Vine copula models have become highly popular and practical tools for modelling multivariate probability distributions due to their flexibility in modelling different kinds of dependences between the random variables involved. However, their flexibility comes with the drawback of a high-dimensional parameter space. To tackle this problem, truncated vine copulas were introduced by Kurowicka (2010) (Gaussian case) and Brechmann and Czado (2013) (general case). Truncated vine copulas contain conditionally independent pair copulas after the truncation level. So far, in the general case, truncated vine constructing algorithms started from the lowest tree in order to encode the largest dependences in the lower trees. The novelty of this paper starts from the observation that a truncated vine is determined by the first tree after the truncation level (see Kovács and Szántai (2017)). This paper introduces a new score for fitting truncated vines to given data, called the Weight of the truncated vine. Then we propose a completely new methodology for constructing truncated vines. We prove theorems which motivate this new approach. While earlier algorithms did not use conditional independences, we give algorithms for constructing and encoding truncated vines which do exploit them. Finally, we illustrate the algorithms on real datasets and compare the results with well-known methods included in R packages. Our method generally compare favorably to previously known methods.

In this paper we introduce the so-called Generalized Naive Bayes structure as an extension of the Naive Bayes structure. We give a new greedy algorithm that finds a good fitting Generalized Naive Bayes (GNB) probability distribution. We prove that this fits the data at least as well as the probability distribution determined by the classical Naive Bayes (NB). Then, under a not very restrictive condition, we give a second algorithm for which we can prove that it finds the optimal GNB probability distribution, i.e. best fitting structure in the sense of KL divergence. Both algorithms are constructed to maximize the information content and aim to minimize redundancy. Based on these algorithms, new methods for feature selection are introduced. We discuss the similarities and differences to other related algorithms in terms of structure, methodology, and complexity. Experimental results show, that the algorithms introduced outperform the related algorithms in many cases.

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.

Vine copulas can efficiently model multivariate probability distributions. This paper focuses on a more thorough understanding of their structures, since in the literature, vine copula representations are often ambiguous. The graph representations include the original, cherry and chordal graph sequence structures, which we show equivalence between. Importantly we also show a new result, namely that when a perfect elimination ordering of a vine structure is given, then it can always be uniquely represented with a matrix. O. M. N\'apoles has shown a way to represent vines in a matrix, and we algorithmify this previous approach, while also showing a new method for constructing such a matrix, through cherry tree sequences. We also calculate the runtime of these algorithms. Lastly, we prove that these two matrix-building algorithms are equivalent if the same perfect elimination ordering is being used.

I don't think sci-fi saw this coming. For so long, futuristic books and films have promised us robots like C-3PO that translate alien languages and assist us in hijinks. Or ones like Rosie that clean our houses. Or, on the other end of the spectrum, robots that level our houses and destroy humanity. The reality of modern robotics couldn't be more different.

AKITA – A northeastern city renowned for its cherry blossom is teaming up with a software developer to film the scene using drones, in a bid to boost tourism and assist in the upkeep of the trees. Senboku in Akita Prefecture, which boasts the cherry trees in its Kakunodate district, is one of the several areas nationwide designated as special zones where the use of drones for public purposes is allowed. This is intended to encourage economic revitalization. Infoteria Corp. will provide the technology and donate 1 million toward the care of the trees. Technicians flew a drone along a river bank in the city Wednesday, shooting vistas of early blossom.