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arXiv2024

Probability Proofs for Stirling (and More): the Ubiquitous Role of $\mathbf{\sqrt{2π}}$

The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and also to approximations of marginal distributions in Bayesian setups. Certain formulae emerge by working through particular instances, some independently verifiable but others perhaps not. A particular case yielding new formulae is that of summing independent uniforms, related to the Irwin--Hall distribution. Yet further proofs of the Stirling flow from examining aspects of limiting normality of the sample median of uniforms, and from these again we find a proof for the Wallis product formula for $π$.

math.PR↗
arXiv2024

Ethical Statistical Practice and Ethical AI

Artificial Intelligence (AI) is a field that utilizes computing and often, data and statistics, intensively together to solve problems or make predictions. AI has been evolving with literally unbelievable speed over the past few years, and this has led to an increase in social, cultural, industrial, scientific, and governmental concerns about the ethical development and use of AI systems worldwide. The ASA has issued a statement on ethical statistical practice and AI (ASA, 2024), which echoes similar statements from other groups. Here we discuss the support for ethical statistical practice and ethical AI that has been established in long-standing human rights law and ethical practice standards for computing and statistics. There are multiple sources of support for ethical statistical practice and ethical AI deriving from these source documents, which are critical for strengthening the operationalization of the "Statement on Ethical AI for Statistics Practitioners". These resources are explicated for interested readers to utilize to guide their development and use of AI in, and through, their statistical practice.

stat.OT↗
arXiv2025

Impossibility results for equating the Youden Index with average scoring rules and Tjur $R^2$-like metrics

We consider the Youden index fas well as measures evaluating predicted probabilities for the maximum-likelihood estimate of a logistic regression model with predictor the classifier. We give impossibility results showing that the Youden index can not equal any average of a real scoring rule nor any metric averaging over binary outcomes (0s and 1s) for any continuous real-valued scoring rule. This shows the obstructions of such potential equivalences and highlights the distinct roles these metrics play in diagnostic assessment.

stat.OT↗
arXiv2025

Frequentist Statistics as Internalist Reliabilism

There has long been an impression that reliabilism implies externalism and that frequentist statistics, due to its reliabilist nature, is inherently externalist. I argue, however, that frequentist statistics can plausibly be understood as a form of internalist reliabilism -- internalist in the conventional sense, yet reliabilist in certain unconventional and intriguing ways. Crucially, in developing the thesis that reliabilism does not imply externalism, my aim is not to stretch the meaning of `reliabilism' merely to sever the implication. Instead, it is to gain a deeper understanding of frequentist statistics, which stands as one of the most sustained attempts by scientists to develop an epistemology for their own use.

stat.OT↗
arXiv2025

Good intentions, unintended consequences: exploring forecasting harms

Organizations worldwide that rely on data-driven approaches regularly employ forecasting methods to enhance their planning and decision-making processes. While extensive research has examined the harms associated with traditional machine learning applications, relatively little attention has been given to the ethical implications of time series forecasting. However, forecasting presents distinct ethical challenges due to its diverse organizational applications, varied objectives, and unique data processing, model development, and evaluation workflows. These distinctions complicate the direct application of existing machine learning harm taxonomies to common forecasting scenarios. To address this gap, we conduct multiple interviews with industry experts and academic researchers, systematically identifying and analyzing underexplored domains, use cases, and potential risks associated with forecasting. Our objective is to develop a novel taxonomy of forecasting-specific harms. Drawing inspiration from Microsoft Azure taxonomy for responsible innovation, we integrate a human-led inductive coding approach with AI-driven analysis to extract key categories of harm in forecasting. This taxonomy aims to support researchers and practitioners by fostering ethical reflection on their decision-making throughout the forecasting process. Additionally, we seek to establish a research agenda focused on identifying measures to mitigate potential harms in forecasting. By highlighting unique risks within forecasting, our work contributes to the broader discourse on machine learning ethics.

stat.OT↗
arXiv2025

Bhirkuti's Test of Bias Acceptance (BTBA): Examining Its Performance in Psychometric Simulations

We introduce Bhirkuti's Test of Bias Acceptance (BTBA), a standardized framework for evaluating estimator bias in Monte Carlo simulation studies. BTBA uses a simulation-specific standardized score (Z*) and a decision matrix to assess bias acceptability based on the mean and variance of Z* distributions. Under ideal conditions, Z* values should approximate a standard normal distribution (Z-distribution) with a mean near zero and variance near one in the context of simulation research. Systematic deviations from these patterns such as shifted means or inflated variances indicate bias or estimator instability in simulation-based research. BTBA visualizes these patterns using ridgeline density plots, which reveal distributional features such as central tendency, spread, skewness, and outliers. Demonstrated in a latent growth modeling context, BTBA offers a reproducible and interpretable method for diagnosing bias across varying simulation conditions. By addressing key limitations of traditional relative bias (RB) metrics, BTBA provides a theoretically grounded, distribution-aware, transparent, and replicable alternative for evaluating estimator quality, particularly in psychometric modeling, structural equation modeling, and missing data research. Through this framework, we aim to enhance methodological decision-making by integrating statistical reasoning with comprehensive visualization techniques.

stat.ME↗
arXiv2024

Contrasting the optimal resource allocation to cybersecurity and cyber insurance using prospect theory versus expected utility theory

Protecting against cyber-threats is vital for every organization and can be done by investing in cybersecurity controls and purchasing cyber insurance. However, these are interlinked since insurance premiums could be reduced by investing more in cybersecurity controls. The expected utility theory and the prospect theory are two alternative theories explaining decision-making under risk and uncertainty, which can inform strategies for optimizing resource allocation. While the former is considered a rational approach, research has shown that most people make decisions consistent with the latter, including on insurance uptakes. We compare and contrast these two approaches to provide important insights into how the two approaches could lead to different optimal allocations resulting in differing risk exposure as well as financial costs. We introduce the concept of a risk curve and show that identifying the nature of the risk curve is a key step in deriving the optimal resource allocation.

econ.EM↗
arXiv2024

Examining Multimodal Gender and Content Bias in ChatGPT-4o

This study investigates ChatGPT-4o's multimodal content generation, highlighting significant disparities in its treatment of sexual content and nudity versus violent and drug-related themes. Detailed analysis reveals that ChatGPT-4o consistently censors sexual content and nudity, while showing leniency towards violence and drug use. Moreover, a pronounced gender bias emerges, with female-specific content facing stricter regulation compared to male-specific content. This disparity likely stems from media scrutiny and public backlash over past AI controversies, prompting tech companies to impose stringent guidelines on sensitive issues to protect their reputations. Our findings emphasize the urgent need for AI systems to uphold genuine ethical standards and accountability, transcending mere political correctness. This research contributes to the understanding of biases in AI-driven language and multimodal models, calling for more balanced and ethical content moderation practices.

cs.CY↗
arXiv2024

Noncommutative Model Selection for Data Clustering and Dimension Reduction Using Relative von Neumann Entropy

We propose a pair of completely data-driven algorithms for unsupervised classification and dimension reduction, and we empirically study their performance on a number of data sets, both simulated data in three-dimensions and images from the COIL-20 data set. The algorithms take as input a set of points sampled from a uniform distribution supported on a metric space, the latter embedded in an ambient metric space, and they output a clustering or reduction of dimension of the data. They work by constructing a natural family of graphs from the data and selecting the graph which maximizes the relative von Neumann entropy of certain normalized heat operators constructed from the graphs. Once the appropriate graph is selected, the eigenvectors of the graph Laplacian may be used to reduce the dimension of the data, and clusters in the data may be identified with the kernel of the associated graph Laplacian. Notably, these algorithms do not require information about the size of a neighborhood or the desired number of clusters as input, in contrast to popular algorithms such as $k$-means, and even more modern spectral methods such as Laplacian eigenmaps, among others. In our computational experiments, our clustering algorithm outperforms $k$-means clustering on data sets with non-trivial geometry and topology, in particular data whose clusters are not concentrated around a specific point, and our dimension reduction algorithm is shown to work well in several simple examples.

stat.ML↗
arXiv2024

Unified Inductive Logic: From Formal Learning to Statistical Inference to Supervised Learning

While the traditional conception of inductive logic is Carnapian, I develop a Peircean alternative and use it to unify formal learning theory, statistics, and a significant part of machine learning: supervised learning. Some crucial standards for evaluating non-deductive inferences have been assumed separately in those areas, but can actually be justified by a unifying principle.

stat.OT↗
arXiv2024

A dynamical measure of algorithmically infused visibility

This work focuses on the nature of visibility in societies where the behaviours of humans and algorithms influence each other - termed algorithmically infused societies. We propose a quantitative measure of visibility, with implications and applications to an array of disciplines including communication studies, political science, marketing, technology design, and social media analytics. The measure captures the basic characteristics of the visibility of a given topic, in algorithm/AI-mediated communication/social media settings. Topics, when trending, are ranked against each other, and the proposed measure combines the following two attributes of a topic: (i) the amount of time a topic spends at different ranks, and (ii) the different ranks the topic attains. The proposed measure incorporates a tunable parameter, termed the discrimination level, whose value determines the relative weights of the two attributes that contribute to visibility. Analysis of a large-scale, real-time dataset of trending topics, from one of the largest social media platforms, demonstrates that the proposed measure can explain a large share of the variability of the accumulated views of a topic.

cs.SI↗
arXiv2025

My Statistics is Better than Yours

Statistical schools-such as Bayesianism and Frequentism-are often presented as competing frameworks, each claiming technical rigour and superiority. Frequentism emphasizes objective inferences through repeated sampling, while Bayesianism incorporates prior beliefs and updates them with new evidence. Despite their strengths, neither school proves universally applicable, and the pursuit of a single "correct" statistical framework is ultimately misguided. Instead, this essay advocates for a context-dependent approach to statistical norms, drawing on Douglas (2004)'s concept of "operational objectivity". The idea is that by aligning the context of the research question with the value judgments inherent to its field, a certain statistical paradigm is warranted. This essay explores the decision-theoretic foundations of Bayesianism, examines its descriptive limitations as highlighted by the Ellsberg paradox, and addresses the challenges of comparing different normative systems.

stat.OT↗
arXiv2024

Scientific Realism vs. Anti-Realism: Toward a Common Ground

The debate between scientific realism and anti-realism remains at a stalemate, making reconciliation seem hopeless. Yet, important work remains: exploring a common ground, even if only to uncover deeper points of disagreement and, ideally, to benefit both sides of the debate. I propose such a common ground. Specifically, many anti-realists, such as instrumentalists, have yet to seriously engage with Sober's call to justify their preferred version of Ockham's razor through a positive account. Meanwhile, realists face a similar challenge: providing a non-circular explanation of how their version of Ockham's razor connects to truth. The common ground I propose addresses these challenges for both sides; the key is to leverage the idea that everyone values some truths and to draw on insights from scientific fields that study scientific inference -- namely, statistics and machine learning. This common ground also isolates a distinctively epistemic root of the irreconcilability in the realism debate.

stat.OT↗
arXiv2024

Deep Learning-based Approaches for State Space Models: A Selective Review

State-space models (SSMs) offer a powerful framework for dynamical system analysis, wherein the temporal dynamics of the system are assumed to be captured through the evolution of the latent states, which govern the values of the observations. This paper provides a selective review of recent advancements in deep neural network-based approaches for SSMs, and presents a unified perspective for discrete time deep state space models and continuous time ones such as latent neural Ordinary Differential and Stochastic Differential Equations. It starts with an overview of the classical maximum likelihood based approach for learning SSMs, reviews variational autoencoder as a general learning pipeline for neural network-based approaches in the presence of latent variables, and discusses in detail representative deep learning models that fall under the SSM framework. Very recent developments, where SSMs are used as standalone architectural modules for improving efficiency in sequence modeling, are also examined. Finally, examples involving mixed frequency and irregularly-spaced time series data are presented to demonstrate the advantage of SSMs in these settings.

stat.ML↗
arXiv2024

Equity in the Use of ChatGPT for the Classroom: A Comparison of the Accuracy and Precision of ChatGPT 3.5 vs. ChatGPT4 with Respect to Statistics and Data Science Exams

A college education historically has been seen as method of moving upward with regards to income brackets and social status. Indeed, many colleges recognize this connection and seek to enroll talented low income students. While these students might have their education, books, room, and board paid; there are other items that they might be expected to use that are not part of most college scholarship packages. One of those items that has recently surfaced is access to generative AI platforms. The most popular of these platforms is ChatGPT, and it has a paid version (ChatGPT4) and a free version (ChatGPT3.5). We seek to explore differences in the free and paid versions in the context of homework questions and data analyses as might be seen in a typical introductory statistics course. We determine the extent to which students who cannot afford newer and faster versions of generative AI programs would be disadvantaged in terms of writing such projects and learning these methods.

stat.OT↗
arXiv2024

A Comprehensive Guide to Item Recovery Using the Multidimensional Graded Response Model in R

The purpose of this study is to provide a step-by-step demonstration of item recovery for the Multidimensional Graded Response Model (MGRM) in R. Within this scope, a sample simulation design was constructed where the test lengths were set to 20 and 40, the interdimensional correlations were varied as 0.3 and 0.7, and the sample size was fixed at 2000. Parameter estimates were derived from the generated datasets for the 3-dimensional GRM, and bias and Root Mean Square Error (RMSE) values were calculated and visualized. In line with the aim of the study, R codes for all these steps were presented along with detailed explanations, enabling researchers to replicate and adapt the procedures for their own analyses. This study is expected to contribute to the literature by serving as a practical guide for implementing item recovery in the MGRM. In addition, the methods presented, including data generation, parameter estimation, and result visualization, are anticipated to benefit researchers even if they are not directly engaged in item recovery.

cs.CY↗
arXiv2024

Towards Strong AI: Transformational Beliefs and Scientific Creativity

Strong artificial intelligence (AI) is envisioned to possess general cognitive abilities and scientific creativity comparable to human intelligence, encompassing both knowledge acquisition and problem-solving. While remarkable progress has been made in weak AI, the realization of strong AI remains a topic of intense debate and critical examination. In this paper, we explore pivotal innovations in the history of astronomy and physics, focusing on the discovery of Neptune and the concept of scientific revolutions as perceived by philosophers of science. Building on these insights, we introduce a simple theoretical and statistical framework of weak beliefs, termed the Transformational Belief (TB) framework, designed as a foundation for modeling scientific creativity. Through selected illustrative examples in statistical science, we demonstrate the TB framework's potential as a promising foundation for understanding, analyzing, and even fostering creativity -- paving the way toward the development of strong AI. We conclude with reflections on future research directions and potential advancements.

stat.OT↗