In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error bound for linear time-invariant (LTI) stochastic dynamical systems with inputs. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. In particular, with the bound derived in this paper relates future average prediction errors with the prediction error generated by the model on the data used for learning. In turn, this allows us to provide finite-sample error bounds for a wide class of learning/system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

The mixed multinomial logit model assumes constant preference parameters of a decision-maker throughout different choice situations, which may be considered too strong for certain choice modelling applications. This paper proposes an effective approach to model context-dependent intra-respondent heterogeneity, thereby introducing the concept of the Context-aware Bayesian mixed multinomial logit model, where a neural network maps contextual information to interpretable shifts in the preference parameters of each individual in each choice occasion. The proposed model offers several key advantages. First, it supports both continuous and discrete variables, as well as complex non-linear interactions between both types of variables. Secondly, each context specification is considered jointly as a whole by the neural network rather than each variable being considered independently. Finally, since the neural network parameters are shared across all decision-makers, it can leverage information from other decision-makers to infer the effect of a particular context on a particular decision-maker. Even though the context-aware Bayesian mixed multinomial logit model allows for flexible interactions between attributes, the increase in computational complexity is minor, compared to the mixed multinomial logit model. We illustrate the concept and interpretation of the proposed model in a simulation study. We furthermore present a real-world case study from the travel behaviour domain - a bicycle route choice model, based on a large-scale, crowdsourced dataset of GPS trajectories including 119,448 trips made by 8,555 cyclists.

Tree-shaped graphical models are widely used for their tractability. However, they unfortunately lack expressive power as they require committing to a particular sparse dependency structure. We propose a novel class of generative models called mixtures of all trees: that is, a mixture over all possible ($n^{n-2}$) tree-shaped graphical models over $n$ variables. We show that it is possible to parameterize this Mixture of All Trees (MoAT) model compactly (using a polynomial-size representation) in a way that allows for tractable likelihood computation and optimization via stochastic gradient descent. Furthermore, by leveraging the tractability of tree-shaped models, we devise fast-converging conditional sampling algorithms for approximate inference, even though our theoretical analysis suggests that exact computation of marginals in the MoAT model is NP-hard. Empirically, MoAT achieves state-of-the-art performance on density estimation benchmarks when compared against powerful probabilistic models including hidden Chow-Liu Trees.

The process of training feedforward neural networks (FFNNs) can benefit from an automated process where the best heuristic to train the network is sought out automatically by means of a high-level probabilistic-based heuristic. This research introduces a novel population-based Bayesian hyper-heuristic (BHH) that is used to train feedforward neural networks (FFNNs). The performance of the BHH is compared to that of ten popular low-level heuristics, each with different search behaviours. The chosen heuristic pool consists of classic gradient-based heuristics as well as meta-heuristics (MHs). The empirical process is executed on fourteen datasets consisting of classification and regression problems with varying characteristics. The BHH is shown to be able to train FFNNs well and provide an automated method for finding the best heuristic to train the FFNNs at various stages of the training process.

We present a hierarchical Bayesian learning approach to infer jointly sparse parameter vectors from multiple measurement vectors. Our model uses separate conditionally Gaussian priors for each parameter vector and common gamma-distributed hyper-parameters to enforce joint sparsity. The resulting joint-sparsity-promoting priors are combined with existing Bayesian inference methods to generate a new family of algorithms. Our numerical experiments, which include a multi-coil magnetic resonance imaging application, demonstrate that our new approach consistently outperforms commonly used hierarchical Bayesian methods.

Segmentation uncertainty models predict a distribution over plausible segmentations for a given input, which they learn from the annotator variation in the training set. However, in practice these annotations can differ systematically in the way they are generated, for example through the use of different labeling tools. This results in datasets that contain both data variability and differing label styles. In this paper, we demonstrate that applying state-of-the-art segmentation uncertainty models on such datasets can lead to model bias caused by the different label styles. We present an updated modelling objective conditioning on labeling style for aleatoric uncertainty estimation, and modify two state-of-the-art-architectures for segmentation uncertainty accordingly. We show with extensive experiments that this method reduces label style bias, while improving segmentation performance, increasing the applicability of segmentation uncertainty models in the wild. We curate two datasets, with annotations in different label styles, which we will make publicly available along with our code upon publication.

This paper revisits classical works of Rauch (1963, et al. 1965) and develops a novel method for maximum likelihood (ML) smoothing estimation from incomplete information/data of stochastic state-space systems. Score function and conditional observed information matrices of incomplete data are introduced and their distributional identities are established. Using these identities, the ML smoother $\widehat{x}_{k\vert n}^s =\argmax_{x_k} \log f(x_k,\widehat{x}_{k+1\vert n}^s, y_{0:n}\vert\theta)$, $k\leq n-1$, is presented. The result shows that the ML smoother gives an estimate of state $x_k$ with more adherence of loglikehood having less standard errors than that of the ML state estimator $\widehat{x}_k=\argmax_{x_k} \log f(x_k,y_{0:k}\vert\theta)$, with $\widehat{x}_{n\vert n}^s=\widehat{x}_n$. Recursive estimation is given in terms of an EM-gradient-particle algorithm which extends the work of \cite{Lange} for ML smoothing estimation. The algorithm has an explicit iteration update which lacks in (\cite{Ramadan}) EM-algorithm for smoothing. A sequential Monte Carlo method is developed for valuation of the score function and observed information matrices. A recursive equation for the covariance matrix of estimation error is developed to calculate the standard errors. In the case of linear systems, the method shows that the Rauch-Tung-Striebel (RTS) smoother is a fully efficient smoothing state-estimator whose covariance matrix coincides with the Cram\'er-Rao lower bound, the inverse of expected information matrix. Furthermore, the RTS smoother coincides with the Kalman filter having less covariance matrix. Numerical studies are performed, confirming the accuracy of the main results.

The maximum likelihood principle is a fundamental method of estimation for a large number of models in data science, machine learning, and artificial intelligence. It is applicable to a range of…

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Abstract--Machine learning methods strive to acquire a robust model during training that can generalize well to test samples, even under distribution shifts. However, these methods often suffer from a performance drop due to unknown test distributions. Test-time adaptation (TTA), an emerging paradigm, has the potential to adapt a pre-trained model to unlabeled data during testing, before making predictions. Recent progress in this paradigm highlights the significant benefits of utilizing unlabeled data for training self-adapted models prior to inference. In this survey, we divide TTA into several distinct categories, namely, test-time (source-free) domain adaptation, test-time batch adaptation, online test-time adaptation, and test-time prior adaptation. For each category, we provide a comprehensive taxonomy of advanced algorithms, followed by a discussion of different learning scenarios. Furthermore, we analyze relevant applications of TTA and discuss open challenges and promising areas for future research. However, when the test distribution (target) differs from the training distribution (source), we face the problem of distribution shifts. Such a shift poses significant challenges for machine learning systems deployed in the wild, such as images captured by different cameras [2], road scenes of different cities [3], and imaging devices in different hospitals [4]. In contrast, TTA only requires access to the pre-trained from one or multiple source domains that can generalize model from the source domain, making it a secure and well to any out-of-distribution target domain. Figure 1: test-time domain adaptation, test-time batch adaptation This survey primarily focuses on test-time adaptation (TTBA), and online test-time adaptation (OTTA). That is to say, test data. Additionally, DA typically necessitates access to the predictions of each mini-batch are independent of the both labeled data from the source domain and (unlabeled) predictions for the other mini-batches. Ran He is also with the School of Artificial Intelligence, University of Chinese Academy of Sciences. In this survey, we use the terms "test data" and "target data" Tieniu Tan is also with Nanjing University, China. DA methods rely on the existence of source applied to OTTA with the assumption of knowledge reuse.