Bayesian approaches for learning deep neural networks (BNN) have been received much attention and successfully applied to various applications. Particularly, BNNs have the merit of having better generalization ability as well as better uncertainty quantification. For the success of BNN, search an appropriate architecture of the neural networks is an important task, and various algorithms to find good sparse neural networks have been proposed. In this paper, we propose a new node-sparse BNN model which has good theoretical properties and is computationally feasible. We prove that the posterior concentration rate to the true model is near minimax optimal and adaptive to the smoothness of the true model. In particular the adaptiveness is the first of its kind for node-sparse BNNs. In addition, we develop a novel MCMC algorithm which makes the Bayesian inference of the node-sparse BNN model feasible in practice.

Many scientific models are composed of multiple discrete components, and scien tists often make heuristic decisions about which components to include. Bayesian inference provides a mathematical framework for systematically selecting model components, but defining prior distributions over model components and developing associated inference schemes has been challenging. We approach this problem in an amortized simulation-based inference framework: We define implicit model priors over a fixed set of candidate components and train neural networks to infer joint probability distributions over both, model components and associated parameters from simulations. To represent distributions over model components, we introduce a conditional mixture of multivariate binary distributions in the Grassmann formalism. Our approach can be applied to any compositional stochastic simulator without requiring access to likelihood evaluations. We first illustrate our method on a simple time series model with redundant components and show that it can retrieve joint posterior distribution over a set of symbolic expressions and their parameters while accurately capturing redundancy with strongly correlated posteriors. We then apply our approach to drift-diffusion models, a commonly used model class in cognitive neuroscience. After validating the method on synthetic data, we show that our approach explains experimental data as well as previous methods, but that our fully probabilistic approach can help to discover multiple data-consistent model configurations, as well as reveal non-identifiable model components and parameters. Our method provides a powerful tool for data-driven scientific inquiry which will allow scientists to systematically identify essential model components and make uncertainty-informed modelling decisions.

We introduce two new particle-based algorithms for learning latent variable models via marginal maximum likelihood estimation, including one which is entirely tuning-free. Our methods are based on the perspective of marginal maximum likelihood estimation as an optimization problem: namely, as the minimization of a free energy functional. One way to solve this problem is to consider the discretization of a gradient flow associated with the free energy. We study one such approach, which resembles an extension of the popular Stein variational gradient descent algorithm. In particular, we establish a descent lemma for this algorithm, which guarantees that the free energy decreases at each iteration. This method, and any other obtained as the discretization of the gradient flow, will necessarily depend on a learning rate which must be carefully tuned by the practitioner in order to ensure convergence at a suitable rate. With this in mind, we also propose another algorithm for optimizing the free energy which is entirely learning rate free, based on coin betting techniques from convex optimization. We validate the performance of our algorithms across a broad range of numerical experiments, including several high-dimensional settings. Our results are competitive with existing particle-based methods, without the need for any hyperparameter tuning.

Previous work on controllable text generation has explored the idea of control from the latent space, such as optimizing a representation with attribute-related classifiers or sampling a representation from relevant discrete samples. However, they are not effective enough in modeling both the latent space and the control, leaving controlled text with low quality and diversity. In this work, we propose a novel control framework using probability density estimation in the latent space. Our method utilizes an invertible transformation function, the Normalizing Flow, that maps the complex distributions in the latent space to simple Gaussian distributions in the prior space. Thus, we can perform sophisticated and flexible control in the prior space and feed the control effects back into the latent space owing to the one-one-mapping property of invertible transformations. Experiments on single-attribute controls and multi-attribute control reveal that our method outperforms several strong baselines on attribute relevance and text quality and achieves the SOTA. Further analysis of control strength adjustment demonstrates the flexibility of our control strategy.

We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.

The Bayesian Synthetic Likelihood (BSL) method is a widely-used tool for likelihood-free Bayesian inference. This method assumes that some summary statistics are normally distributed, which can be incorrect in many applications. We propose a transformation, called the Wasserstein Gaussianization transformation, that uses a Wasserstein gradient flow to approximately transform the distribution of the summary statistics into a Gaussian distribution. BSL also implicitly requires compatibility between simulated summary statistics under the working model and the observed summary statistics. A robust BSL variant which achieves this has been developed in the recent literature. We combine the Wasserstein Gaussianization transformation with robust BSL, and an efficient Variational Bayes procedure for posterior approximation, to develop a highly efficient and reliable approximate Bayesian inference method for likelihood-free problems.

Design of de novo biological sequences with desired properties, like protein and DNA sequences, often involves an active loop with several rounds of molecule ideation and expensive wet-lab evaluations. These experiments can consist of multiple stages, with increasing levels of precision and cost of evaluation, where candidates are filtered. This makes the diversity of proposed candidates a key consideration in the ideation phase. In this work, we propose an active learning algorithm leveraging epistemic uncertainty estimation and the recently proposed GFlowNets as a generator of diverse candidate solutions, with the objective to obtain a diverse batch of useful (as defined by some utility function, for example, the predicted anti-microbial activity of a peptide) and informative candidates after each round. We also propose a scheme to incorporate existing labeled datasets of candidates, in addition to a reward function, to speed up learning in GFlowNets. We present empirical results on several biological sequence design tasks, and we find that our method generates more diverse and novel batches with high scoring candidates compared to existing approaches.

Bayesian inference usually requires running potentially costly inference procedures separately for every new observation. In contrast, the idea of amortized Bayesian inference is to initially invest computational cost in training an inference network on simulated data, which can subsequently be used to rapidly perform inference (i.e., to return estimates of posterior distributions) for new observations. This approach has been applied to many real-world models in the sciences and engineering, but it is unclear how robust the approach is to adversarial perturbations in the observed data. Here, we study the adversarial robustness of amortized Bayesian inference, focusing on simulation-based estimation of multi-dimensional posterior distributions. We show that almost unrecognizable, targeted perturbations of the observations can lead to drastic changes in the predicted posterior and highly unrealistic posterior predictive samples, across several benchmark tasks and a real-world example from neuroscience. We propose a computationally efficient regularization scheme based on penalizing the Fisher information of the conditional density estimator, and show how it improves the adversarial robustness of amortized Bayesian inference.

The intrinsically infinite-dimensional features of the functional observations over multidimensional domains render the standard classification methods effectively inapplicable. To address this problem, we introduce a novel multiclass functional deep neural network (mfDNN) classifier as an innovative data mining and classification tool. Specifically, we consider sparse deep neural network architecture with rectifier linear unit (ReLU) activation function and minimize the cross-entropy loss in the multiclass classification setup. This neural network architecture allows us to employ modern computational tools in the implementation. The convergence rates of the misclassification risk functions are also derived for both fully observed and discretely observed multidimensional functional data. We demonstrate the performance of mfDNN on simulated data and several benchmark datasets from different application domains.

Deep kernel processes are a recently introduced class of deep Bayesian models that have the flexibility of neural networks, but work entirely with Gram matrices. They operate by alternately sampling a Gram matrix from a distribution over positive semi-definite matrices, and applying a deterministic transformation. When the distribution is chosen to be Wishart, the model is called a deep Wishart process (DWP). This particular model is of interest because its prior is equivalent to a deep Gaussian process (DGP) prior, but at the same time it is invariant to rotational symmetries, leading to a simpler posterior distribution. Practical inference in the DWP was made possible in recent work ("A variational approximate posterior for the deep Wishart process" Ober and Aitchison 2021a) where the authors used a generalisation of the Bartlett decomposition of the Wishart distribution as the variational approximate posterior. However, predictive performance in that paper was less impressive than one might expect, with the DWP only beating a DGP on a few of the UCI datasets used for comparison. In this paper, we show that further generalising their distribution to allow linear combinations of rows and columns in the Bartlett decomposition results in better predictive performance, while incurring negligible additional computation cost.