Avsnitt
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In the 20th episode, we review the seminal paper by Rao which introduced the Cramer Rao bound:
Rao, Calyampudi Radakrishna (1945). "Information and the accuracy attainable in the estimation of statistical parameters". Bulletin of the Calcutta Mathematical Society. 37. Calcutta Mathematical Society: 81–89.
The Cramér-Rao Bound (CRB) sets a theoretical lower limit on the variance of any unbiased estimator for a parameter.
It is derived from the Fisher information, which quantifies how much the data tells us about the parameter. This bound provides a benchmark for assessing the precision of estimators and helps identify efficient estimators that achieve this minimum variance.
The CRB connects to key statistical concepts we have covered previously:
Consistency: Estimators approach the true parameter as the sample size grows, ensuring they become arbitrarily accurate in the limit. While consistency guarantees convergence, it does not necessarily imply the estimator achieves the CRB in finite samples.
Efficiency: An estimator is efficient if it reaches the CRB, minimizing variance while remaining unbiased. Efficiency represents the optimal use of data to achieve the smallest possible estimation error.
Sufficiency: Working with sufficient statistics ensures no loss of information about the parameter, increasing the chances of achieving the CRB. Additionally, the CRB relates to KL divergence, as Fisher information reflects the curvature of the likelihood function and the divergence between true and estimated distributions.
In modern DD and AI, the CRB plays a foundational role in uncertainty quantification, probabilistic modeling, and optimization. It informs the design of Bayesian inference systems, regularized estimators, and gradient-based methods like natural gradient descent. By highlighting the tradeoffs between bias, variance, and information, the CRB provides theoretical guidance for building efficient and robust machine learning models -
In this episode with go over the Kullback-Leibler (KL) divergence paper, "On Information and Sufficiency" (1951).It introduced a measure of the difference between two probability distributions, quantifying the cost of assuming one distribution when another is true.
This concept, rooted in Shannon's information theory (which we reviewed in previous episodes), became fundamental in hypothesis testing, model evaluation, and statistical inference.KL divergence has profoundly impacted data science and AI, forming the basis for techniques like maximum likelihood estimation, Bayesian inference, and generative models such as variational autoencoders (VAEs).
It measures distributional differences, enabling optimization in clustering, density estimation, and natural language processing.In AI, KL divergence ensures models generalize well by aligning training and real-world data distributions. Its role in probabilistic reasoning and adaptive decision-making bridges theoretical information theory and practical machine learning, cementing its relevance in modern technologies.
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Saknas det avsnitt?
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In the 18th episode we go over the original k-nearest neighbors algorithm; Fix, Evelyn; Hodges, Joseph L. (1951). Discriminatory Analysis. Nonparametric Discrimination: Consistency Properties USAF School of Aviation Medicine, Randolph Field, TexasThey introduces a nonparametric method for classifying a new observation 𝑧 z as belonging to one of two distributions, 𝐹 F or 𝐺 G, without assuming specific parametric forms. Using 𝑘 k-nearest neighbor density estimates, the paper implements a likelihood ratio test for classification and rigorously proves the method's consistency.
The work is a precursor to the modern 𝑘 k-Nearest Neighbors (KNN) algorithm and established nonparametric approaches as viable alternatives to parametric methods. Its focus on consistency and data-driven learning influenced many modern machine learning techniques, including kernel density estimation and decision trees.
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.
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We review the original Monte Carlo paper from 1949 by
Metropolis, Nicholas, and Stanislaw Ulam. "The monte carlo method." Journal of the American statistical association 44.247 (1949): 335-341.
The Monte Carlo method uses random sampling to approximate solutions for problems that are too complex for analytical methods, such as integration, optimization, and simulation.
Its power lies in leveraging randomness to solve high-dimensional and nonlinear problems, making it a fundamental tool in computational science.
In modern data science and AI, Monte Carlo drives key techniques like Bayesian inference (via MCMC) for probabilistic modeling, reinforcement learning for policy evaluation, and uncertainty quantification in predictions.
It is essential for handling intractable computations in machine learning and AI systems.
By combining scalability and flexibility, Monte Carlo methods enable breakthroughs in areas like natural language processing, computer vision, and autonomous systems.
Its ability to approximate solutions underpins advancements in probabilistic reasoning, decision-making, and optimization in the era of AI and big data. -
In the 16th episode we go over the seminal the 1952 paper titled:"A stochastic approximation method." The annals of mathematical statistics (1951): 400-407, by Robbins, Herbert and Sutton Monro.The paper introduced the stochastic approximation method, a groundbreaking iterative technique for finding the root of an unknown function using noisy observations.
This method enabled real-time, adaptive estimation without requiring the function’s explicit form, revolutionizing statistical practices in fields like bioassay and engineering.Robbins and Monro’s work laid the ideas behind stochastic gradient descent (SGD), the primary optimization algorithm in modern machine learning and deep learning. SGD’s efficiency in training neural networks through iterative updates is directly rooted in this method.
Additionally, their approach to handling binary feedback inspired early concepts in reinforcement learning, where algorithms learn from sparse rewards and adapt over time.The paper's principles are fundamental to nonparametric methods, online learning, and dynamic optimization in data science and AI today.
By enabling sequential, probabilistic updates, the Robbins-Monro method supports adaptive decision-making in real-time applications such as recommender systems, autonomous systems, and financial trading, making it a cornerstone of modern AI’s ability to learn in complex, uncertain environments.
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the 15th episode we went over the paper "Problems in the Analysis of Survey Data, and a Proposal" by James N. Morgan and John A. Sonquist from 1963.It highlights seven key issues in analyzing complex survey data, such as high dimensionality, categorical variables, measurement errors, sample variability, intercorrelations, interaction effects, and causal chains.
These challenges complicate efforts to draw meaningful conclusions about relationships between factors like income, education, and occupation.To address these problems, the authors propose a method that sequentially splits data by identifying features that reduce unexplained variance, much like modern decision trees.
The method focuses on maximizing explained variance (SSE), capturing interaction effects, and accounting for sample variability.
It handles both categorical and continuous variables while respecting logical causal priorities.This paper has had a significant influence on modern data science and AI, laying the groundwork for decision trees, CART, random forests, and boosting algorithms.
Its method of splitting data to reduce error, handle interactions, and respect feature hierarchies is foundational in many machine learning models used today.Link to full paper at our website:
https://datasciencedecodedpodcast.com/episode-15-the-first-decision-tree-algorithm-1963
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At the 14th episode we go over the Stuart Lloyd's 1957 paper, "Least Squares Quantization in PCM," (which was published only at 1982)The k-means algorithm can be traced back to this paper.Loyd introduces an approach to quantization in pulse-code modulation (PCM). Which is like a 1-D k means clustering. Lloyd discusses how quantization intervals and corresponding quantum values should be adjusted based on signal amplitude distributions to minimize noise, improving efficiency in PCM systems.
He derives an optimization framework that minimizes quantization noise under finite quantization schemes. Lloyd’s algorithm bears significant resemblance to the k-means clustering algorithm, both seeking to minimize a sum of squared errors.
In Lloyd's method, the quantization process is analogous to assigning data points (signal amplitudes) to clusters (quantization intervals) based on proximity to centroids (quantum values), with the centroids updated iteratively based on the mean of the assigned points.
This iterative process of recalculating quantization values mirrors k-means’ recalculation of cluster centroids. While Lloyd’s work focuses on signal processing in telecommunications, its underlying principles of optimizing quantization have clear parallels with the k-means method used in clustering tasks in data science.The paper's influence on modern data science is profound. Lloyd's algorithm not only laid the groundwork for k-means but also provided a fundamental understanding of quantization error minimization, critical in fields such as machine learning, image compression, and signal processing.
The algorithm's simplicity, combined with its iterative nature, has led to its wide adoption in various data science applications. Lloyd's work remains a cornerstone in both the theory of clustering algorithms and practical applications in signal and data compression technologies.
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In the 14th episode we review the second part of Kolmogorov's seminal paper:Three approaches to the quantitative definition of information’." Problems of information transmission 1.1 (1965): 1-7.The paper introduces algorithmic complexity (or Kolmogorov complexity), which measures the amount of information in an object based on the length of the shortest program that can describe it.
This shifts focus from Shannon entropy, which measures uncertainty probabilistically, to understanding the complexity of structured objects.
Kolmogorov argues that systems like texts or biological data, governed by rules and patterns, are better analyzed by their compressibility—how efficiently they can be described—rather than by random probabilistic models.In modern data science and AI, these ideas are crucial. Machine learning models, like neural networks, aim to compress data into efficient representations to generalize and predict. Kolmogorov complexity underpins the idea of minimizing model complexity while preserving key information, which is essential for preventing overfitting and improving generalization.
In AI, tasks such as text generation and data compression directly apply Kolmogorov's concept of finding the most compact representation, making his work foundational for building efficient, powerful models.This is part 2 out of 2 episodes covering this paper (the first one is in Episode 12).
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In the 12th episode we review the first part of Kolmogorov's seminal paper:
"3 approaches to the quantitative definition of information’." Problems of information transmission 1.1 (1965): 1-7.The paper introduces algorithmic complexity (or Kolmogorov complexity), which measures the amount of information in an object based on the length of the shortest program that can describe it.
This shifts focus from Shannon entropy, which measures uncertainty probabilistically, to understanding the complexity of structured objects.
Kolmogorov argues that systems like texts or biological data, governed by rules and patterns, are better analyzed by their compressibility—how efficiently they can be described—rather than by random probabilistic models.In modern data science and AI, these ideas are crucial. Machine learning models, like neural networks, aim to compress data into efficient representations to generalize and predict. Kolmogorov complexity underpins the idea of minimizing model complexity while preserving key information, which is essential for preventing overfitting and improving generalization.
In AI, tasks such as text generation and data compression directly apply Kolmogorov's concept of finding the most compact representation, making his work foundational for building efficient, powerful models.This is part 1 out of 2 episodes covering this paper
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Frank Rosenblatt's 1958 paper, "The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain," introduces the perceptron, an early neural network model inspired by how the brain stores and processes information.Rosenblatt explores two theories: one where sensory data is stored as coded representations, and another, which he advocates, where learning occurs through forming new neural connections.
The perceptron illustrates this connectionist approach by mimicking how neurons process input and reinforce connections based on experience.The perceptron operates by passing sensory input through a network of neurons, where weights on connections adjust with each stimulus, enabling the system to recognize patterns and classify information. Rosenblatt emphasizes the probabilistic nature of learning in the perceptron, which mirrors how biological systems might generalize and adapt based on exposure to different stimuli. His model serves as a theoretical framework for understanding both biological and artificial neural systems.The paper's significance to modern data science lies in its foundational role in developing machine learning. The perceptron model directly influenced the creation of more advanced neural networks, including today's deep learning models.
Though limited in handling complex, non-linear data, the perceptron established key principles—such as weighted connections and learning from data.
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Hotelling, Harold. "Analysis of a complex of statistical variables into principal components." Journal of educational psychology 24.6 (1933): 417.
This seminal work by Harold Hotelling on PCA remains highly relevant to modern data science because PCA is still widely used for dimensionality reduction, feature extraction, and data visualization. The foundational concepts of eigenvalue decomposition and maximizing variance in orthogonal directions form the backbone of PCA, which is now automated through numerical methods such as Singular Value Decomposition (SVD). Modern PCA handles much larger datasets with advanced variants (e.g., Kernel PCA, Sparse PCA), but the core ideas from the paper—identifying and interpreting key components to reduce dimensionality while preserving the most important information—are still crucial in handling high-dimensional data efficiently today.
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In this special episode, Daniel Aronovich joins forces with the 632 nm podcast.In this timeless paper Wigner reflects on how mathematical concepts, often developed independently of any concern for the physical world, turn out to be remarkably effective in describing natural phenomena.
This effectiveness is "unreasonable" because there is no clear reason why abstract mathematical constructs should align so well with the laws governing the universe.Full paper is at our website:
https://datasciencedecodedpodcast.com/episode-9-the-unreasonable-effectiveness-of-mathematics-in-natural-sciences-eugene-wigner-1960
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This paper is a foundational text in the field of artificial intelligence (AI) and explores the question: "Can machines think?" Turing introduces what is now known as the "Turing Test" as a way to operationalize this question, he called it the imitation game.Are there imaginable digital computers that could perform well in the imitation game?The imitation game involves an interrogator trying to distinguish between a human and a machine based on their responses to various questions.Turing argues that if a machine could perform well enough in this game to be indistinguishable from a human, then it could be said to "think." He explores various objections to the idea that machines can think, including theological, mathematical, and arguments from consciousness.
Turing addresses each objection, ultimately suggesting that machines can indeed be said to think if they can perform human-like tasks, especially those that involve reasoning, learning, and language.
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This paper introduced linear discriminant analysis(LDA), a statistical technique that revolutionized classification in biology and beyond.
Fisher demonstrated how to use multiple measurements to distinguish between different species of iris flowers, laying the foundation for modern multivariate statistics.
His work showed that combining several characteristics could provide more accurate classification than relying on any single trait.
This paper not only solved a practical problem in botany but also opened up new avenues for statistical analysis across various fields.
Fisher's method became a cornerstone of pattern recognition and machine learning, influencing diverse areas from medical diagnostics to AI.
The iris dataset he used, now known as the "Fisher iris" or "Anderson iris" dataset, remains a popular example in data science education and research.
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This paper is considered one of the foundational works in modern statistical hypothesis testing.
Key insights and influences:
Neyman-Pearson Lemma: The paper introduced the Neyman-Pearson Lemma, which provides a method for finding the most powerful test for a simple hypothesis against a simple alternative. Type I and Type II errors: It formalized the concepts of Type I (false positive) and Type II (false negative) errors in hypothesis testing. Power of a test: The paper introduced the concept of the power of a statistical test, which is the probability of correctly rejecting a false null hypothesis. Likelihood ratio tests: It laid the groundwork for likelihood ratio tests, which are widely used in modern statistics. Optimal testing: The paper provided a framework for finding optimal statistical tests, balancing the tradeoff between Type I and Type II errors.These concepts have had a profound influence on modern statistical theory and practice, forming the basis of much of classical hypothesis testing used today in various fields of science and research.
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Shannon, Claude Elwood. "A mathematical theory of communication." The Bell system technical journal 27.3 (1948): 379-423.Part 3/3.The paper fundamentally reshapes how we understand communication.The paper introduces a formal framework for analyzing communication systems, addressing the transmission of information with and without noise. Key concepts include the definition of information entropy, the logarithmic measure of information, and the capacity of communication channels. In the third part we go over the Fundamental theorem of the noisy and noiseless channel!Full breakdown of the paper with math and python code is at our website:https://datasciencedecodedpodcast.com/
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Shannon, Claude Elwood. "A mathematical theory of communication." The Bell system technical journal 27.3 (1948): 379-423.Part 2/3.The paper fundamentally reshapes how we understand communication.The paper introduces a formal framework for analyzing communication systems, addressing the transmission of information with and without noise. Key concepts include the definition of information entropy, the logarithmic measure of information, and the capacity of communication channels. Shannon demonstrates that information can be efficiently encoded and decoded to maximize the transmission rate while minimizing errors introduced by noise. This work is pivotal today as it underpins digital communication technologies, from data compression to error correction in modern telecommunication systems.Full breakdown of the paper with math and python code is at our website:https://datasciencedecodedpodcast.com...This is the second part out of 3, as the paper is quite long!
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Shannon, Claude Elwood. "A mathematical theory of communication." The Bell system technical journal 27.3 (1948): 379-423.Part 1/3.The paper fundamentally reshapes how we understand communication.The paper introduces a formal framework for analyzing communication systems, addressing the transmission of information with and without noise. Key concepts include the definition of information entropy, the logarithmic measure of information, and the capacity of communication channels. Shannon demonstrates that information can be efficiently encoded and decoded to maximize the transmission rate while minimizing errors introduced by noise. This work is pivotal today as it underpins digital communication technologies, from data compression to error correction in modern telecommunication systems.Full breakdown of the paper with math and python code is at our website:https://datasciencedecodedpodcast.com...This is the first part out of 3, as the paper is quite long!
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"Application of the Logistic Function to Bio-Assays" (1944), Berkson Joseph
It gained further prominence in the 20th century through applications in various fields, including biology and bio-assay. Joseph Berkson's 1944 paper, 'Application of the Logistic Function to Bio-Assay,' was pivotal in popularizing its use for estimating drug potency.
Berkson argued that the logistic function was a more statistically manageable and theoretically sound alternative to the probit function, which assumed that individual susceptibilities to a drug follow a normal distribution.
The logistic function's ability to be easily linearized via the logit transformation simplifies parameter estimation, making it an attractive choice for analyzing dose-response data.
