Data Science Decoded

• Data Science #27 - The History of Least Squares (1877)

  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.

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• Data Science #26 - The First Gradient decent algorithm by Cauchy (1847)

  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.

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• Data Science #24 - The Expectation Maximization (EM) algorithm Paper review (1977)

  At the 24th episode we go over the paper titled: Dempster, Arthur P., Nan M. Laird, and Donald B. Rubin. "Maximum likelihood from incomplete data via the EM algorithm." Journal of the royal statistical society: series B (methodological) 39.1 (1977): 1-22. The Expectation-Maximization (EM) algorithm is an iterative method for finding Maximum Likelihood Estimates (MLEs) when data is incomplete or contains latent variables. It alternates between the E-step, where it computes the expected value of the missing data given current parameter estimates, and the M-step, where it maximizes the expected complete-data log-likelihood to update the parameters. This process repeats until convergence, ensuring a monotonic increase in the likelihood function. EM is widely used in statistics and machine learning, especially in Gaussian Mixture Models (GMMs), hidden Markov models (HMMs), and missing data imputation. 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. In modern data science and AI, EM has had a profound impact, enabling unsupervised learning in natural language processing (NLP), computer vision, and speech recognition. 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.

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• Data Science #23- The Markov Chain Monte Carl MCMC Paper review (1953)

  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. The work pioneered the Metropolis algorithm, a key development in what later became known as Markov Chain Monte Carlo (MCMC) methods. 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. This breakthrough has had a profound impact on modern AI and ML, where MCMC methods are now central to probabilistic modeling, Bayesian inference, and optimization. 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

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• Data Science #22 - The theory of dynamic programming, Paper review 1954

  We review Richard Bellman's "The Theory of Dynamic Programming" paper from 1954 which revolutionized how we approach complex decision-making problems through two key innovations. First, his Principle of Optimality established that optimal solutions have a recursive structure - each sub-decision must be optimal given the state resulting from previous decisions. Second, he introduced the concept of focusing on immediate states rather than complete historical sequences, providing a practical way to tackle what he termed the "curse of dimensionality." These foundational ideas directly shaped modern artificial intelligence, particularly reinforcement learning. The mathematical framework Bellman developed - breaking complex problems into smaller, manageable subproblems and making decisions based on current state - underpins many contemporary AI achievements, from game-playing agents like AlphaGo to autonomous systems and robotics. His work essentially created the theoretical backbone that enables modern AI systems to handle sequential decision-making under uncertainty. The principles established in this 1954 paper continue to influence how we design AI systems today, particularly in reinforcement learning and neural network architectures dealing with sequential decision problems.

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