Road traffic congestion is a persistent problem. Focusing resources on the causes of congestion is a potentially efficient strategy for reducing slowdowns. We present NEXICA, an algorithm to discover which parts of the highway system tend to cause slowdowns on other parts of the highway. We use time series of road speeds as inputs to our causal discovery algorithm. Finding other algorithms inadequate, we develop a new approach that is novel in three ways. First, it concentrates on just the presence or absence of events in the time series, where an event indicates the temporal beginning of a traffic slowdown. Second, we develop a probabilistic model using maximum likelihood estimation to compute the probabilities of spontaneous and caused slowdowns between two locations on the highway. Third, we train a binary classifier to identify pairs of cause/effect locations trained on pairs of road locations where we are reasonably certain a priori of their causal connections, both positive and negative. We test our approach on six months of road speed data from 195 different highway speed sensors in the Los Angeles area, showing that our approach is superior to state-of-the-art baselines in both accuracy and computation speed.

The second inequality can be proved by resorting to the first one.

Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees.

Backpropagation (BP) of errors is the backbone training algorithm for artificial neural networks (ANNs). It updates network weights through gradient descent to minimize a loss function representing the mismatch between predictions and desired outputs. BP uses the chain rule to propagate the loss gradient backward through the network hierarchy, allowing efficient weight updates. However, this process requires weight updates at every layer to rely on a global error signal generated at the network's output. In contrast, the Hebbian model of synaptic plasticity states that weight updates are local, depending only on the activity of pre- and post-synaptic neurons. This suggests biological brains likely do not implement BP directly. Recently, Predictive Coding (PC) has gained interest as a biologically plausible alternative that updates weights using only local information. Originating from 1950s work on signal compression, PC was later proposed as a model of the visual cortex and formalized under the free energy principle, linking it to Bayesian inference and dynamical systems. PC weight updates rely solely on local information and provide theoretical advantages such as automatic scaling of gradients based on uncertainty. This lecture notes column offers a novel, tutorial-style introduction to PC, focusing on its formulation, derivation, and connections to well-known optimization and signal processing algorithms such as BP and the Kalman Filter (KF). It aims to support existing literature by guiding readers from the mathematical foundations of PC to practical implementation, including Python examples using PyTorch.

The scientific process is a means for turning the results of experiments into knowledge about the world in which we live. Much research effort has been directed toward automating this process. To do this, one needs to formulate the scientific process in a precise mathematical language. This paper outlines one such language. What is presented here is hardly new. The material leans much on great thinkers of times past as well as more modern contributions. The novel contributions of this paper are: A new, general data processing inequality, a bias variance decomposition for canonical losses, Streamlined proofs of the Blackwell-Sherman-Stein and Randomization Theorems, and Means to calculate deficiency via linear programming.

We propose a novel method for sensitivity analysis to unobserved confounding in causal inference. The method builds on a copula-based causal graphical normalizing flow that we term $ρ$-GNF, where $ρ\in [-1,+1]$ is the sensitivity parameter. The parameter represents the non-causal association between exposure and outcome due to unobserved confounding, which is modeled as a Gaussian copula. In other words, the $ρ$-GNF enables scholars to estimate the average causal effect (ACE) as a function of $ρ$, accounting for various confounding strengths. The output of the $ρ$-GNF is what we term the $ρ_{curve}$, which provides the bounds for the ACE given an interval of assumed $ρ$ values. The $ρ_{curve}$ also enables scholars to identify the confounding strength required to nullify the ACE. We also propose a Bayesian version of our sensitivity analysis method. Assuming a prior over the sensitivity parameter $ρ$ enables us to derive the posterior distribution over the ACE, which enables us to derive credible intervals. Finally, leveraging on experiments from simulated and real-world data, we show the benefits of our sensitivity analysis method.

Causal machine learning has the potential to revolutionize decision-making by combining the predictive power of machine learning algorithms with the theory of causal inference. However, these methods remain underutilized by the broader machine learning community, in part because current empirical evaluations do not permit assessment of their reliability and robustness, undermining their practical utility. Specifically, one of the principal criticisms made by the community is the extensive use of synthetic experiments. We argue, on the contrary, that synthetic experiments are essential and necessary to precisely assess and understand the capabilities of causal machine learning methods. To substantiate our position, we critically review the current evaluation practices, spotlight their shortcomings, and propose a set of principles for conducting rigorous empirical analyses with synthetic data. Adopting the proposed principles will enable comprehensive evaluations that build trust in causal machine learning methods, driving their broader adoption and impactful real-world use.

Understanding causal relationships between variables is fundamental across scientific disciplines. Most causal discovery algorithms rely on two key assumptions: (i) all variables are observed, and (ii) the underlying causal graph is acyclic. While these assumptions simplify theoretical analysis, they are often violated in real-world systems, such as biological networks. Existing methods that account for confounders either assume linearity or struggle with scalability. To address these limitations, we propose DCCD-CONF, a novel framework for differentiable learning of nonlinear cyclic causal graphs in the presence of unmeasured confounders using interventional data. Our approach alternates between optimizing the graph structure and estimating the confounder distribution by maximizing the log-likelihood of the data. Through experiments on synthetic data and real-world gene perturbation datasets, we show that DCCD-CONF outperforms state-of-the-art methods in both causal graph recovery and confounder identification. Additionally, we also provide consistency guarantees for our framework, reinforcing its theoretical soundness.

-- Deep generative models, particularly diffusion and flow matching models, have recently shown remarkable potential in learning complex policies through imitation learning. However, the safety of generated motions remains overlooked, particularly in complex environments with inherent obstacles. In this work, we address this critical gap by proposing Potential Field-Guided Flow Matching Policy (PF2MP), a novel approach that simultaneously learns task policies and extracts obstacle-related information, represented as a potential field, from the same set of successful demonstrations. During inference, PF2MP modulates the flow matching vector field via the learned potential field, enabling safe motion generation. By leveraging these complementary fields, our approach achieves improved safety without compromising task success across diverse environments, such as navigation tasks and robotic manipulation scenarios. We evaluate PF2MP in both simulation and real-world settings, demonstrating its effectiveness in task space and joint space control. Experimental results demonstrate that PF2MP enhances safety, achieving a significant reduction of collisions compared to baseline policies. This work paves the way for safer motion generation in unstructured and obstacle-rich environments.