The Levy walk in which the frequency of occurrence of step lengths follows a power-law distribution, can be observed in the migratory behavior of organisms at various levels. Levy walks with power exponents close to 2 are observed, and the reasons are unclear. This study aims to propose a model that universally generates inverse square Levy walks (called Cauchy walks) and to identify the conditions under which Cauchy walks appear. We demonstrate that Cauchy walks emerge universally in goal-oriented tasks. We use the term "goal-oriented" when the goal is clear, but this can be achieved in different ways, which cannot be uniquely determined. We performed a simulation in which an agent observed the data generated from a probability distribution in a two-dimensional space and successively estimated the central coordinates of that probability distribution. The agent has a model of probability distribution as a hypothesis for data-generating distribution and can modify the model such that each time a data point is observed, thereby increasing the estimated probability of occurrence of the observed data. To achieve this, the center coordinates of the model must be moved closer to those of the observed data. However, in the case of a two-dimensional space, arbitrariness arises in the direction of correction of the center; this task is goal oriented. We analyze two cases: a strategy that allocates the amount of modification randomly in the x- and y-directions, and a strategy that determines allocation such that movement is minimized. The results reveal that when a random strategy is used, the Cauchy walk appears. When the minimum strategy is used, the Brownian walk appears. The presence or absence of the constraint of minimizing the amount of movement may be a factor that causes the difference between Brownian and Levy walks.

Semi-supervised learning by self-training heavily relies on pseudo-label selection (PLS). The selection often depends on the initial model fit on labeled data. Early overfitting might thus be propagated to the final model by selecting instances with overconfident but erroneous predictions, often referred to as confirmation bias. This paper introduces BPLS, a Bayesian framework for PLS that aims to mitigate this issue. At its core lies a criterion for selecting instances to label: an analytical approximation of the posterior predictive of pseudo-samples. We derive this selection criterion by proving Bayes optimality of the posterior predictive of pseudo-samples. We further overcome computational hurdles by approximating the criterion analytically. Its relation to the marginal likelihood allows us to come up with an approximation based on Laplace's method and the Gaussian integral. We empirically assess BPLS for parametric generalized linear and non-parametric generalized additive models on simulated and real-world data. When faced with high-dimensional data prone to overfitting, BPLS outperforms traditional PLS methods.

Discovering causal relations from observational data becomes possible with additional assumptions such as considering the functional relations to be constrained as nonlinear with additive noise (ANM). Even with strong assumptions, causal discovery involves an expensive search problem over the space of directed acyclic graphs (DAGs). \emph{Topological ordering} approaches reduce the optimisation space of causal discovery by searching over a permutation rather than graph space. For ANMs, the \emph{Hessian} of the data log-likelihood can be used for finding leaf nodes in a causal graph, allowing its topological ordering. However, existing computational methods for obtaining the Hessian still do not scale as the number of variables and the number of samples increase. Therefore, inspired by recent innovations in diffusion probabilistic models (DPMs), we propose \emph{DiffAN}\footnote{Implementation is available at \url{https://github.com/vios-s/DiffAN} .}, a topological ordering algorithm that leverages DPMs for learning a Hessian function. We introduce theory for updating the learned Hessian without re-training the neural network, and we show that computing with a subset of samples gives an accurate approximation of the ordering, which allows scaling to datasets with more samples and variables. We show empirically that our method scales exceptionally well to datasets with up to $500$ nodes and up to $10^5$ samples while still performing on par over small datasets with state-of-the-art causal discovery methods. Implementation is available at https://github.com/vios-s/DiffAN .

This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log-likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set.

While a number of metrics--such as log-likelihood, userspecified This paper proposes a new metric to measure the scoring functions, and the area under the receiver calibration error of probabilistic binary classifiers, operating characteristic (ROC) curve--are used to assess the called test-based calibration error (TCE). TCE incorporates quality of probabilistic classifiers, it is usually hard or even a novel loss function based on a statistical impossible to gauge whether predictions are well-calibrated test to examine the extent to which model predictions from the values of these metrics. For assessment of calibration, differ from probabilities estimated from it is typically necessary to use a metric that measures data. It offers (i) a clear interpretation, (ii) a consistent calibration error, that is, a deviation between model predictions scale that is unaffected by class imbalance, and and probabilities of target occurrences estimated from (iii) an enhanced visual representation with repect data. The importance of assessing calibration error has been to the standard reliability diagram. In addition, we long emphasised in machine learning [Nixon et al., 2019, introduce an optimality criterion for the binning Minderer et al., 2021] and in probabilistic forecasting more procedure of calibration error metrics based on a broadly [Dawid, 1982, Degroot and Fienberg, 1983].

The quantification of aleatoric and epistemic uncertainty in terms of conditional entropy and mutual information, respectively, has recently become quite common in machine learning. While the properties of these measures, which are rooted in information theory, seem appealing at first glance, we identify various incoherencies that call their appropriateness into question. In addition to the measures themselves, we critically discuss the idea of an additive decomposition of total uncertainty into its aleatoric and epistemic constituents. Experiments across different computer vision tasks support our theoretical findings and raise concerns about current practice in uncertainty quantification.

One of the challenges practitioners face when applying structure learning algorithms to their data involves determining a set of hyperparameters; otherwise, a set of hyperparameter defaults is assumed. The optimal hyperparameter configuration often depends on multiple factors, including the size and density of the usually unknown underlying true graph, the sample size of the input data, and the structure learning algorithm. We propose a novel hyperparameter tuning method, called the Out-of-sample Tuning for Structure Learning (OTSL), that employs out-of-sample and resampling strategies to estimate the optimal hyperparameter configuration for structure learning, given the input data set and structure learning algorithm. Synthetic experiments show that employing OTSL as a means to tune the hyperparameters of hybrid and score-based structure learning algorithms leads to improvements in graphical accuracy compared to the state-of-the-art. We also illustrate the applicability of this approach to real datasets from different disciplines.

We present a model-agnostic algorithm for generating post-hoc explanations and uncertainty intervals for a machine learning model when only a static sample of inputs and outputs from the model is available, rather than direct access to the model itself. This situation may arise when model evaluations are expensive; when privacy, security and bandwidth constraints are imposed; or when there is a need for real-time, on-device explanations. Our algorithm uses a bootstrapping approach to quantify the uncertainty that inevitably arises when generating explanations from a finite sample of model queries. Through a simulation study, we show that the uncertainty intervals generated by our algorithm exhibit a favorable trade-off between interval width and coverage probability compared to the naive confidence intervals from classical regression analysis as well as current Bayesian approaches for quantifying explanation uncertainty. We further demonstrate the capabilities of our method by applying it to black-box models, including a deep neural network, trained on three real-world datasets.

Existing causal discovery methods typically require the data to be available in a centralized location. However, many practical domains, such as healthcare, limit access to the data gathered by local entities, primarily for privacy and regulatory constraints. To address this, we propose FED-CD, a federated framework for inferring causal structures from distributed datasets containing observational and interventional data. By exchanging updates instead of data samples, FED-CD ensures privacy while enabling decentralized discovery of the underlying directed acyclic graph (DAG). We accommodate scenarios with shared or disjoint intervened covariates, and mitigate the adverse effects of interventional data heterogeneity. We provide empirical evidence for the performance and scalability of FED-CD for decentralized causal discovery using synthetic and real-world DAGs.

Recent work in Dialogue Act (DA) classification approaches the task as a sequence labeling problem, using neural network models coupled with a Conditional Random Field (CRF) as the last layer. CRF models the conditional probability of the target DA label sequence given the input utterance sequence. However, the task involves another important input sequence, that of speakers, which is ignored by previous work. To address this limitation, this paper proposes a simple modification of the CRF layer that takes speaker-change into account. Experiments on the SwDA corpus show that our modified CRF layer outperforms the original one, with very wide margins for some DA labels. Further, visualizations demonstrate that our CRF layer can learn meaningful, sophisticated transition patterns between DA label pairs conditioned on speaker-change in an end-to-end way. Code is publicly available.