Mansfield Merriman's 1877 paper traces the historical development of the Method of Least Squares, crediting Legendre (1805) for introducing the method, Adrain (1808) for the first formal probabilistic proof, and Gauss (1809) for linking it to the normal distribution.

He evaluates multiple proofs, including Laplace’s (1810) general probability-based derivation, and highlights later refinements by various mathematicians.

The paper underscores the method’s fundamental role in statistical estimation, probability theory, and error minimization, solidifying its place in scientific and engineering applications.

In this episode, we review Cauchy’s 1847 paper, which introduced an iterative method for solving simultaneous equations by minimizing a function using its partial derivatives. Instead of elimination, he proposed progressively reducing the function’s value through small updates, forming an early version of gradient descent. His approach allowed systematic approximation of solutions, influencing numerical optimization.This work laid the foundation for machine learning and AI, where gradient-based methods are essential. Modern stochastic gradient descent (SGD) and deep learning training algorithms follow Cauchy’s principle of stepwise minimization. His ideas power optimization in neural networks, making AI training efficient and scalable.

At the 24th episode we go over the paper titled:

This process repeats until convergence, ensuring a monotonic increase in the likelihood function.

Its ability to handle incomplete data makes it invaluable for problems in clustering, anomaly detection, and probabilistic modeling. The algorithm guarantees stable convergence, though it may reach local maxima, depending on initialization.

It serves as a foundation for probabilistic graphical models like Bayesian networks and Variational Inference, which power applications such as chatbots, recommendation systems, and deep generative models.

Its iterative nature has also inspired optimization techniques in deep learning, such as Expectation-Maximization inspired variational autoencoders (VAEs), demonstrating its ongoing influence in AI advancements.

In the 23rd episode we review the The 1953 paper Metropolis, Nicholas, et al. "Equation of state calculations by fast computing machines."

The journal of chemical physics 21.6 (1953): 1087-1092 which introduced the Monte Carlo method for simulating molecular systems, particularly focusing on two-dimensional rigid-sphere models.

The study used random sampling to compute equilibrium properties like pressure and density, demonstrating a feasible approach for solving analytically intractable statistical mechanics problems.

By validating the Monte Carlo technique against free volume theories and virial expansions, the study showcased its accuracy and set the stage for MCMC as a powerful tool for exploring complex probability distributions.

These techniques enable applications like generative models, reinforcement learning, and neural network training, supporting the development of robust, data-driven AI systems.

Youtube: https://www.youtube.com/watch?v=gWOawt7hc88&t

In this episode with go over the Kullback-Leibler (KL) divergence paper, "On Information and Sufficiency" (1951).

This concept, rooted in Shannon's information theory (which we reviewed in previous episodes), became fundamental in hypothesis testing, model evaluation, and statistical inference.

It measures distributional differences, enabling optimization in clustering, density estimation, and natural language processing.

In the 18th episode we go over the original k-nearest neighbors algorithm;

The work is a precursor to the modern 𝑘 k-Nearest Neighbors (KNN) algorithm and established nonparametric approaches as viable alternatives to parametric methods.

This paper's impact on data science is significant, introducing concepts like neighborhood-based learning and flexible discrimination.

These ideas underpin algorithms widely used today in healthcare, finance, and artificial intelligence, where robust and interpretable models are critical.