuss its expected benefits and challenges: a unified framework with functional biological plausibility using variational Bayesian inference.
In this paper we present a new algorithm for online (sequential) inference in Bayesian neural networks, and show its suitability for tackling contextual bandit problems. The key idea is to combine the extended Kalman filter (which locally linearizes the likelihood function at each time step) with a (learned or random) low-dimensional affine subspace for the parameters; the use of a subspace enables us to scale our algorithm to models with $\sim 1M$ parameters. While most other neural bandit methods need to store the entire past dataset in order to avoid the problem of "catastrophic forgetting", our approach uses constant memory. This is possible because we represent uncertainty about all the parameters in the model, not just the final linear layer. We show good results on the "Deep Bayesian Bandit Showdown" benchmark, as well as MNIST and a recommender system.
Inspired by the foundational works by Spivak and Fong and Cruttwell et al., we introduce a categorical framework to formalize Bayesian inference and learning. The two key ideas at play here are the notions of Bayesian inversions and the functor GL as constructed by Cruttwell et al.. In this context, we find that Bayesian learning is the simplest case of the learning paradigm. We then obtain categorical formulations of batch and sequential Bayes updates while also verifying that the two coincide in a specific example.
We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.
Variational particle-based Bayesian learning methods have the advantage of not being limited by the bias affecting more conventional parametric techniques. This paper proposes to leverage the flexibility of non-parametric Bayesian approximate inference to develop a novel Bayesian federated unlearning method, referred to as Forget-Stein Variational Gradient Descent (Forget-SVGD). Forget-SVGD builds on SVGD - a particle-based approximate Bayesian inference scheme using gradient-based deterministic updates - and on its distributed (federated) extension known as Distributed SVGD (DSVGD). Upon the completion of federated learning, as one or more participating agents request for their data to be "forgotten", Forget-SVGD carries out local SVGD updates at the agents whose data need to be "unlearned", which are interleaved with communication rounds with a parameter server. The proposed method is validated via performance comparisons with non-parametric schemes that train from scratch by excluding data to be forgotten, as well as with existing parametric Bayesian unlearning methods.
The discrepancies between reality and simulation impede the optimisation and scalability of solid-state quantum devices. Disorder induced by the unpredictable distribution of material defects is one of the major contributions to the reality gap. We bridge this gap using physics-aware machine learning, in particular, using an approach combining a physical model, deep learning, Gaussian random field, and Bayesian inference. This approach has enabled us to infer the disorder potential of a nanoscale electronic device from electron transport data. This inference is validated by verifying the algorithm's predictions about the gate voltage values required for a laterally-defined quantum dot device in AlGaAs/GaAs to produce current features corresponding to a double quantum dot regime.
Seismic impedance inversion can be performed with a semi-supervised learning algorithm, which only needs a few logs as labels and is less likely to get overfitted. However, classical semi-supervised learning algorithm usually leads to artifacts on the predicted impedance image. In this artical, we improve the semi-supervised learning from two aspects. First, by replacing 1-d convolutional neural network (CNN) layers in deep learning structure with 2-d CNN layers and 2-d maxpooling layers, the prediction accuracy is improved. Second, prediction uncertainty can also be estimated by embedding the network into a Bayesian inference framework. Local reparameterization trick is used during forward propagation of the network to reduce sampling cost. Tests with Marmousi2 model and SEAM model validate the feasibility of the proposed strategy.
The Kalman filter is an algorithm for the estimation of hidden variables in dynamical systems under linear Gauss-Markov assumptions with widespread applications across different fields. Recently, its Bayesian interpretation has received a growing amount of attention especially in neuroscience, robotics and machine learning. In neuroscience, in particular, models of perception and control under the banners of predictive coding, optimal feedback control, active inference and more generally the so-called Bayesian brain hypothesis, have all heavily relied on ideas behind the Kalman filter. Active inference, an algorithmic theory based on the free energy principle, specifically builds on approximate Bayesian inference methods proposing a variational account of neural computation and behaviour in terms of gradients of variational free energy. Using this ambitious framework, several works have discussed different possible relations between free energy minimisation and standard Kalman filters. With a few exceptions, however, such relations point at a mere qualitative resemblance or are built on a set of very diverse comparisons based on purported differences between free energy minimisation and Kalman filtering. In this work, we present a straightforward derivation of Kalman filters consistent with active inference via a variational treatment of free energy minimisation in terms of gradient descent. The approach considered here offers a more direct link between models of neural dynamics as gradient descent and standard accounts of perception and decision making based on probabilistic inference, further bridging the gap between hypotheses about neural implementation and computational principles in brain and behavioural sciences.