The free energy principle (FEP) states that any dynamical system can be interpreted as performing Bayesian inference upon its surrounding environment. In this work, we examine in depth the assumptions required to derive the FEP in the simplest possible set of systems -- weakly-coupled non-equilibrium linear stochastic systems. Specifically, we explore (i) how general the requirements imposed on the statistical structure of a system are and (ii) how informative the FEP is about the behaviour of such systems. We discover that two requirements of the FEP -- the Markov blanket condition (i.e. a statistical boundary precluding direct coupling between internal and external states) and stringent restrictions on its solenoidal flows (i.e. tendencies driving a system out of equilibrium) -- are only valid for a very narrow space of parameters. Suitable systems require an absence of perception-action asymmetries that is highly unusual for living systems interacting with an environment. More importantly, we observe that a mathematically central step in the argument, connecting the behaviour of a system to variational inference, relies on an implicit equivalence between the dynamics of the average states of a system with the average of the dynamics of those states. This equivalence does not hold in general even for linear systems, since it requires an effective decoupling from the system's history of interactions. These observations are critical for evaluating the generality and applicability of the FEP and indicate the existence of significant problems of the theory in its current form. These issues make the FEP, as it stands, not straightforwardly applicable to the simple linear systems studied here and suggest that more development is needed before the theory could be applied to the kind of complex systems that describe living and cognitive processes.
With uncertain changes of the economic environment, macroeconomic downturns during recessions and crises can hardly be explained by a Gaussian structural shock. There is evidence that the distribution of macroeconomic variables is skewed and heavy tailed. In this paper, we contribute to the literature by extending a vector autoregression (VAR) model to account for a more realistic assumption of the multivariate distribution of the macroeconomic variables. We propose a general class of generalized hyperbolic skew Student's t distribution with stochastic volatility for the error term in the VAR model that allows us to take into account skewness and heavy tails. Tools for Bayesian inference and model selection using a Gibbs sampler are provided. In an empirical study, we present evidence of skewness and heavy tails for monthly macroeconomic variables. The analysis also gives a clear message that skewness should be taken into account for better predictions during recessions and crises.
Balancing privacy and accuracy is a major challenge in designing differentially private machine learning algorithms. One way to improve this tradeoff for free is to leverage the noise in common data operations that already use randomness. Such operations include noisy SGD and data subsampling. The additional noise in these operations may amplify the privacy guarantee of the overall algorithm, a phenomenon known as privacy amplification. In this paper, we analyze the privacy amplification of sampling from a multidimensional Bernoulli distribution family given the parameter from a private algorithm. This setup has applications to Bayesian inference and to data compression. We provide an algorithm to compute the amplification factor, and we establish upper and lower bounds on this factor.
We explore the notion of uncertainty in the context of modern abstractive summarization models, using the tools of Bayesian Deep Learning. Our approach approximates Bayesian inference by first extending state-of-the-art summarization models with Monte Carlo dropout and then using them to perform multiple stochastic forward passes. Based on Bayesian inference we are able to effectively quantify uncertainty at prediction time. Having a reliable uncertainty measure, we can improve the experience of the end user by filtering out generated summaries of high uncertainty. Furthermore, uncertainty estimation could be used as a criterion for selecting samples for annotation, and can be paired nicely with active learning and human-in-the-loop approaches. Finally, Bayesian inference enables us to find a Bayesian summary which performs better than a deterministic one and is more robust to uncertainty. In practice, we show that our Variational Bayesian equivalents of BART and PEGASUS can outperform their deterministic counterparts on multiple benchmark datasets.
Bayesian inference allows to obtain useful information on the parameters of models, either in computational statistics or more recently in the context of Bayesian Neural Networks. The computational cost of usual Monte Carlo methods for sampling posterior laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional noise in the dynamics and hence a bias on the invariant measure which is sampled by the Markov chain. We advocate using the so-called Adaptive Langevin dynamics, which is a modification of standard inertial Langevin dynamics with a dynamical friction which automatically corrects for the increased noise arising from mini-batching. We investigate the practical relevance of the assumptions underpinning Adaptive Langevin (constant covariance for the estimation of the gradient, Gaussian minibatching noise), which are not satisfied in typical models of Bayesian inference, and quantify the bias induced by minibatching in this case. We also suggest a possible extension of AdL to further reduce the bias on the posterior distribution, by considering a dynamical friction depending on the current value of the parameter to sample.
Neuronal ensemble inference is a significant problem in the study of biological neural networks. Various methods have been proposed for ensemble inference from experimental data of neuronal activity. Among them, Bayesian inference approach with generative model was proposed recently. However, this method requires large computational cost for appropriate inference. In this work, we give an improved Bayesian inference algorithm by modifying update rule in Markov chain Monte Carlo method and introducing the idea of simulated annealing for hyperparameter control. We compare the performance of ensemble inference between our algorithm and the original one, and discuss the advantage of our method.
Traditionally, Hawkes processes are used to model time--continuous point processes with history dependence. Here we propose an extended model where the self--effects are of both excitatory and inhibitory type and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and our approach dispenses with the necessity of estimating a branching structure for the posterior, as we perform inference on an aggregated sum of Gaussian Processes. Efficient approximate Bayesian inference is achieved via data augmentation, and we describe a mean--field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from three different domains and compare it to previously reported results.
Neural networks (NNs) have demonstrated their potential in a wide range of applications such as image recognition, decision making or recommendation systems. However, standard NNs are unable to capture their model uncertainty which is crucial for many safety-critical applications including healthcare and autonomous vehicles. In comparison, Bayesian neural networks (BNNs) are able to express uncertainty in their prediction via a mathematical grounding. Nevertheless, BNNs have not been as widely used in industrial practice, mainly because of their expensive computational cost and limited hardware performance. This work proposes a novel FPGA-based hardware architecture to accelerate BNNs inferred through Monte Carlo Dropout. Compared with other state-of-the-art BNN accelerators, the proposed accelerator can achieve up to 4 times higher energy efficiency and 9 times better compute efficiency. Considering partial Bayesian inference, an automatic framework is proposed, which explores the trade-off between hardware and algorithmic performance. Extensive experiments are conducted to demonstrate that our proposed framework can effectively find the optimal points in the design space.
In this paper we leverage on probability over Riemannian manifolds to rethink the interpretation of priors and posteriors in Bayesian inference. The main mindshift is to move away from the idea that "a prior distribution establishes a probability distribution over the parameters of our model" to the idea that "a prior distribution establishes a probability distribution over probability distributions". To do that we assume that our probabilistic model is a Riemannian manifold with the Fisher metric. Under this mindset, any distribution over probability distributions should be "intrinsic", that is, invariant to the specific parametrization which is selected for the manifold. We exemplify our ideas through a simple analysis of distributions over the manifold of Bernoulli distributions. One of the major shortcomings of maximum a posteriori estimates is that they depend on the parametrization. Based on the understanding developed here, we can define the maximum a posteriori estimate which is independent of the parametrization.
The advent of comprehensive synaptic wiring diagrams of large neural circuits has created the field of connectomics and given rise to a number of open research questions. One such question is whether it is possible to reconstruct the information stored in a recurrent network of neurons, given its synaptic connectivity matrix. Here, we address this question by determining when solving such an inference problem is theoretically possible in specific attractor network models and by providing a practical algorithm to do so. The algorithm builds on ideas from statistical physics to perform approximate Bayesian inference and is amenable to exact analysis. We study its performance on three different models, compare the algorithm to standard algorithms such as PCA, and explore the limitations of reconstructing stored patterns from synaptic connectivity.
While the choice of prior is one of the most critical parts of the Bayesian inference workflow, recent Bayesian deep learning models have often fallen back on vague priors, such as standard Gaussians. In this review, we highlight the importance of prior choices for Bayesian deep learning and present an overview of different priors that have been proposed for (deep) Gaussian processes, variational autoencoders, and Bayesian neural networks. We also outline different methods of learning priors for these models from data. We hope to motivate practitioners in Bayesian deep learning to think more carefully about the prior specification for their models and to provide them with some inspiration in this regard.
This paper aims to address few-shot segmentation. While existing prototype-based methods have achieved considerable success, they suffer from uncertainty and ambiguity caused by limited labeled examples. In this work, we propose attentional prototype inference (API), a probabilistic latent variable framework for few-shot segmentation. We define a global latent variable to represent the prototype of each object category, which we model as a probabilistic distribution. The probabilistic modeling of the prototype enhances the model's generalization ability by handling the inherent uncertainty caused by limited data and intra-class variations of objects. To further enhance the model, we introduce a local latent variable to represent the attention map of each query image, which enables the model to attend to foreground objects while suppressing the background. The optimization of the proposed model is formulated as a variational Bayesian inference problem, which is established by amortized inference networks. We conduct extensive experiments on four benchmarks, where our proposal obtains at least competitive and often better performance than state-of-the-art prototype-based methods. We also provide comprehensive analyses and ablation studies to gain insight into the effectiveness of our method for few-shot segmentation.
In this contribution, we predict the wellhead protection area (WHPA, target), the shape and extent of which is influenced by the distribution of hydraulic conductivity (K), from a small number of tracing experiments (predictor). Our first objective is to make stochastic predictions of the WHPA within the Bayesian Evidential Learning (BEL) framework, which aims to find a direct relationship between predictor and target using machine learning. This relationship is learned from a small set of training models (400) sampled from the prior distribution of K. The associated 400 pairs of simulated predictors and targets are obtained through forward modelling. Newly collected field data can then be directly used to predict the approximate posterior distribution of the corresponding WHPA. The uncertainty range of the posterior WHPA distribution is affected by the number and position of data sources (injection wells). Our second objective is to extend BEL to identify the optimal design of data source locations that minimizes the posterior uncertainty of the WHPA. This can be done explicitly, without averaging or approximating because once trained, the BEL model allows the computation of the posterior uncertainty corresponding to any new input data. We use the Modified Hausdorff Distance and the Structural Similarity index metrics to estimate the posterior uncertainty range of the WHPA. Increasing the number of injection wells effectively reduces the derived posterior WHPA uncertainty. Our approach can also estimate which injection wells are more informative than others, as validated through a k-fold cross-validation procedure. Overall, the application of BEL to experimental design makes it possible to identify the data sources maximizing the information content of any measurement data.
We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior performance of MsIGN over previous approaches in posterior approximation and multiple mode capture. On the natural image synthesis task, MsIGN achieves superior performance in bits-per-dimension over baseline models and yields great interpret-ability of its neurons in intermediate layers.
One of the standardized features of financial data is that log-returns are uncorrelated, but absolute log-returns or their squares namely the fluctuating volatility are correlated and is characterized by heavy tailed in the sense that some moment of the absolute log-returns is infinite and typically non-Gaussian [20]. And this last characteristic change accordantly to different timescales. We propose to model this long-memory phenomenon by superstatistical dynamics and provide a Bayesian Inference methodology drawing on Metropolis-Hasting random walk sampling to determine which superstatistics among inverse-Gamma and log-Normal describe the best log-returns complexity on different timescales, from high to low frequency. We show that on smaller timescales (minutes) even though the Inverse-Gamma superstatistics works the best, the log-Normal model remains very reliable and suitable to fit the absolute log-returns probability density distribution with strong capacity of describing heavy tails and power law decays. On larger timescales (daily), we show in terms of Bayes factor that the inverse-Gamma superstatistics is preferred to the log-Normal model. We also show evidence of a transition of statistics from power law decay on small timescales to exponential decay on large scale with less heavy tails meaning that on larger time scales the fluctuating volatility tend to be memoryless, consequently superstatistics becomes less relevant.
Neural networks and Gaussian processes are complementary in their strengths and weaknesses. Having a better understanding of their relationship comes with the promise to make each method benefit from the strengths of the other. In this work, we establish an equivalence between the forward passes of neural networks and (deep) sparse Gaussian process models. The theory we develop is based on interpreting activation functions as interdomain inducing features through a rigorous analysis of the interplay between activation functions and kernels. This results in models that can either be seen as neural networks with improved uncertainty prediction or deep Gaussian processes with increased prediction accuracy. These claims are supported by experimental results on regression and classification datasets.
Domain generalization is challenging due to the domain shift and the uncertainty caused by the inaccessibility of target domain data. In this paper, we address both challenges with a probabilistic framework based on variational Bayesian inference, by incorporating uncertainty into neural network weights. We couple domain invariance in a probabilistic formula with the variational Bayesian inference. This enables us to explore domain-invariant learning in a principled way. Specifically, we derive domain-invariant representations and classifiers, which are jointly established in a two-layer Bayesian neural network. We empirically demonstrate the effectiveness of our proposal on four widely used cross-domain visual recognition benchmarks. Ablation studies validate the synergistic benefits of our Bayesian treatment when jointly learning domain-invariant representations and classifiers for domain generalization. Further, our method consistently delivers state-of-the-art mean accuracy on all benchmarks.
Bayesian inference on non-Gaussian data is often non-analytic and requires computationally expensive approximations such as sampling or variational inference. We propose an approximate inference framework primarily designed to be computationally cheap while still achieving high approximation quality. The concept, which we call Laplace Matching, involves closed-form, approximate, bi-directional transformations between the parameter spaces of exponential families. These are constructed from Laplace approximations under custom-designed basis transformations. The mappings can then be leveraged to effectively turn a latent Gaussian distribution into an approximate conjugate prior to a rich class of observable variables. This allows us to train latent Gaussian models such as Gaussian Processes on non-Gaussian data at nearly no additional cost. The method can be thought of as a pre-processing step which can be implemented in <5 lines of code and runs in less than a second. Furthermore, Laplace Matching yields a simple way to group similar data points together, e.g. to produce inducing points for GPs. We empirically evaluate the method with experiments for four different exponential distributions, namely the Beta, Gamma, Dirichlet and inverse Wishart, showing approximation quality comparable to state-of-the-art approximate inference techniques at a drastic reduction in computational cost.
Monte Carlo sampling methods are the standard procedure for approximating complicated integrals of multidimensional posterior distributions in Bayesian inference. In this work, we focus on the class of Layered Adaptive Importance Sampling (LAIS) scheme, which is a family of adaptive importance samplers where Markov chain Monte Carlo algorithms are employed to drive an underlying multiple importance sampling scheme. The modular nature of LAIS allows for different possible implementations, yielding a variety of different performance and computational costs. In this work, we propose different enhancements of the classical LAIS setting in order to increase the efficiency and reduce the computational cost, of both upper and lower layers. The different variants address computational challenges arising in real-world applications, for instance with highly concentrated posterior distributions. Furthermore, we introduce different strategies for designing cheaper schemes, for instance, recycling samples generated in the upper layer and using them in the final estimators in the lower layer. Different numerical experiments, considering several challenging scenarios, show the benefits of the proposed schemes comparing with benchmark methods presented in the literature.
Neural networks are powerful function approximators with tremendous potential in learning complex distributions. However, they are prone to overfitting on spurious patterns. Bayesian inference provides a principled way to regularize neural networks and give well-calibrated uncertainty estimates. It allows us to specify prior knowledge on weights. However, specifying domain knowledge via distributions over weights is infeasible. Furthermore, it is unable to correct models when they focus on spurious or irrelevant features. New methods within explainable artificial intelligence allow us to regularize explanations in the form of feature importance to add domain knowledge and correct the models' focus. Nevertheless, they are incompatible with Bayesian neural networks, as they require us to modify the loss function. We propose a new explanation regularization method that is compatible with Bayesian inference. Consequently, we can quantify uncertainty and, at the same time, have correct explanations. We test our method using four different datasets. The results show that our method improves predictive performance when models overfit on spurious features or are uncertain of which features to focus on. Moreover, our method performs better than augmenting training data with samples where spurious features are removed through masking. We provide code, data, trained weights, and hyperparameters.