A counterfactual distribution is the probability distribution of a random variable under a hypothetical scenario that differs from the observed reality. "What would have been the outcome for this individual if they had received a different treatment?" is an example of a counterfactual question. Here the personal data of the individual constitute the evidence that specifies the observed reality, and the interest lies in the distribution of the outcome under a hypothetical treatment. Counterfactual questions belong to the third and highest level in the causal hierarchy (Shpitser and Pearl, 2008) and are in general more difficult than associational (first level) or interventional (second level) questions. Algorithms for checking the identifiability of counterfactual queries from observational and experimental data have been developed (Shpitser and Pearl, 2007; Shpitser and Sherman, 2018; Correa et al., 2021) and implemented (Tikka, 2022).

Dueling bandits are widely used to model preferential feedback prevalent in many applications such as recommendation systems and ranking. In this paper, we study the Borda regret minimization problem for dueling bandits, which aims to identify the item with the highest Borda score while minimizing the cumulative regret. We propose a rich class of generalized linear dueling bandit models, which cover many existing models. We first prove a regret lower bound of order $\Omega(d^{2/3} T^{2/3})$ for the Borda regret minimization problem, where $d$ is the dimension of contextual vectors and $T$ is the time horizon. To attain this lower bound, we propose an explore-then-commit type algorithm for the stochastic setting, which has a nearly matching regret upper bound $\tilde{O}(d^{2/3} T^{2/3})$. We also propose an EXP3-type algorithm for the adversarial linear setting, where the underlying model parameter can change at each round. Our algorithm achieves an $\tilde{O}(d^{2/3} T^{2/3})$ regret, which is also optimal. Empirical evaluations on both synthetic data and a simulated real-world environment are conducted to corroborate our theoretical analysis.

To achieve the highest perceptual quality, state-of-the-art diffusion models are optimized with objectives that typically look very different from the maximum likelihood and the Evidence Lower Bound (ELBO) objectives. In this work, we reveal that diffusion model objectives are actually closely related to the ELBO. Specifically, we show that all commonly used diffusion model objectives equate to a weighted integral of ELBOs over different noise levels, where the weighting depends on the specific objective used. Under the condition of monotonic weighting, the connection is even closer: the diffusion objective then equals the ELBO, combined with simple data augmentation, namely Gaussian noise perturbation. We show that this condition holds for a number of state-of-the-art diffusion models. In experiments, we explore new monotonic weightings and demonstrate their effectiveness, achieving state-of-the-art FID scores on the high-resolution ImageNet benchmark.

We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data in the form of an undirected graph called the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr\"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr\"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the $20\%$ HPD credible sets include the true structure at a high rate for data-generating graphs with density at least $0.5$.

In this paper, we use a probabilistic model to estimate the number of uncorrelated features in a large dataset. Our model allows for both pairwise feature correlation (collinearity) and interdependency of multiple features (multicollinearity) and we use the probabilistic method to obtain upper and lower bounds of the same order, for the size of a feature set that exhibits low collinearity and low multicollinearity. We also prove an auxiliary result regarding mutually good constrained sets that is of independent interest.

The quintessential learning algorithm of empirical risk minimization (ERM) is known to fail in various settings for which uniform convergence does not characterize learning. It is therefore unsurprising that the practice of machine learning is rife with considerably richer algorithmic techniques for successfully controlling model capacity. Nevertheless, no such technique or principle has broken away from the pack to characterize optimal learning in these more general settings. The purpose of this work is to characterize the role of regularization in perhaps the simplest setting for which ERM fails: multiclass learning with arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit optimal learning algorithms that dovetail with tried-and-true algorithmic principles: Occam's Razor as embodied by structural risk minimization (SRM), the principle of maximum entropy, and Bayesian reasoning. Most notably, we introduce an optimal learner which relaxes structural risk minimization on two dimensions: it allows the regularization function to be "local" to datapoints, and uses an unsupervised learning stage to learn this regularizer at the outset. We justify these relaxations by showing that they are necessary: removing either dimension fails to yield a near-optimal learner. We also extract from OIGs a combinatorial sequence we term the Hall complexity, which is the first to characterize a problem's transductive error rate exactly. Lastly, we introduce a generalization of OIGs and the transductive learning setting to the agnostic case, where we show that optimal orientations of Hamming graphs -- judged using nodes' outdegrees minus a system of node-dependent credits -- characterize optimal learners exactly. We demonstrate that an agnostic version of the Hall complexity again characterizes error rates exactly, and exhibit an optimal learner using maximum entropy programs.

Most of today's wearable technology provides seamless cardiac activity monitoring. Specifically, the vast majority employ Photoplethysmography (PPG) sensors to acquire blood volume pulse information, which is further analysed to extract useful and physiologically related features. Nevertheless, PPG-based signal reliability presents different challenges that strongly affect such data processing. This is mainly related to the fact of PPG morphological wave distortion due to motion artefacts, which can lead to erroneous interpretation of the extracted cardiac-related features. On this basis, in this paper, we propose a novel personalised and adjustable Interval Type-2 Fuzzy Logic System (IT2FLS) for assessing the quality of PPG signals. The proposed system employs a personalised approach to adapt the IT2FLS parameters to the unique characteristics of each individual's PPG signals.Additionally, the system provides adjustable levels of personalisation, allowing healthcare providers to adjust the system to meet specific requirements for different applications. The proposed system obtained up to 93.72\% for average accuracy during validation. The presented system has the potential to enable ultra-low complexity and real-time PPG quality assessment, improving the accuracy and reliability of PPG-based health monitoring systems at the edge.

Financial performance management is at the core of business management and has historically relied on financial ratio analysis using Balance Sheet and Income Statement data to assess company performance as compared with competitors. Little progress has been made in predicting how a company will perform or in assessing the risks (probabilities) of financial underperformance. In this study I introduce a new set of financial and macroeconomic ratios that supplement standard ratios of Balance Sheet and Income Statement. I also provide a set of supervised learning models (ML Regressors and Neural Networks) and Bayesian models to predict company performance. I conclude that the new proposed variables improve model accuracy when used in tandem with standard industry ratios. I also conclude that Feedforward Neural Networks (FNN) are simpler to implement and perform best across 6 predictive tasks (ROA, ROE, Net Margin, Op Margin, Cash Ratio and Op Cash Generation); although Bayesian Networks (BN) can outperform FNN under very specific conditions. BNs have the additional benefit of providing a probability density function in addition to the predicted (expected) value. The study findings have significant potential helping CFOs and CEOs assess risks of financial underperformance to steer companies in more profitable directions; supporting lenders in better assessing the condition of a company and providing investors with tools to dissect financial statements of public companies more accurately.

Human knowledge is subject to uncertainties, imprecision, incompleteness and inconsistencies. Moreover, the meaning of many everyday terms is dependent on the context. That poses a huge challenge for the Semantic Web. This paper introduces work on an intuitive notation and model for defeasible reasoning with imperfect knowledge, and relates it to previous work on argumentation theory. PKN is to N3 as defeasible reasoning is to deductive logic. Further work is needed on an intuitive syntax for describing reasoning strategies and tactics in declarative terms, drawing upon the AIF ontology for inspiration. The paper closes with observations on symbolic approaches in the era of large language models.

Choosing an appropriate representation of the environment for the underlying decision-making process of the RL agent is not always straightforward. The state representation should be inclusive enough to allow the agent to informatively decide on its actions and compact enough to increase sample efficiency for policy training. Given this outlook, this work examines the effect of various state representations in incentivizing the agent to solve a specific robotic task: antipodal and planar object grasping. A continuum of state representation abstractions is defined, starting from a model-based approach with complete system knowledge, through hand-crafted numerical, to image-based representations with decreasing level of induced task-specific knowledge. We examine the effects of each representation in the ability of the agent to solve the task in simulation and the transferability of the learned policy to the real robot. The results show that RL agents using numerical states can perform on par with non-learning baselines. Furthermore, we find that agents using image-based representations from pre-trained environment embedding vectors perform better than end-to-end trained agents, and hypothesize that task-specific knowledge is necessary for achieving convergence and high success rates in robot control.