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Covariance estimation and selection for high-dimensional multivariate datasets is a fundamental problem in modern statistics. Gaussian directed acyclic graph (DAG) models are a popular class of models used for this purpose. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying variables. A variety of priors have been developed in recent years for Bayesian inference in DAG models, yet crucial convergence and sparsity selection properties for these models have not been thoroughly investigated. Most of these priors are adaptations/generalizations of the Wishart distribution in the DAG context. In this paper, we consider a flexible and general class of these 'DAG-Wishart' priors with multiple shape parameters. Under mild regularity assumptions, we establish strong graph selection consistency and establish posterior convergence rates for estimation when the number of variables p is allowed to grow at an appropriate sub-exponential rate with the sample size n.MSC 2010 subject classifications: Primary 62F15; secondary 62G20 P (|S ij − (Σ 0 ) ij | ≥ t) ≤ m 1 exp{−m 2 n(tǫ 0,n ) 2 }, |t| ≤ δ.

Objective To explore the influences of digital media use on the core symptoms, emotional state, life events, learning motivation, executive function (EF) and family environment of children and adolescents diagnosed with attention deficit hyperactivity disorder (ADHD) during the novel coronavirus disease 2019 (COVID-19) pandemic. Method A total of 192 participants aged 8–16 years who met the diagnostic criteria for ADHD were included in the study. Children scoring higher than predetermined cut-off point in self-rating questionnaires for problematic mobile phone use (SQPMPU) or Young’s internet addiction test (IAT), were defined as ADHD with problematic digital media use (PDMU), otherwise were defined as ADHD without PDMU. The differences between the two groups in ADHD symptoms, EF, anxiety and depression, stress from life events, learning motivation and family environment were compared respectively. Results When compared with ADHD group without PDMU, the group with PDMU showed significant worse symptoms of inattention, oppositional defiant, behavior and emotional problems by Swanson, Nolan, and Pelham Rating Scale (SNAP), more self-reported anxiety by screening child anxiety-related emotional disorders (SCARED) and depression by depression self-rating scale for children (DSRSC), more severe EF deficits by behavior rating scale of executive function (BRIEF), more stress from life events by adolescent self-rating life events checklist (ASLEC), lower learning motivation by students learning motivation scale (SLMS), and more impairment on cohesion by Chinese version of family environment scale (FES-CV). The ADHD with PDMU group spent significantly more time on both video game and social media with significantly less time spend on physical exercise as compared to the ADHD without PDMU group. Conclusion The ADHD children with PDMU suffered from more severe core symptoms, negative emotions, EF deficits, damage on family environment, pressure from life events, and a lower motivation to learn. Supervision of digital media usage, especially video game and social media, along with increased physical exercise, is essential to the management of core symptoms and associated problems encountered with ADHD.