Following from Section 2.1, we review several relaxations of faithfulness, which give rise to different We provide further simulation results for the analysis of the SUCF assumption in Section 3.2 . Different number of nodes and expected degrees are considered. X-axes are visualized in log scale. With the above lemmas, we now provide the proofs of the main results. We proceed by contraposition in both parts of the proof.

We present a method for learning Bayesian networks from data sets containing thousands of variables without the need for structure constraints. Our approach is made of two parts. The first is a novel algorithm that effectively explores the space of possible parent sets of a node. It guides the exploration towards the most promising parent sets on the basis of an approximated score function that is computed in constant time. The second part is an improvement of an existing ordering-based algorithm for structure optimization. The new algorithm provably achieves a higher score compared to its original formulation. Our novel approach consistently outperforms the state of the art on very large data sets.

GNN and assign important scores on explained features. Here, we consider the simplest gradient-based explanation method in which the score of each feature is associated with the gradient of the GNN's loss function with respect to that feature. The proof that this explanation method falls into the class of additive feature attribution methods is quite straight-forward. S is a good explanation for the target prediction. This experiment setup is the same as that in experiment of Figure 1.

Additional Feedback: - At equation (2) M should be defined.

Existing score-based methods for directed acyclic graph (DAG) learning from observational data struggle to recover the causal graph accurately and sample-efficiently. To overcome this, in this study, we propose DrBO (DAG recovery via Bayesian Optimization)-a novel DAG learning framework leveraging Bayesian optimization (BO) to find high-scoring DAGs. We show that, by sophisticatedly choosing the promising DAGs to explore, we can find higher-scoring ones much more efficiently. To address the scalability issues of conventional BO in DAG learning, we replace Gaussian Processes commonly employed in BO with dropout neural networks, trained in a continual manner, which allows for (i) flexibly modeling the DAG scores without overfitting, (ii) incorporation of uncertainty into the estimated scores, and (iii) scaling with the number of evaluations. As a result, DrBO is computationally efficient and can find the accurate DAG in fewer trials and less time than existing state-of-the-art methods. This is demonstrated through an extensive set of empirical evaluations on many challenging settings with both synthetic and real data. Our implementation is available at https://github.com/baosws/DrBO.

Causal Bayesian networks (CBN) are popular graphical probabilistic models that encode causal relations among variables. Learning their graphical structure from observational data has received a lot of attention in the literature. When there exists no latent (unobserved) confounder, i.e., no unobserved direct common cause of some observed variables, learning algorithms can be divided essentially into two classes: constraint-based and score-based approaches. The latter are often thought to be more robust than the former and to produce better results. However, to the best of our knowledge, when variables are discrete, no score-based algorithm is capable of dealing with latent confounders. This paper introduces the first fully score-based structure learning algorithm searching the space of DAGs (directed acyclic graphs) that is capable of identifying the presence of some latent confounders. It is justified mathematically and experiments highlight its effectiveness.

Sepsis is a life-threatening and serious global health issue. This study combines knowledge with available hospital data to investigate the potential causes of Sepsis that can be affected by policy decisions. We investigate the underlying causal structure of this problem by combining clinical expertise with score-based, constraint-based, and hybrid structure learning algorithms. A novel approach to model averaging and knowledge-based constraints was implemented to arrive at a consensus structure for causal inference. The structure learning process highlighted the importance of exploring data-driven approaches alongside clinical expertise. This includes discovering unexpected, although reasonable, relationships from a clinical perspective. Hypothetical interventions on Chronic Obstructive Pulmonary Disease, Alcohol dependence, and Diabetes suggest that the presence of any of these risk factors in patients increases the likelihood of Sepsis. This finding, alongside measuring the effect of these risk factors on Sepsis, has potential policy implications. Recognising the importance of prediction in improving Sepsis related health outcomes, the model built is also assessed in its ability to predict Sepsis. The predictions generated by the consensus model were assessed for their accuracy, sensitivity, and specificity. These three indicators all had results around 70%, and the AUC was 80%, which means the causal structure of the model is reasonably accurate given that the models were trained on data available for commissioning purposes only.

We present a method for learning Bayesian networks from data sets containing thousands of variables without the need for structure constraints. Our approach is made of two parts. The first is a novel algorithm that effectively explores the space of possible parent sets of a node. It guides the exploration towards the most promising parent sets on the basis of an approximated score function that is computed in constant time. The second part is an improvement of an existing ordering-based algorithm for structure optimization. The new algorithm provably achieves a higher score compared to its original formulation. Our novel approach consistently outperforms the state of the art on very large data sets.

In this study, we examine the fluctuation in the value of the Great Britain Pound (GBP). We focus particularly on its relationship with the United States Dollar (USD) and the Euro (EUR) currency pairs. Utilizing data from June 15, 2018, to June 15, 2023, we apply various mathematical models to assess their effectiveness in predicting the 20-day variation in the pairs' daily returns. Our analysis involves the implementation of Exponentially Weighted Moving Average (EWMA), Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, and Implied Volatility (IV) models. To evaluate their performance, we compare the accuracy of their predictions using Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) metrics. We delve into the intricacies of GARCH models, examining their statistical characteristics when applied to the provided dataset. Our findings suggest the existence of asymmetric returns in the EUR/GBP pair, while such evidence is inconclusive for the GBP/USD pair. Additionally, we observe that GARCH-type models better fit the data when assuming residuals follow a standard t-distribution rather than a standard normal distribution. Furthermore, we investigate the efficacy of different forecasting techniques within GARCH-type models. Comparing rolling window forecasts to expanding window forecasts, we find no definitive superiority in either approach across the tested scenarios. Our experiments reveal that for the GBP/USD pair, the most accurate volatility forecasts stem from the utilization of GARCH models employing a rolling window methodology. Conversely, for the EUR/GBP pair, optimal forecasts are derived from GARCH models and Ordinary Least Squares (OLS) models incorporating the annualized implied volatility of the exchange rate as an independent variable. Keywords: Euro, Exponentially Weighted Moving Average, Generalized Autoregressive Conditional Heteroskedasticity, Great Britain Pound, Implied Volatility, United States Dollar 1. Introduction As defined by the [1], a volatility of a financial asset is simply how much its price changes over time.