The frequency of the preferred order for a noun phrase formed by demonstrative, numeral, adjective and noun has received significant attention over the last two decades. We investigate the actual distribution of the preferred 24 possible orders. There is no consensus on whether it can be well-fitted by an exponential or a power law distribution. We find that an exponential distribution is a much better model. This finding and other circumstances where an exponential-like distribution is found challenge the view that power-law distributions, e.g., Zipf's law for word frequencies, are inevitable. We also investigate which of two exponential distributions gives a better fit: an exponential model where the 24 orders have non-zero probability or an exponential model where the number of orders that can have non-zero probability is variable. When parsimony and generalizability are prioritized, we find strong support for the exponential model where all 24 orders have non-zero probability. This finding suggests that there is no hard constraint on word order variation and then unattested orders merely result from undersampling, consistently with Cysouw's view.

``When a measure becomes a target, it ceases to be a good measure'', this adage is known as {\it Goodhart's law}. In this paper, we investigate formally this law and prove that it critically depends on the tail distribution of the discrepancy between the true goal and the measure that is optimized. Discrepancies with long-tail distributions favor a Goodhart's law, that is, the optimization of the measure can have a counter-productive effect on the goal. We provide a formal setting to assess Goodhart's law by studying the asymptotic behavior of the correlation between the goal and the measure, as the measure is optimized. Moreover, we introduce a distinction between a {\it weak} Goodhart's law, when over-optimizing the metric is useless for the true goal, and a {\it strong} Goodhart's law, when over-optimizing the metric is harmful for the true goal. A distinction which we prove to depend on the tail distribution. We stress the implications of this result to large-scale decision making and policies that are (and have to be) based on metrics, and propose numerous research directions to better assess the safety of such policies in general, and to the particularly concerning case where these policies are automated with algorithms.

The study of word co-occurrence networks has attracted the attention of researchers due to their potential significance as well as applications. Understanding the structure of word co-occurrence networks is therefore important to fully realize their significance and usages. In past studies, word co-occurrence networks built on well-formed texts have been found to possess certain characteristics, including being small-world, following a two-regime power law distribution, and being generally disassortative. On the flip side, past studies have found that word co-occurrence networks built from ill-formed texts such as microblog posts may behave differently from those built from well-formed documents. While both kinds of word co-occurrence networks are small-world and disassortative, word co-occurrence networks built from ill-formed texts are scale-free and follow the power law distribution instead of the two-regime power law distribution. However, since past studies on the behavior of word co-occurrence networks built from ill-formed texts only investigated English, the universality of such characteristics remains to be seen among different languages. In addition, it is yet to be investigated whether there could be possible similitude/differences between word co-occurrence networks and other potentially comparable networks. This study therefore investigates and compares the structure of word co-occurrence networks and word similarity networks based on Taiwan Mandarin ill-formed internet forum posts and compare them with those built with well-formed judicial judgments, and seeks to find out whether the three aforementioned properties (scale-free, small-world, and disassortative) for ill-formed and well-formed texts are universal among different languages and between word co-occurrence and word similarity networks.

Graph Neural Networks (GNNs) excel in diverse tasks, yet their applications in high-stakes domains are often hampered by unreliable predictions. Although numerous uncertainty quantification methods have been proposed to address this limitation, they often lack \textit{rigorous} uncertainty estimates. This work makes the first attempt to introduce a distribution-free and model-agnostic uncertainty quantification approach to construct a predictive interval with a statistical guarantee for GNN-based link prediction. We term it as \textit{conformalized link prediction.} Our approach builds upon conformal prediction (CP), a framework that promises to construct statistically robust prediction sets or intervals. We first theoretically and empirically establish a permutation invariance condition for the application of CP in link prediction tasks, along with an exact test-time coverage. Leveraging the important structural information in graphs, we then identify a novel and crucial connection between a graph's adherence to the power law distribution and the efficiency of CP. This insight leads to the development of a simple yet effective sampling-based method to align the graph structure with a power law distribution prior to the standard CP procedure. Extensive experiments demonstrate that for conformalized link prediction, our approach achieves the desired marginal coverage while significantly improving the efficiency of CP compared to baseline methods.

The COVID-19 epidemic has had a great impact on social media conversation, especially on sites like Twitter, which has emerged as a hub for public reaction and information sharing. This paper deals by analyzing a vast dataset of Twitter messages related to this disease, starting from January 2020. Two approaches were used: a statistical analysis of word frequencies and a sentiment analysis to gauge user attitudes. Word frequencies are modeled using unigrams, bigrams, and trigrams, with power law distribution as the fitting model. The validity of the model is confirmed through metrics like Sum of Squared Errors (SSE), R-squared ($R^2$), and Root Mean Squared Error (RMSE). High $R^2$ and low SSE/RMSE values indicate a good fit for the model. Sentiment analysis is conducted to understand the general emotional tone of Twitter users messages. The results reveal that a majority of tweets exhibit neutral sentiment polarity, with only 2.57\% expressing negative polarity.

It turns out that some empirical facts in Big Data are the effects of properties of large numbers. Zipf's law 'noise' is an example of such an artefact. We expose several properties of the power law distributions and of similar distribution that occur when the population is finite and the rank and counts of elements in the population are natural numbers. We are particularly concerned with the low-rank end of the graph of the law, the potential of noise in the law, and with the approximation of the number of types of objects at various ranks. Approximations instead of exact solutions are the center of attention. Consequences in the interpretation of Zipf's law are discussed.

Learning to rank is an effective recommendation approach since its introduction around 2010. Famous algorithms such as Bayesian Personalized Ranking and Collaborative Less is More Filtering have left deep impact in both academia and industry. However, most learning to rank approaches focus on improving technical accuracy metrics such as AUC, MRR and NDCG. Other evaluation metrics of recommender systems like fairness have been largely overlooked until in recent years. In this paper, we propose a new learning to rank algorithm named Pareto Pairwise Ranking. We are inspired by the idea of Bayesian Personalized Ranking and power law distribution. We show that our algorithm is competitive with other algorithms when evaluated on technical accuracy metrics. What is more important, in our experiment section we demonstrate that Pareto Pairwise Ranking is the most fair algorithm in comparison with 9 other contemporary algorithms.

I use Kaggle public dataset "Insurance Premium Prediction" in this exercise. The data includes independent variables: age, sex, bmi, children, smoker, region, and target variable -- expenses. Firstly, let's load the data and have a preliminary examination of the data using df.info() EDA is essential to both investigate the data quality and reveal hidden correlations among variables. In this exercise, I cover three techniques relevant to linear regression.

Although recommender systems have received great success, We categorize recommender systems as shallow it is well known for highly skewed datasets, models and deep models. The first class engineers and researchers need to adjust their incorporates shallow machine learning technologies methods to tackle the specific problem to yield good such as matrix factorization and learning to rank, results. Inability to deal with highly skewed dataset while the second class are deep learning models like usually generates hard computational problems for Wide and Deep [6]. Although a bit of out-of-dated, big data clusters and unsatisfactory results for shallow models are still widely used in small customers. In this paper, we propose a new companies and projects where agility, usability and algorithm solving the problem in the framework of matrix factorization. We model the data skewness efficiency far outweighs boost of performance which factors in the theoretic modeling of the approach is only economically visible for huge datasets. It is with easy to interpret and easy to implement well known since the invention of the first shallow formulas. We prove in experiments our method model, that data skewness and sparsity poses generates comparably favorite results with popular serious challenges for recommender system recommender system algorithms such as Learning performance. The setbacks are two folds: data to Rank, Alternating Least Squares and Deep Matrix skewness causes problems that need special Factorization.

This paper presents a discovery that the length of the entities follows a family of scale-free power law distributions. The concept of entity here broadly includes the named entity, entity mention, time expression, and domain-specific entity that are well investigated in natural language processing and related areas. The power law distributions in entity length have well-defined means and finite variances and possess the scale-free property. We explain the phenomenon of power laws in entity length by the principle of least effort in communication and the preferential mechanism.