For instance, penalties are used to improve generalization, prune neurons or reduce the rank of tensors of weights. Therefore, usual penalties are mostly empirical and user-defined, and integrated to the loss as follows: L( w) null( w) r (w), with w the vector of all parameters in the network, null( w) the error term and r (w) the penalty term. From a Bayesian point of view, optimizing such a loss L is equivalent to finding the Maximum A Posteriori (MAP) of the parameters w given the training data and a prior α exp( r). Indeed, assuming that the loss null is a log-likelihood loss, namely, null(w) ln p w( D) with dataset D, then minimizing L is equivalent to minimizing L MAP(w) ln p w(D) ln(α (w)). Thus, within the MAP framework, we can interpret the penalty term r as the influence of a prior α [14]. However, the MAP approximates the Bayesian posterior very roughly, by taking its maximum. Variational Inference (VI) provides a variational posterior distribution rather than a single value, hopefully representing the Bayesian posterior much better. VI looks for the best posterior approximation within a family β u(w) of approximate posteriors over w, parameterized Inria, Team TAU, Gif-sur-Yvette, France † Facebook, France 1 arXiv:2002.00178v1

Nested sampling is an efficient algorithm for the calculation of the Bayesian evidence and posterior parameter probability distributions. It is based on the step-by-step exploration of the parameter space by Monte Carlo sampling with a series of values sets called live points that evolve towards the region of interest, i.e. where the likelihood function is maximal. In presence of several local likelihood maxima, the algorithm converges with difficulty. Some systematic errors can also be introduced by unexplored parameter volume regions. In order to avoid this, different methods are proposed in the literature for an efficient search of new live points, even in presence of local maxima. Here we present a new solution based on the mean shift cluster recognition method implemented in a random walk search algorithm. The clustering recognition is integrated within the Bayesian analysis program NestedFit. It is tested with the analysis of some difficult cases. Compared to the analysis results without cluster recognition, the computation time is considerably reduced. At the same time, the entire parameter space is efficiently explored, which translates into a smaller uncertainty of the extracted value of the Bayesian evidence.

Optimal Bayesian feature selection (OBFS) is a multivariat e supervised screening method designed from the ground up for bioma rker discovery. In this work, we prove that Gaussian OBFS is strongly consisten t under mild conditions, and provide rates of convergence for key posteriors i n the framework. These results are of enormous importance, since they identify pre cisely what features are selected by OBFS asymptotically, characterize the relativ e rates of convergence for posteriors on different types of features, provide condi tions that guarantee convergence, justify the use of OBFS when its internal assum ptions are invalid, and set the stage for understanding the asymptotic behavior of other algorithms based on the OBFS framework.

Approximate Bayesian Computation is widely used in systems biology for inferring parameters in stochastic gene regulatory network models. Its performance hinges critically on the ability to summarize high-dimensional system responses such as time series into a few informative, low-dimensional summary statistics. The quality of those statistics critically affect the accuracy of the inference. Existing methods to select the best subset out of a pool of candidate statistics do not scale well with large pools. Since it is imperative for good performance this becomes a serious bottleneck when doing inference on complex and high-dimensional problems. This paper proposes a convolutional neural network architecture for automatically learning informative summary statistics of temporal responses. We show that the proposed network can effectively circumvent the statistics selection problem as a preprocessing step to ABC for a challenging inference problem learning parameters in a high-dimensional stochastic genetic oscillator. We also study the impact of experimental design on network performance by comparing different data richness and different data acquisition strategies.

Classifiers built with neural networks handle large-scale high dimensional data, such as facial images from computer vision, extremely well while traditional statistical methods often fail miserably. In this paper, we attempt to understand this empirical success in high dimensional classification by deriving the convergence rates of excess risk. In particular, a teacher-student framework is proposed that assumes the Bayes classifier to be expressed as ReLU neural networks. In this setup, we obtain a sharp rate of convergence, i.e., $\tilde{O}_d(n^{-2/3})$, for classifiers trained using either 0-1 loss or hinge loss. This rate can be further improved to $\tilde{O}_d(n^{-1})$ when the data distribution is separable. Here, $n$ denotes the sample size. An interesting observation is that the data dimension only contributes to the $\log(n)$ term in the above rates. This may provide one theoretical explanation for the empirical successes of deep neural networks in high dimensional classification, particularly for structured data.

Anna Sergeevna Bosman Department of Computer Science University of Pretoria Pretoria, South Africa anna.bosman@up.ac.za Abstract --Saliency maps are often used in computer vision to provide intuitive interpretations of what input regions a model has used to produce a specific prediction. A number of approaches to saliency map generation are available, but most require access to model parameters. This work proposes an approach for saliency map generation for black-box models, where no access to model parameters is available, using a Bayesian optimisation sampling method. The approach aims to find the global salient image region responsible for a particular (black-box) model's prediction. This is achieved by a sampling-based approach to model perturbations that seeks to localise salient regions of an image to the black-box model. Results show that the proposed approach to saliency map generation outperforms grid-based perturbation approaches, and performs similarly to gradient-based approaches which require access to model parameters. I NTRODUCTION Deep learning (DL) techniques have become a standard approach in computer vision. Specifically, the convolutional neural network (CNN) architecture has shown exceptional performance, achieving results comparable to human performance on image recognition tasks [1]-[3]. As a result, the CNN models are often deployed in real life as efficient black-box tools.

Predicting stochastic cellular dynamics as emerging from the mechanistic models of molecular interactions is a long-standing challenge in systems biology: low-level chemical reaction network (CRN) models give raise to a highly-dimensional continuous-time Markov chain (CTMC) which is computationally demanding and often prohibitive to analyse in practice. A recently proposed abstraction method uses deep learning to replace this CTMC with a discrete-time continuous-space process, by training a mixture density deep neural network with traces sampled at regular time intervals (which can obtained either by simulating a given CRN or as time-series data from experiment). The major advantage of such abstraction is that it produces a computational model that is dramatically cheaper to execute, while preserving the statistical features of the training data. In general, the abstraction accuracy improves with the amount of training data. However, depending on a CRN, the overall quality of the method -- the efficiency gain and abstraction accuracy -- will also depend on the choice of neural network architecture given by hyper-parameters such as the layer types and connections between them. As a consequence, in practice, the modeller would have to take care of finding the suitable architecture manually, for each given CRN, through a tedious and time-consuming trial-and-error cycle. In this paper, we propose to further automatise deep abstractions for stochastic CRNs, through learning the optimal neural network architecture along with learning the transition kernel of the abstract process. Automated search of the architecture makes the method applicable directly to any given CRN, which is time-saving for deep learning experts and crucial for non-specialists. We implement the method and demonstrate its performance on a number of representative CRNs with multi-modal emergent phenotypes.

Many classification applications require accurate probability estimates in addition to good class separation but often classifiers are designed focusing only on the latter. Calibration is the process of improving probability estimates by post-processing but commonly used calibration algorithms work poorly on small data sets and assume the classification task to be binary. Both of these restrictions limit their real-world applicability. Previously introduced Data Generation and Grouping algorithm alleviates the problem posed by small data sets and in this article, we will demonstrate that its application to multi-class problems is also possible which solves the other limitation. Our experiments show that calibration error can be decreased using the proposed approach and the additional computational cost is acceptable.

Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the dataset. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless dataset. Specifically, we consider the scenario where the scale parameter of a Sobolev kernel (such as a Mat\'ern kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become "slowly" overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model. The analysis is based on a combination of techniques from nonparametric regression and scattered data interpolation. Empirical results are provided in support of the theoretical findings.

Incorporating constraints is a major concern in probabilistic machine learning. A wide variety of problems require predictions to be integrated with reasoning about constraints, from modelling routes on maps to approving loan predictions. In the former, we may require the prediction model to respect the presence of physical paths between the nodes on the map, and in the latter, we may require that the prediction model respect fairness constraints that ensure that outcomes are not subject to bias. Broadly speaking, constraints may be probabilistic, logical or causal, but the overarching challenge is to determine if and how a model can be learnt that handles all the declared constraints. To the best of our knowledge, this is largely an open problem. In this paper, we consider a mathematical inquiry on how the learning of tractable probabilistic models, such as sum-product networks, is possible while incorporating constraints.