Introduction to probability and statistics: a computational framework of randomness (2401.08622v2)

Abstract: This text presents an unified approach of probability and statistics in the pursuit of understanding and computation of randomness in engineering or physical or social system with prediction with generalizability. Starting from elementary probability and theory of distributions, the material progresses towards conceptual and advances in prediction and generalization in statistical models and large sample theory. We also pay special attention to unified derivation approach and one-shot proof of each and every probabilistic concept. Our presentation of intuitive and computation framework of conditional distribution and probability are strongly influenced by unified patterns of linear models for regression and for classification. The text ends with a future note on the unified approximation of the linear models, the generalized linear models and the discovery models to neural networks and a summarized ML system.

• The paper presents a unified computational framework that simplifies derivations in probability and statistics.
• The paper demonstrates one-shot proof techniques and in-depth exploration of conditional distributions with practical Python simulations.
• The paper bridges classical linear models with advanced machine learning paradigms, opening avenues for future integration.

An Analytical Overview of "Introduction to Probability and Statistics: A Computational Framework of Randomness"

The text, authored by Lakshman Mahto, proposes a unified approach to understanding and computing randomness in diverse systems, including engineering, physical, and social domains. Starting from elementary probability, the material advances through conceptual developments in statistical models and large sample theory, culminating in a comprehensive view of linear models and their generalizations to machine learning paradigms.

Core Aspects and Contributions

Unified Derivation and Proof Techniques

The paper places a strong emphasis on a unified derivation approach and one-shot proof techniques for probabilistic concepts. This methodological choice aims to create a cohesive narrative that eases comprehension and application across varied contexts. The paper illustrates this through examples and simulations, ultimately forming a computational framework that encapsulates probability, statistics, optimization, and algorithmic methods.

Framework for Conditional Distribution and Probability

A significant portion of the text is devoted to exploring conditional distributions and probabilities, influenced by linear models for regression and classification. The uniform approach to deriving statistical properties for both simple and complex models, including embedded multivariate and multiple multivariate linear models, signifies an important contribution to the literature.

Generalization to Machine Learning

In a forward-looking section, the text touches upon how traditional linear models, such as generalized linear models and discovery models, transition to neural networks and machine learning systems. This bridging between classical statistics and modern AI techniques is pivotal and suggests pathways for further integration and research.

Key Statistical Topics

Part I focuses on probability with topics such as distribution of random experiments, probabilistic modeling, joint distributions, and expectation and variance of random variables. Part II expands towards statistical inference, covering topics like point estimation, hypothesis testing, linear models for prediction, and experimental simulations, predominantly utilizing Python.

Probabilistic Inequalities and Asymptotic Theory

The paper discusses probabilistic inequalities as foundational to non-asymptotic large sample theory, offering practical bounds on probabilities without necessitating a full distributional specification. It reinforces understanding through discussion on Markov, Chebyshev, and Chernoff inequalities and emphasizes their applicability in bounding sample statistics.

Point Estimation Techniques

A variety of point estimation techniques are discussed, including maximum likelihood estimation, method of moments, and Bayesian approaches through maximum a posteriori (MAP) estimation. Each technique is elucidated with relevant examples from classic distributions like Bernoulli, Poisson, and Exponential, providing insights into their practical deployment.

Goodness-of-Fit and Statistical Sufficiency

Sufficiency and efficiency of estimators are explored through a detailed examination of statistical properties such as bias and variance. The criteria for minimal variance unbiased estimation (MVUE) and the use of sufficient statistics in ensuring reduction without loss of relevant information are noteworthy sections that reinforce the statistical rigor of the text.

Future Directions and Implications

The text concludes with a perspective on the future approximation methodologies that unify linear and generalized models with neural networks, suggesting a continuous evolution of statistical models in the field of machine learning and modern data science.

Numerical Simulations

The practical sections involving Python demonstrations serve to solidify theoretical understanding and application. This emphasis on computational exemplars aligns with the modern pedagogical approaches in statistics and probability education, encouraging a hands-on, experimental interaction with the subject matter.

In summary, Lakshman Mahto's comprehensive exploration of probability and statistics within a computational framework provides both theoretical insights and practical tools. It bridges classical methodologies with modern computational demands and opens avenues for integrating traditional statistical methods with contemporary machine learning models, underscoring its utility in a wide array of scientific and engineering applications.