🔭Data Science: Overfitting (Just the Quotes)

"[...] with four parameters I can fit an elephant, and with five I can make him wiggle his trunk." (John von Neymann) [attributed, Freeman Dyson first wrote down this quote in a 2004 article published in the journal Nature, recounting a 1953 meeting where Enrico Fermi quoted John von Neumann]

"When training a neural network, it is important to understand when to stop. […] If the same training patterns or examples are given to the neural network over and over, and the weights are adjusted to match the desired outputs, we are essentially telling the network to memorize the patterns, rather than to extract the essence of the relationships. What happens is that the neural network performs extremely well on the training data. However, when it is presented with patterns it hasn't seen before, it cannot generalize and does not perform well. What is the problem? It is called overtraining." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

[Over-fitting fallacy:] "The error of designing an over-complex trading strategy with too many parameters that performs well on the in-sample-data, but is actually no more than a close description of the past data. This is a problem often encountered in time-series analysis and modelling." (Kermit Zieg &Heinrich Weber, "The Complete Guide to Point-and-Figure Charting", 2003)

"A smaller model with fewer covariates has two advantages: it might give better predictions than a big model and it is more parsimonious (simpler). Generally, as you add more variables to a regression, the bias of the predictions decreases and the variance increases. Too few covariates yields high bias; this called underfitting. Too many covariates yields high variance; this called overfitting. Good predictions result from achieving a good balance between bias and variance. […] finding a good model involves trading of fit and complexity." (Larry A Wasserman, "All of Statistics: A concise course in statistical inference", 2004)

"Learning a complicated function that matches the training data closely but fails to recognize the underlying process that generates the data. As a result of overfitting, the model performs poor on new input. Overfitting occurs when the training patterns are sparse in input space and/or the trained networks are too complex." (Frank Padberg, "Counting the Hidden Defects in Software Documents", 2010)

"A forecaster should almost never ignore data, especially when she is studying rare events […]. Ignoring data is often a tip-off that the forecaster is overconfident, or is overfitting her model - that she is interested in showing off rather than trying to be accurate."  (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"A problem in data mining when random variations in data are misclassified as important patterns. Overfitting often occurs when the data set is too small to represent the real world." (Microsoft, "SQL Server 2012 Glossary", 2012)

"If you look too hard at a set of data, you will find something - but it might not generalize beyond the data you’re looking at. This is referred to as overfitting a dataset. Data mining techniques can be very powerful, and the need to detect and avoid overfitting is one of the most important concepts to grasp when applying data mining to real problems. The concept of overfitting and its avoidance permeates data science processes, algorithms, and evaluation methods." (Foster Provost & Tom Fawcett, "Data Science for Business", 2013)

"Overfitting occurs when a formula describes a set of data very closely, but does not lead to any sensible explanation for the behavior of the data and does not predict the behavior of comparable data sets. In the case of overfitting, the formula is said to describe the noise of the system rather than the characteristic behavior of the system. Overfitting occurs frequently with models that perform iterative approximations on training data, coming closer and closer to the training data set with each iteration. Neural networks are an example of a data modeling strategy that is prone to overfitting." (Jules H Berman, "Principles of Big Data: Preparing, Sharing, and Analyzing Complex Information", 2013)

"Briefly speaking, to solve a Machine Learning problem means you optimize a model to fit all the data from your training set, and then you use the model to predict the results you want. Therefore, evaluating a model need to see how well it can be used to predict the data out of the training set. Usually there are three types of the models: underfitting, fair and overfitting model [...]. If we want to predict a value, both (a) and (c) in this figure cannot work well. The underfitting model does not capture the structure of the problem at all, and we say it has high bias. The overfitting model tries to fit every sample in the training set and it did it, but we say it is of high variance. In other words, it fails to generalize new data." (Shudong Hao, "A Beginner’s Tutorial for Machine Learning Beginners", 2014)

"Neural networks can model very complex patterns and decision boundaries in the data and, as such, are very powerful. In fact, they are so powerful that they can even model the noise in the training data, which is something that definitely should be avoided. One way to avoid this overfitting is by using a validation set in a similar way as with decision trees.[...] Another scheme to prevent a neural network from overfitting is weight regularization, whereby the idea is to keep the weights small in absolute sense because otherwise they may be fitting the noise in the data. This is then implemented by adding a weight size term (e.g., Euclidean norm) to the objective function of the neural network." (Bart Baesens, "Analytics in a Big Data World: The Essential Guide to Data Science and Its Applications", 2014)

"Underfitting is when a model doesn’t take into account enough information to accurately model real life. For example, if we observed only two points on an exponential curve, we would probably assert that there is a linear relationship there. But there may not be a pattern, because there are only two points to reference. [...] It seems that the best way to mitigate underfitting a model is to give it more information, but this actually can be a problem as well. More data can mean more noise and more problems. Using too much data and too complex of a model will yield something that works for that particular data set and nothing else." (Matthew Kirk, "Thoughtful Machine Learning", 2015)

"Neural nets are typically over-parametrized, and hence are prone to overfitting. Originally early stopping was set up as the primary tuning parameter, and the stopping time was determined using a held-out set of validation data. In modern networks the regularization is tuned adaptively to avoid overfitting, and hence it is less of a problem." (Bradley Efron & Trevor Hastie, "Computer Age Statistical Inference: Algorithms, Evidence, and Data Science", 2016)

"The greater the uncertainty, the bigger the gap between what you can measure and what matters, the more you should watch out for overfitting - that is, the more you should prefer simplicity." (Brian Christian & Thomas L Griffiths, "Algorithms to Live By: The Computer Science of Human Decisions", 2016)

"When memorization happens, you may have the illusion that everything is working well because your machine learning algorithm seems to have fitted the in sample data so well. Instead, problems can quickly become evident when you start having it work with out-of-sample data and you notice that it produces errors in its predictions as well as errors that actually change a lot when you relearn from the same data with a slightly different approach. Overfitting occurs when your algorithm has learned too much from your data, up to the point of mapping curve shapes and rules that do not exist [...]. Any slight change in the procedure or in the training data produces erratic predictions." (John P Mueller & Luca Massaron, Machine Learning for Dummies, 2016)

"By far the greatest headache in machine learning is the problem of overfitting. This means that your results look great for the data you trained them on, but they don’t generalize to other data in the future. [...] The solution is to train on some of your data and assess performance on other data." (Field Cady, "The Data Science Handbook", 2017) 

"Cross-validation means we split our data into test and training sets, and then train the model on the training set before testing it on the test set. Cross-validation prevents overfitting, which is when a model seems quite accurate but fails to actually predict future events well." (Russell Jurney, "Agile Data Science 2.0: Building Full-Stack Data Analytics Applications with Spark", 2017)

"Multilayer perceptrons share with polynomial classifiers one unpleasant property. Theoretically speaking, they are capable of modeling any decision surface, and this makes them prone to overfitting the training data."  (Miroslav Kubat," An Introduction to Machine Learning" 2nd Ed., 2017)

"The main reason why pruning tends to improve classification performance on future examples is that the removal of low-level tests, which have poor statistical support, usually reduces the danger of overfitting. This, however, works only up to a certain point. If overdone, a very high extent of pruning can (in the extreme) result in the decision being replaced with a single leaf labeled with the majority class." (Miroslav Kubat," An Introduction to Machine Learning" 2nd Ed., 2017)

"From a typical training set, many alternative decision trees can be created. As a rule, smaller trees are to be preferred, their main advantages being interpretability, removal of irrelevant and redundant attributes, and lower danger of overfitting noisy training data." (Miroslav Kubat, "An Introduction to Machine Learning" 2nd Ed., 2017)

"High-bias models typically produce simpler models that do not overfit and in those cases the danger is that of underfitting. Models with low-bias are typically more complex and that complexity enables us to represent the training data in a more accurate way. The danger here is that the flexibility provided by higher complexity may end up representing not only a relationship in the data but also the noise. Another way of portraying the bias-variance trade-off is in terms of complexity v simplicity." (Jesús Rogel-Salazar, "Data Science and Analytics with Python", 2017) 

"If either bias or variance is high, the model can be very far off from reality. In general, there is a trade-off between bias and variance. The goal of any machine-learning algorithm is to achieve low bias and low variance such that it gives good prediction performance. In reality, because of so many other hidden parameters in the model, it is hard to calculate the real bias and variance error. Nevertheless, the bias and variance provide a measure to understand the behavior of the machine-learning algorithm so that the model model can be adjusted to provide good prediction performance." (Umesh R Hodeghatta & Umesha Nayak, "Business Analytics Using R: A Practical Approach", 2017)

"It’s important to note that model complexity is intimately tied to the variation of inputs contained in your training dataset: the larger variety of data points your data‐ set contains, the more complex a model you can use without overfitting. Usually, collecting more data points will yield more variety, so larger datasets allow building more complex models. However, simply duplicating the same data points or collecting very similar data will not help." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"Overfitting and underfitting are two important factors that could impact the performance of machine-learning models. Overfitting occurs when the model performs well with training data and poorly with test data. Underfitting occurs when the model is so simple that it performs poorly with both training and test data. [...]  When the model does not capture and fit the data, it results in poor performance. We call this underfitting. Underfitting is the result of a poor model that typically does not perform well for any data." (Umesh R Hodeghatta & Umesha Nayak, "Business Analytics Using R: A Practical Approach", 2017)

"Overfitting refers to the phenomenon where a model is highly fitted on a dataset. This generalization thus deprives the model from making highly accurate predictions about unseen data. [...] Underfitting is a phenomenon where the model is not trained with high precision on data at hand. The treatment of underfitting is subject to bias and variance. A model will have a high bias if both train and test errors are high [...] If a model has a high bias type underfitting, then the remedy can be to increase the model complexity, and if a model is suffering from high variance type underfitting, then the cure can be to bring in more data or otherwise make the model less complex." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"The danger of overfitting is particularly severe when the training data is not a perfect gold standard. Human class annotations are often subjective and inconsistent, leading boosting to amplify the noise at the expense of the signal. The best boosting algorithms will deal with overfitting though regularization. The goal will be to minimize the number of non-zero coefficients, and avoid large coefficients that place too much faith in any one classifier in the ensemble." (Steven S Skiena, "The Data Science Design Manual", 2017)

"The tension between bias and variance, simplicity and complexity, or underfitting and overfitting is an area in the data science and analytics process that can be closer to a craft than a fixed rule. The main challenge is that not only is each dataset different, but also there are data points that we have not yet seen at the moment of constructing the model. Instead, we are interested in building a strategy that enables us to tell something about data from the sample used in building the model." (Jesús Rogel-Salazar, "Data Science and Analytics with Python", 2017) 

"Variance is a prediction error due to different sets of training samples. Ideally, the error should not vary from one training sample to another sample, and the model should be stable enough to handle hidden variations between input and output variables. Normally this occurs with the overfitted model." (Umesh R Hodeghatta & Umesha Nayak, "Business Analytics Using R: A Practical Approach", 2017)

"Variance is error from sensitivity to fluctuations in the training set. If our training set contains sampling or measurement error, this noise introduces variance into the resulting model. [...] Errors of variance result in overfit models: their quest for accuracy causes them to mistake noise for signal, and they adjust so well to the training data that noise leads them astray. Models that do much better on testing data than training data are overfit." (Steven S Skiena, "The Data Science Design Manual", 2017)

"Even though a natural way of avoiding overfitting is to simply build smaller networks (with fewer units and parameters), it has often been observed that it is better to build large networks and then regularize them in order to avoid overfitting. This is because large networks retain the option of building a more complex model if it is truly warranted. At the same time, the regularization process can smooth out the random artifacts that are not supported by sufficient data. By using this approach, we are giving the model the choice to decide what complexity it needs, rather than making a rigid decision for the model up front (which might even underfit the data)." (Charu C Aggarwal, "Neural Networks and Deep Learning: A Textbook", 2018)

"One of the most common problems that you will encounter when training deep neural networks will be overfitting. What can happen is that your network may, owing to its flexibility, learn patterns that are due to noise, errors, or simply wrong data. [...] The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e., the noise) as if that variation represented the underlying model structure. The opposite is called underfitting - when the model cannot capture the structure of the data." (Umberto Michelucci, "Applied Deep Learning: A Case-Based Approach to Understanding Deep Neural Networks", 2018)

"The high generalization error in a neural network may be caused by several reasons. First, the data itself might have a lot of noise, in which case there is little one can do in order to improve accuracy. Second, neural networks are hard to train, and the large error might be caused by the poor convergence behavior of the algorithm. The error might also be caused by high bias, which is referred to as underfitting. Finally, overfitting (i.e., high variance) may cause a large part of the generalization error. In most cases, the error is a combination of more than one of these different factors." (Charu C Aggarwal, "Neural Networks and Deep Learning: A Textbook", 2018)

"The trick is to walk the line between underfitting and overfitting. An underfit model has low variance, generally making the same predictions every time, but with extremely high bias, because the model deviates from the correct answer by a significant amount. Underfitting is symptomatic of not having enough data points, or not training a complex enough model. An overfit model, on the other hand, has memorized the training data and is completely accurate on data it has seen before, but varies widely on unseen data. Neither an overfit nor underfit model is generalizable - that is, able to make meaningful predictions on unseen data." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

"Any fool can fit a statistical model, given the data and some software. The real challenge is to decide whether it actually fits the data adequately. It might be the best that can be obtained, but still not good enough to use." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)

"The classifier accuracy would be extra ordinary when the test data and the training data are overlapping. But when the model is applied to a new data it will fail to show acceptable accuracy. This condition is called as overfitting." (Jesu V  Nayahi J & Gokulakrishnan K, "Medical Image Classification", 2019)

"We over-fit when we go too far in adapting to local circumstances, in a worthy but misguided effort to be ‘unbiased’ and take into account all the available information. Usually we would applaud the aim of being unbiased, but this refinement means we have less data to work on, and so the reliability goes down. Over-fitting therefore leads to less bias but at a cost of more uncertainty or variation in the estimates, which is why protection against over-fitting is sometimes known as the bias/variance trade-off." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"Well, in statistics we develop models from a sample of data and are trying to make inferences to a broader population. […] If you use a lot of parameters to explain the data in hand (the sample), you may have captured your particular dataset but completely miss the mark for the population as a whole! This is known as 'overfitting'." (Therese M Donovan & Ruth M Mickey, "Bayesian Statistics for Beginners: A Step-by-Step Approach", 2019)

"In machine learning, our data has biases as well as useful information for our task. The more exactly our machine learning model fits the data, the more it reflects these biases. This means that the predictions may be based on spurious relationships that incidentally occur in the training data." (Alex Thomas, "Natural Language Processing with Spark NLP", 2020)

"Overfitting in machine learning is a condition in which a model learns the training data so well that it performs poorly on unseen or test data. The model essentially memorizes the noise or random fluctuations in the training data and fails to generalize its learning to new data. In terms of LLMs, overfitting could occur if the model excessively adjusts to the specifics of the training data, thereby losing its ability to generate sensible responses for unseen prompts. This could lead to the model generating too specific or narrowly tailored responses that do not correctly address the new prompts." (Sinan Ozdemir, "Quick Start Guide to Large Language Models: Strategies and Best Practices for Using ChatGPT and Other LLMs", 2024) 

"Generative AI has a lot of problems because it needs a lot of data to work. Generative models like GPT and GANs need a lot of training data to learn patterns and make good outputs. This data dependency can cause problems like overfitting, which happens when the model does well on training data but doesn’t work well with new, unseen data. Also, the quality of the content that is generated is directly related to the diversity and representativeness of the training data. This means that biased or incomplete datasets can lead to outputs that are wrong or unfair." (Saloni Garg et al, "RAG Artificial Intelligence: Retrieval-Augmented Generation in Generative AI", 2026)

See also the definitions on overfitting

🔭Data Science: Principles (Just the Quotes)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"It is superfluous to suppose that what can be accounted for by a few principles has been produced by many." (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Men are often led into errors by the love of simplicity, which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"Every science has for its basis a system of principles as fixed and unalterable as those by which the universe is regulated and governed. Man cannot make principles; he can only discover them." (Thomas Paine, "The Age of Reason", 1794)

"The aim of every science is foresight. For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte, "Plan des travaux scientifiques nécessaires pour réorganiser la société", 1822)

"To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth." (David Hume, "Of the passions", 1826)

"[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us." (Auguste Comte, "Cours de Philosophie Positive", 1830-1842)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"It is of the nature of true science to take nothing on trust or on authority. Every fact must be established by accurate observation, experiment, or calculation. Every law and principle must rest on inductive argument. The apostolic motto, ‘Prove all things, hold fast that which is good’, is thoroughly scientific. It is true that the mere reader of popular science must often be content to take that on testimony which he cannot personally verify; but it is desirable that even the most cursory reader should fully comprehend the modes in which facts are ascertained and the reasons on which the conclusions are based." (Sir John W Dawson, "The Chain of Life in Geological Time", 1880)

"It goes without saying that the laws of nature are in themselves independent of the properties of the instruments with which they are measured. Therefore in every observation of natural phenomena we must remember the principle that the reliability of the measuring apparatus must always play an important role." (Max Planck,"Where is Science Going?", 1932)

"The fundamental gospel of statistics is to push back the domain of ignorance, prejudice, rule-of-thumb, arbitrary or premature decisions, tradition, and dogmatism and to increase the domain in which decisions are made and principles are formulated on the basis of analyzed quantitative facts." (Robert W Burgess, "The Whole Duty of the Statistical Forecaster", Journal of the American Statistical Association , Vol. 32, No. 200, 1937)  

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"It is always more easy to discover and proclaim general principles than it is to apply them." (Winston Churchill, "The Second World War: The gathering storm", 1948)

"It is more important to have a clear understanding of general principles, without, however, thinking of them as fixed laws, than to load the mind with a mass of detailed technical information which can readily be found in reference books or card indexes." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or 'forces' between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, "General System Theory: Foundations, Development, Applications", 1968)

"No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"Real progress in understanding nature is rarely incremental. All important advances are sudden intuitions, new principles, new ways of seeing." (Marilyn Ferguson, "The Aquarian Conspiracy: Personal and Social Transformation in the 1980s", 1980)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

More quotes on "Principles" at the-web-of-knowledge.blogspot.com.

🔭Data Science: Dimensionality (Just the Quotes)

"[…] the intrinsic value of a small-scale model is that it compensates for the renunciation of sensible dimensions by the acquisition of intelligible dimensions." (Claude Levi- Strauss, "The Savage Mind", 1962)

"The idea of knowledge as an improbable structure is still a good place to start. Knowledge, however, has a dimension which goes beyond that of mere information or improbability. This is a dimension of significance which is very hard to reduce to quantitative form. Two knowledge structures might be equally improbable but one might be much more significant than the other." (Kenneth E Boulding, "Beyond Economics: Essays on Society", 1968)

"A time series is a sequence of observations, usually ordered in time, although in some cases the ordering may be according to another dimension. The feature of time series analysis which distinguishes it from other statistical analysis is the explicit recognition of the importance of the order in which the observations are made. While in many problems the observations are statistically independent, in time series successive observations may be dependent, and the dependence may depend on the positions in the sequence. The nature of a series and the structure of its generating process also may involve in other ways the sequence in which the observations are taken." (Theodore W Anderson, "The Statistical Analysis of Time Series", 1971)

"The number of information-carrying (variable) dimensions depicted should not exceed the number of dimensions in the data." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"In addition to dimensionality requirements, chaos can occur only in nonlinear situations. In multidimensional settings, this means that at least one term in one equation must be nonlinear while also involving several of the variables. With all linear models, solutions can be expressed as combinations of regular and linear periodic processes, but nonlinearities in a model allow for instabilities in such periodic solutions within certain value ranges for some of the parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"The dimensionality and nonlinearity requirements of chaos do not guarantee its appearance. At best, these conditions allow it to occur, and even then under limited conditions relating to particular parameter values. But this does not imply that chaos is rare in the real world. Indeed, discoveries are being made constantly of either the clearly identifiable or arguably persuasive appearance of chaos. Most of these discoveries are being made with regard to physical systems, but the lack of similar discoveries involving human behavior is almost certainly due to the still developing nature of nonlinear analyses in the social sciences rather than the absence of chaos in the human setting."  (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"A system may be called complex here if its dimension (order) is too high and its model (if available) is nonlinear, interconnected, and information on the system is uncertain such that classical techniques can not easily handle the problem." (M Jamshidi, "Autonomous Control on Complex Systems: Robotic Applications", Current Advances in Mechanical Design and Production VII, 2000)

"The greatest plus of data modeling is that it produces a simple and understandable picture of the relationship between the input variables and responses [...] different models, all of them equally good, may give different pictures of the relation between the predictor and response variables [...] One reason for this multiplicity is that goodness-of-fit tests and other methods for checking fit give a yes–no answer. With the lack of power of these tests with data having more than a small number of dimensions, there will be a large number of models whose fit is acceptable. There is no way, among the yes–no methods for gauging fit, of determining which is the better model." (Leo Breiman, "Statistical Modeling: The two cultures" Statistical Science 16(3), 2001)

"Three key aspects of presenting high dimensional data are: rendering, manipulation, and linking. Rendering determines what is to be plotted, manipulation determines the structure of the relationships, and linking determines what information will be shared between plots or sections of the graph." (Gerald van Belle, "Statistical Rules of Thumb", 2002)

"With the ever increasing amount of empirical information that scientists from all disciplines are dealing with, there exists a great need for robust, scalable and easy to use clustering techniques for data abstraction, dimensionality reduction or visualization to cope with and manage this avalanche of data."  (Jörg Reichardt, "Structure in Complex Networks", 2009)

"The more dimensions used in quantitative comparisons, the larger are the disparities that can be accommodated. As irony would have it, however, the ease of comparison generally diminishes in direct proportion to the number of dimensions involved." (Joel Katz, "Designing Information: Human factors and common sense in information design", 2012)

"Dimensionality reduction and regression modeling are particularly hard to interpret in terms of original attributes, when the underlying data dimensionality is high. This is because the subspace embedding is defined as a linear combination of attributes with positive or negative coefficients. This cannot easily be intuitively interpreted in terms specific properties of the data attributes." (Charu C Aggarwal, "Outlier Analysis", 2013)

"In the family of idioms typically called dimensionality reduction, the goal is to preserve the meaningful structure of a dataset while using fewer attributes to represent the items. The rationale that a small set of new synthetic attributes might be able to faithfully represent most of the structure or variance in the dataset hinges on the assumption that there is hidden structure and significant redundancy in the original dataset because the underlying latent variables could not be measured directly." (Tamara Munzner, "Visualization Analysis and Design", 2014)

"Dimensionality reduction is essential for coping with big data - like the data coming in through your senses every second. A picture may be worth a thousand words, but it’s also a million times more costly to process and remember. [...] A common complaint about big data is that the more data you have, the easier it is to find spurious patterns in it. This may be true if the data is just a huge set of disconnected entities, but if they’re interrelated, the picture changes." (Pedro Domingos, "The Master Algorithm", 2015)

"The correlational technique known as multiple regression is used frequently in medical and social science research. This technique essentially correlates many independent (or predictor) variables simultaneously with a given dependent variable (outcome or output). It asks, 'Net of the effects of all the other variables, what is the effect of variable A on the dependent variable?' Despite its popularity, the technique is inherently weak and often yields misleading results. The problem is due to self-selection. If we don’t assign cases to a particular treatment, the cases may differ in any number of ways that could be causing them to differ along some dimension related to the dependent variable. We can know that the answer given by a multiple regression analysis is wrong because randomized control experiments, frequently referred to as the gold standard of research techniques, may give answers that are quite different from those obtained by multiple regression analysis." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

"Understanding reduces the complexity of data by collapsing the dimensionality of information to a lower set of known variables. s revolutions, be they tiny or vast, technological or social." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

"Dimensionality reduction is a way of reducing a large number of different measures into a smaller set of metrics. The intent is that the reduced metrics are a simpler description of the complex space that retains most of the meaning." (Danyel Fisher & Miriah Meyer, "Making Data Visual", 2018)

"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)

"This problem with adding additional variables is referred to as the curse of dimensionality. If you add enough variables into your black box, you will eventually find a combination of variables that performs well - but it may do so by chance. As you increase the number of variables you use to make your predictions, you need exponentially more data to distinguish true predictive capacity from luck." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

"We all know that the numerical values on each side of an equation have to be the same. The key to dimensional analysis is that the units have to be the same as well. This provides a convenient way to keep careful track of units when making calculations in engineering and other quantitative disciplines, to make sure one is computing what one thinks one is computing. When an equation exists only for the sake of mathiness, dimensional analysis often makes no sense." (Carl T Bergstrom & Jevin D West, "Calling Bullshit: The Art of Skepticism in a Data-Driven World", 2020)

"Smoothing and aggregating can help us see important features and relationships, but when we have only a handful of observations, smoothing techniques can be misleading. With just a few observations, we prefer rug plots over histograms, box plots, and density curves, and we use scatterplots rather than smooth curves and density contours. This may seem obvious, but when we have a large amount of data, the amount of data in a subgroup can quickly dwindle. This phenomenon is an example of the curse of dimensionality." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"A common myth is that non-linear dimension reduction captures non-linear patterns in the high-dimensional data. It may or may not do this. The term means that the methods transform the data non-linearly into a useful (or not) visual representation." (Dianne Cook & Ursula Laa, "Interactively Exploring High-Dimensional Data and Models in R", 2026)

🔭Data Science: Software (Just the Quotes)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system (the mind) can be mapped onto the other (the computer)." (George Johnson, Machinery of the Mind: Inside the New Science of Artificial Intelligence, 1986)

"Most computer software is not yet intelligent enough to stop the user doing something stupid. The old adage ‘Garbage In -> Garbage Out’ still hold good, and it must be realized that careful thought and close inspection of the data are vital preliminaries to complicated computer analysis." (Christopher Chatfield, "Problem solving: A statistician’s guide", 1988)

"Bayesian methods are complicated enough, that giving researchers user-friendly software could be like handing a loaded gun to a toddler; if the data is crap, you won’t get anything out of it regardless of your political bent." (Brad Carlin, "Bayes offers a new way to make sense of numbers", Science 19, 1999)

"[…] when software systems become so intractable that they can no longer be controlled, swarm intelligence offers an alternative way of designing an ‘intelligent’ systems, in which autonomy, emergence, and distributed functioning replace control, preprogramming, and centralization." (Eric Bonabeau et al, "Swarm Intelligence: From Natural to Artificial Systems", 1999)

"With the growing interest in complex adaptive systems, artificial life, swarms and simulated societies, the concept of 'collective intelligence' is coming more and more to the fore. The basic idea is that a group of individuals (e. g. people, insects, robots, or software agents) can be smart in a way that none of its members is. Complex, apparently intelligent behavior may emerge from the synergy created by simple interactions between individuals that follow simple rules." (Francis Heylighen, "Collective Intelligence and its Implementation on the Web", 1999)

"Today [...] we have high-speed computers and prepackaged statistical routines to perform the necessary calculations [...] statistical software will no more make one a statistician than would a scalpel turn one into a neurosurgeon. Allowing these tools to do our thinking for us is a sure recipe for disaster." (Phillip Good & Hardin James, "Common Errors in Statistics and How to Avoid Them", 2003)

"There are several key issues in the field of statistics that impact our analyses once data have been imported into a software program. These data issues are commonly referred to as the measurement scale of variables, restriction in the range of data, missing data values, outliers, linearity, and nonnormality." (Randall E Schumacker & Richard G Lomax, "A Beginner’s Guide to Structural Equation Modeling" 3rd Ed., 2010)

"A notable difference between many fields and data science is that in data science, if a customer has a wish, even an experienced data scientist may not know whether it’s possible. Whereas a software engineer usually knows what tasks software tools are capable of performing, and a biologist knows more or less what the laboratory can do, a data scientist who has not yet seen or worked with the relevant data is faced with a large amount of uncertainty, principally about what specific data is available and about how much evidence it can provide to answer any given question. Uncertainty is, again, a major factor in the data scientific process and should be kept at the forefront of your mind when talking with customers about their wishes."  (Brian Godsey, "Think Like a Data Scientist", 2017)

"A learning algorithm for a robot or a software agent to take actions in an environment so as to maximize the sum of rewards through trial and error." (Tomohiro Yamaguchi et al, "Analyzing the Goal-Finding Process of Human Learning With the Reflection Subtask", 2018)

"Artificial intelligence is defined as the branch of science and technology that is concerned with the study of software and hardware to provide machines the ability to learn insights from data and the environment, and the ability to adapt in changing situations with high precision, accuracy and speed." (Amit Ray, "Compassionate Artificial Intelligence", 2018)

"One of the biggest myths is the belief that data science is an autonomous process that we can let loose on our data to find the answers to our problems. In reality, data science requires skilled human oversight throughout the different stages of the process. [...] The second big myth of data science is that every data science project needs big data and needs to use deep learning. In general, having more data helps, but having the right data is the more important requirement. [...] A third data science myth is that modern data science software is easy to use, and so data science is easy to do. [...] The last myth about data science [...] is the belief that data science pays for itself quickly. The truth of this belief depends on the context of the organization. Adopting data science can require significant investment in terms of developing data infrastructure and hiring staff with data science expertise. Furthermore, data science will not give positive results on every project." (John D Kelleher & Brendan Tierney, "Data Science", 2018)

"Any fool can fit a statistical model, given the data and some software. The real challenge is to decide whether it actually fits the data adequately. It might be the best that can be obtained, but still not good enough to use." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)

"With the growing availability of massive data sets and user-friendly analysis software, it might be thought that there is less need for training in statistical methods. This would be naïve in the extreme. Far from freeing us from the need for statistical skills, bigger data and the rise in the number and complexity of scientific studies makes it even more difficult to draw appropriate conclusions. More data means that we need to be even more aware of what the evidence is actually worth." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

🔭Data Science: Standard Error (Just the Quotes)

"Precision is expressed by an international standard, viz., the standard error. It measures the average of the difference between a complete coverage and a long series of estimates formed from samples drawn from this complete coverage by a particular procedure or drawing, and processed by a particular estimating formula." (W Edwards Deming,On the Presentation of the Results of Sample Surveys as Legal Evidence", Journal of the American Statistical Association Vol 49" (268), 1954)

"Tests of the null hypothesis that there is no difference between certain treatments are often made in the analysis of agricultural or industrial experiments in which alternative methods or processes are compared. Such tests are [...] totally irrelevant. What are needed are estimates of magnitudes of effects, with standard errors." (Francis J Anscombe,Discussion on Dr. David’s and Dr. Johnson’s Paper", Journal of the Royal Statistical Society B 18, 1956)

"Evaluation of the statistical reliability of a set of results is not mere calculation of standard errors and confidence limits. The statistician must go far beyond the statistical methods in textbooks. He must evaluate uncertainty in terms of possible uses of the data. Some of this writing is not statistical but draws on assistance from the expert in the subject-matter." (W Edwards Deming,Principles of Professional Statistical Practice", Annals of Mathematical Statistics, 36(6), 1965)

"Pencil and paper for construction of distributions, scatter diagrams, and run-charts to compare small groups and to detect trends are more efficient methods of estimation than statistical inference that depends on variances and standard errors, as the simple techniques preserve the information in the original data." (William E Deming,On Probability as Basis for Action" American Statistician Vol. 29" (4), 1975)

"I contend that the general acceptance of statistical hypothesis testing is one of the most unfortunate aspects of 20th century applied science. Tests for the identity of population distributions, for equality of treatment means, for presence of interactions, for the nullity of a correlation coefficient, and so on, have been responsible for much bad science, much lazy science, and much silly science. A good scientist can manage with, and will not be misled by, parameter estimates and their associated standard errors or confidence limits." (Marks Nester,A Myopic View and History of Hypothesis Testing", 1996)

"The standard error of most statistics is proportional to 1 over the square root of the sample size. God did this, and there is nothing we can do to change it." (Howard Wainer,Improving Tabular Displays, With NAEP Tables as Examples and Inspirations", Journal of Educational and Behavioral Statistics Vol 22" (1), 1997)

"The bounds on the standard deviation are pretty crude but it is surprising how often the rule will pick up gross errors such as confusing the standard error and standard deviation, confusing the variance and the standard deviation, or reporting the mean in one scale and the standard deviation in another scale." (Gerald van Belle,Statistical Rules of Thumb", 2002)

 "The estimate of the standard error (SE) is more sensitive to the Normality assumption than the estimate of the coefficients. There are two options available which do not require this assumption; these are the bootstrap and the robust standard error." (Michael J Campbell, "Statistics at Square Two: Understanding modern statistical applications in medicine" 2nd Ed., 2006)

"Scholars feel the need to present tables of model parameters in academic articles" (perhaps just as evidence that they ran the analysis they claimed to have run), but these tables are rarely interpreted other than for their sign and statistical significance. Most of the numbers in these tables are never even discussed in the text. From the perspective of the applied data analyst, R packages without procedures to compute quantities of scientific interest are woefully incomplete. A better approach focuses on quantities of direct scientific interest rather than uninterpretable model parameters. [...] For each quantity of interest, the user needs some summary that includes a point estimate and a measure of uncertainty such as a standard error, confidence interval, or a distribution. The methods of calculating these differ greatly across theories of inference and methods of analysis. However, from the user’s perspective, the result is almost always the same: the point estimate and uncertainty of some quantity of interest." (Kousuke Imai et al,Toward a Common Framework for Statistical Analysis and Development", Journal of Computational and Graphical Statistics vol. 17, 2008)

"Overlapping confidence intervals do not mean two values are not significantly different. Checking confidence intervals or standard errors will mislead. It’s always best to use the appropriate hypothesis test instead. Your eyeball is not a well-defined statistical procedure." (Alex Reinhart,Statistics Done Wrong: The Woefully Complete Guide", 2015)

"We thus echo the classical Bayesian literature in concluding that ‘noninformative prior information’ is a contradiction in terms. The flat prior carries information just like any other; it represents the assumption that the effect is likely to be large. This is often not true. Indeed, the signal-to-noise ratio s is often very low and then it is necessary to shrink the unbiased estimate. Failure to do so by inappropriately using the flat prior causes overestimation of effects and subsequent failure to replicate them." (Erik van Zwet & Andrew Gelman,A proposal for informative default priors scaled by the standard error of estimates", The American Statistician 76, 2022)

🔭Data Science: Supervised Learning (Just the Quotes)

"Consider any of the heuristics that people have come up with for supervised learning: avoid overfitting, prefer simpler to more complex models, boost your algorithm, bag it, etc. The no free lunch theorems say that all such heuristics fail as often" (appropriately weighted) as they succeed. This is true despite formal arguments some have offered trying to prove the validity of some of these heuristics." (David H Wolpert,The lack of a priori distinctions between learning algorithms", Neural Computation Vol. 8(7), 1996)

"A sub-area of machine learning concerned with how an agent ought to take actions in an environment so as to maximize some notion of long-term reward. Reinforcement learning algorithms attempt to find a policy that maps states of the world to the actions the agent ought to take in those states. Differently from supervised learning, in this case there is no target value for each input pattern, only a reward based of how good or bad was the action taken by the agent in the existent environment." (Marley Vellasco et al,Hierarchical Neuro-Fuzzy Systems" Part II, Encyclopedia of Artificial Intelligence, 2009)

"Supervised learning, or function approximation, is simply fitting data to a function of any variety.  […] Unsupervised learning involves figuring out what makes the data special. […] Reinforcement learning involves figuring out how to play a multistage game with rewards and payoffs. Think of it as the algorithms that optimize the life of something." (Matthew Kirk,Thoughtful Machine Learning", 2015)

"A major challenge in unsupervised learning is evaluating whether the algorithm learned something useful. Unsupervised learning algorithms are usually applied to data that does not contain any label information, so we don’t know what the right output should be. Therefore, it is very hard to say whether a model 'did well'. [...] As a consequence, unsupervised algorithms are used often in an exploratory setting, when a data scientist wants to understand the data better, rather than as part of a larger automatic system. Another common application for unsupervised algorithms is as a preprocessing step for supervised algorithms. Learning a new representation of the data can sometimes improve the accuracy of supervised algorithms, or can lead to reduced memory and time consumption." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"Gradient boosted decision trees are among the most powerful and widely used models for supervised learning. Their main drawback is that they require careful tuning of the parameters and may take a long time to train. Similarly to other tree-based models, the algorithm works well without scaling and on a mixture of binary and continuous features. As with other tree-based models, it also often does not work well on high-dimensional sparse data." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"A form of machine learning in which the goal is to identify regularities in the data. These regularities may include clusters of similar instances within the data or regularities between attributes. In contrast to supervised learning, in unsupervised learning no target attribute is defined in the data set." (John D Kelleher & Brendan Tierney,Data science", 2018)

"The premise of classification is simple: given a categorical target variable, learn patterns that exist between instances composed of independent variables and their relationship to the target. Because the target is given ahead of time, classification is said to be supervised machine learning because a model can be trained to minimize error between predicted and actual categories in the training data. Once a classification model is fit, it assigns categorical labels to new instances based on the patterns detected during training." (Benjamin Bengfort et al,Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

"Whereas in supervised learning the training, validation, and test data sets are assumed to exist before the training begins, in RL the agent generates its data itself by interacting with the environment. In many contexts, such as in games, this is a huge simplification. Consider the game of chess: instead of loading thousands of histor‐ ical human-played chess games into a computer, an RL agent can generate thousands or millions of games itself by playing against another chess engine or another version of itself, for instance." (Yves Hilpisch, "Artificial Intelligence in Finance A Python-Based Guide", 2021)

"Machine Learning (ML) is a subfield of AI that enables computers to acquire knowledge and improve performance without explicit programming. Different from traditional programming, where you must write the code line by line to provide instructions, ML allows machines to learn from data and make predictions or take actions based on that learned information. This approach empowers machines to discover patterns, relationships, and insights that may not be immediately apparent to human programmers. ML relies on two types of learning: supervised learning and unsupervised learning." (Angelica Lo Duca, "Data Storytelling with Generative AI: Using Python and Altair", 2024) 

"The label spreading algorithm is another semi-supervised shallow embedding algorithm.  It was built in order to overcome one big limitation of the label propagation method: the   initial labeling. Indeed, according to the label propagation algorithm, the initial labels cannot be modified in the training process and, in each iteration, they are forced to be equal to their original value. This constraint could generate incorrect results when the initial labeling is affected by errors or noise. As a consequence, the error will be propagated in all nodes of the input graph." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"The main difference between unsupervised and supervised embedding methods essentially lies in the task they attempt to solve. Indeed, if unsupervised shallow embedding algorithms try to learn a good graph, node, or edge representation in order to understand the underlying structure, the supervised algorithms try to find the best solution for a prediction task such as node classification, label prediction, or graph classification." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

🔭Data Science: Convergence (Just the Quotes)

"A good estimator will be unbiased and will converge more and more closely (in the long run) on the true value as the sample size increases. Such estimators are known as consistent. But consistency is not all we can ask of an estimator. In estimating the central tendency of a distribution, we are not confined to using the arithmetic mean; we might just as well use the median. Given a choice of possible estimators, all consistent in the sense just defined, we can see whether there is anything which recommends the choice of one rather than another. The thing which at once suggests itself is the sampling variance of the different estimators, since an estimator with a small sampling variance will be less likely to differ from the true value by a large amount than an estimator whose sampling variance is large." (Michael J Moroney, "Facts from Figures", 1951)

"Sometimes the most important fit statistic you can get is ‘convergence not met’ - it can tell you something is wrong with your model." (Oliver Schabenberger, "Applied Statistics in Agriculture Conference", 2006) 

"The central limit theorem differs from laws of large numbers because random variables vary and so they differ from constants such as population means. The central limit theorem says that certain independent random effects converge not to a constant population value such as the mean rate of unemployment but rather they converge to a random variable that has its own Gaussian bell-curve description." (Bart Kosko, "Noise", 2006)

"Each learning algorithm dictates a certain model that comes with a set of assumptions. This inductive bias leads to error if the assumptions do not hold for the data. Learning is an ill-posed problem and with finite data, each algorithm converges to a different solution and fails under different circumstances. The performance of a learner may be fine-tuned to get the highest possible accuracy on a validation set, but this finetuning is a complex task and still there are instances on which even the best learner is not accurate enough. The idea is that there may be another base-learner learner that is accurate on these. By suitably combining multiple base learners then, accuracy can be improved." (Ethem Alpaydin, "Introduction to Machine Learning" 2nd Ed, 2010)

"Regularization works because it is the sum of the coefficients of the predictor variables, therefore it’s important that they’re on the same scale or the regularization may find it difficult to converge, and variables with larger absolute coefficient values will greatly influence it, generating an infective regularization. It’s good practice to standardize the predictor values or bind them to a common min‐max, such as the [‐1,+1] range." (Luca Massaron & John P Mueller, "Python for Data Science For Dummies", 2015)

"Cluster analysis refers to the grouping of observations so that the objects within each cluster share similar properties, and properties of all clusters are independent of each other. Cluster algorithms usually optimize by maximizing the distance among clusters and minimizing the distance between objects in a cluster. Cluster analysis does not complete in a single iteration but goes through several iterations until the model converges. Model convergence means that the cluster memberships of all objects converge and don’t change with every new iteration." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"Theoretically, the normal distribution is most famous because many distributions converge to it, if you sample from them enough times and average the results. This applies to the binomial distribution, Poisson distribution and pretty much any other distribution you’re likely to encounter (technically, any one for which the mean and standard deviation are finite)." (Field Cady, "The Data Science Handbook", 2017)

"Early stopping and regularization can ensure network generalization when you apply them properly. [...] With early stopping, the choice of the validation set is also important. The validation set should be representative of all points in the training set. When you use Bayesian regularization, it is important to train the network until it reaches convergence. The sum-squared error, the sum-squared weights, and the effective number of parameters should reach constant values when the network has converged. With both early stopping and regularization, it is a good idea to train the network starting from several different initial conditions. It is possible for either method to fail in certain circumstances. By testing several different initial conditions, you can verify robust network performance." (Mark H Beale et al, "Neural Network Toolbox™ User's Guide", 2017)

"The high generalization error in a neural network may be caused by several reasons. First, the data itself might have a lot of noise, in which case there is little one can do in order to improve accuracy. Second, neural networks are hard to train, and the large error might be caused by the poor convergence behavior of the algorithm. The error might also be caused by high bias, which is referred to as underfitting. Finally, overfitting (i.e., high variance) may cause a large part of the generalization error. In most cases, the error is a combination of more than one of these different factors." (Charu C Aggarwal, "Neural Networks and Deep Learning: A Textbook", 2018)

🔭Data Science: Normality (Just the Quotes)

"Logging size transforms the original skewed distribution into a more symmetrical one by pulling in the long right tail of the distribution toward the mean. The short left tail is, in addition, stretched. The shift toward symmetrical distribution produced by the log transform is not, of course, merely for convenience. Symmetrical distributions, especially those that resemble the normal distribution, fulfill statistical assumptions that form the basis of statistical significance testing in the regression model." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"When the statistician looks at the outside world, he cannot, for example, rely on finding errors that are independently and identically distributed in approximately normal distributions. In particular, most economic and business data are collected serially and can be expected, therefore, to be heavily serially dependent. So is much of the data collected from the automatic instruments which are becoming so common in laboratories these days. Analysis of such data, using procedures such as standard regression analysis which assume independence, can lead to gross error. Furthermore, the possibility of contamination of the error distribution by outliers is always present and has recently received much attention. More generally, real data sets, especially if they are long, usually show inhomogeneity in the mean, the variance, or both, and it is not always possible to randomize." (George E P Box, "Some Problems of Statistics and Everyday Life", Journal of the American Statistical Association, Vol. 74 (365), 1979)

"Linear regression assumes that in the population a normal distribution of error values around the predicted Y is associated with each X value, and that the dispersion of the error values for each X value is the same. The assumptions imply normal and similarly dispersed error distributions." (Fred C Pampel, "Linear Regression: A primer", 2000)

"The central limit theorem says that, under conditions almost always satisfied in the real world of experimentation, the distribution of such a linear function of errors will tend to normality as the number of its components becomes large. The tendency to normality occurs almost regardless of the individual distributions of the component errors. An important proviso is that several sources of error must make important contributions to the overall error and that no particular source of error dominate the rest." (George E P Box et al, "Statistics for Experimenters: Design, discovery, and innovation" 2nd Ed., 2005)

"Two things explain the importance of the normal distribution: (1) The central limit effect that produces a tendency for real error distributions to be 'normal like'. (2) The robustness to nonnormality of some common statistical procedures, where 'robustness' means insensitivity to deviations from theoretical normality." (George E P Box et al, "Statistics for Experimenters: Design, discovery, and innovation" 2nd Ed., 2005)

"In error analysis the so-called 'chi-squared' is a measure of the agreement between the uncorrelated internal and the external uncertainties of a measured functional relation. The simplest such relation would be time independence. Theory of the chi-squared requires that the uncertainties be normally distributed. Nevertheless, it was found that the test can be applied to most probability distributions encountered in practice." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"To fulfill the requirements of the theory underlying uncertainties, variables with random uncertainties must be independent of each other and identically distributed. In the limiting case of an infinite number of such variables, these are called normally distributed. However, one usually speaks of normally distributed variables even if their number is finite." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"[…] the Central Limit Theorem says that if we take any sequence of small independent random quantities, then in the limit their sum (or average) will be distributed according to the normal distribution. In other words, any quantity that can be viewed as the sum of many small independent random effects. will be well approximated by the normal distribution. Thus, for example, if one performs repeated measurements of a fixed physical quantity, and if the variations in the measurements across trials are the cumulative result of many independent sources of error in each trial, then the distribution of measured values should be approximately normal." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Statistical inference is really just the marriage of two concepts that we’ve already discussed: data and probability (with a little help from the central limit theorem)." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"The central limit theorem tells us that in repeated samples, the difference between the two means will be distributed roughly as a normal distribution." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"According to the central limit theorem, it doesn’t matter what the raw data look like, the sample variance should be proportional to the number of observations and if I have enough of them, the sample mean should be normal." (Kristin H Jarman, "The Art of Data Analysis: How to answer almost any question using basic statistics", 2013)

"For a confidence interval, the central limit theorem plays a role in the reliability of the interval because the sample mean is often approximately normal even when the underlying data is not. A prediction interval has no such protection. The shape of the interval reflects the shape of the underlying distribution. It is more important to examine carefully the normality assumption by checking the residuals […]." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"When data is not normal, the reason the formulas are working is usually the central limit theorem. For large sample sizes, the formulas are producing parameter estimates that are approximately normal even when the data is not itself normal. The central limit theorem does make some assumptions and one is that the mean and variance of the population exist. Outliers in the data are evidence that these assumptions may not be true. Persistent outliers in the data, ones that are not errors and cannot be otherwise explained, suggest that the usual procedures based on the central limit theorem are not applicable." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"Why are you testing your data for normality? For large sample sizes the normality tests often give a meaningful answer to a meaningless question (for small samples they give a meaningless answer to a meaningful question)." (Greg Snow, "R-Help", 2014)

"The variance, the standard deviation, mean absolute deviation, and median absolute deviation from the median are not equivalent estimates, even in the case where the data comes from a normal distribution. In fact, the standard deviation is always greater than the mean absolute deviation, which itself is greater than the median absolute deviation. Sometimes, the median absolute deviation is multiplied by a constant scaling factor (it happens to work out to 1.4826) to put MAD on the same scale as the standard deviation in the case of a normal distribution." (Peter C Bruce & Andrew G Bruce, "Statistics for Data Scientists: 50 Essential Concepts", 2016)

"At very small time scales, the motion of a particle is more like a random walk, as it gets jostled about by discrete collisions with water molecules. But virtually any random movement on small time scales will give rise to Brownian motion on large time scales, just so long as the motion is unbiased. This is because of the Central Limit Theorem, which tells us that the aggregate of many small, independent motions will be normally distributed." (Field Cady, "The Data Science Handbook", 2017)

"Theoretically, the normal distribution is most famous because many distributions converge to it, if you sample from them enough times and average the results. This applies to the binomial distribution, Poisson distribution and pretty much any other distribution you’re likely to encounter (technically, any one for which the mean and standard deviation are finite)." (Field Cady, "The Data Science Handbook", 2017)

"Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their 'raw' form they are often asymmetric." (Terence C Mills, "Applied Time Series Analysis: A practical guide to modeling and forecasting", 2019)

"[...] the Central Limit Theorem [...] says that the distribution of sample means tends towards the form of a normal distribution with increasing sample size, almost regardless of the shape of the original data distribution." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"Although the normality assumption is a good approximation for many real-world phenomena, such as in physics, it is not appropri‐ ate and can even be dangerous when it comes to financial returns." (Yves Hilpisch, "Artificial Intelligence in Finance A Python-Based Guide", 2021)

"Almost no financial return sample data set passes statistical normality tests. Beyond the fact that it has proven useful in other domains, a major reason why this assumption is found in so many financial models is that it leads to elegant and relatively simple mathematical models, calculations, and proofs." (Yves Hilpisch, "Artificial Intelligence in Finance A Python-Based Guide", 2021)

🔭Data Science: Similarity (Just the Quotes)

"Many of the basic functions performed by neural networks are mirrored by human abilities. These include making distinctions between items (classification), dividing similar things into groups (clustering), associating two or more things (associative memory), learning to predict outcomes based on examples (modeling), being able to predict into the future (time-series forecasting), and finally juggling multiple goals and coming up with a good- enough solution (constraint satisfaction)." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"While classification is important, it can certainly be overdone. Making too fine a distinction between things can be as serious a problem as not being able to decide at all. Because we have limited storage capacity in our brain (we still haven't figured out how to add an extender card), it is important for us to be able to cluster similar items or things together. Not only is clustering useful from an efficiency standpoint, but the ability to group like things together (called chunking by artificial intelligence practitioners) is a very important reasoning tool. It is through clustering that we can think in terms of higher abstractions, solving broader problems by getting above all of the nitty-gritty details." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory." (Salvatore Greco et al, "Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach", 2009)

"Cluster analysis refers to the grouping of observations so that the objects within each cluster share similar properties, and properties of all clusters are independent of each other. Cluster algorithms usually optimize by maximizing the distance among clusters and minimizing the distance between objects in a cluster. Cluster analysis does not complete in a single iteration but goes through several iterations until the model converges. Model convergence means that the cluster memberships of all objects converge and don’t change with every new iteration." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"Understanding transcends context, since the different contexts collapse according to their previously unknown similarity, which the principle contains. That is what understanding does. And you actually feel it in your brain when it happens. Your 'cognitive load' decreases, your level of stress and anxiety decrease, and your emotional state improves." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

"Graph embedding is the representation of the nodes and relationships in a graph asfeature vectors. [...] Graph embedding uses graph data slightly differently than in connected feature extraction. It enables us to represent entire graphs, or subsets of graph data, in a numerical format ready for machine learning tasks. This is especially useful for unsu‐pervised learning, where the data is not categorized because it pulls in more contextual information through relationships. Graph embedding is also useful for data exploration, computing similarity between entities, and reducing dimensionality to aid in statistical analysis." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"When using community detection algorithms, be conscious of the density of the relationships. If the graph is very dense, you may end up with all nodes congregating in one or just a few clusters. You can counteract this by filtering by degree, relationship weights, or similarity metrics. On the other hand, if the graph is too sparse with few connected nodes, you may end up with each node in its own cluster. In this case, try to incorporate additional relationship types that carry more relevant information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"To compute the similarity between items, we must define a similarity measure. Cosine similarity is the standard metric in item-based recommendation approaches: it determines the similarity between two vectors by calculating the cosine of the angle between them In machine learning applications, this measure is often used to compare two text documents, which are represented as vectors of terms [...] prediction represents a small portion of possible uses). In this learning path, data visualization plays a key role because it allows us to access and analyze data from a different perspective." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

"Data drift manifests in several distinct ways. Input drift typically shows up as an increase in adversarial or malformed queries that deviate from the original training or design expectations. This can stress the system’s robustness and degrade output quality. Retriever drift occurs when the relevance of the documents returned by retrieval components declines, even if the retrieval algorithms and configurations remain unchanged. Similarly, embedding drift arises when the vector representations used to compare semantic similarity become less effective, causing retrieval systems to fail despite stable system parameters." (Abi Aryan, "LLMOps: Managing Large Language Models in Production", 2025)

"LLM-centric workloads change everything. Now the raw material is heterogeneous text, code, images, audio, and chat logs whose value depends on semantic richness - that is, the informational value of the content - rather than a rigid structure. Pipelines must tokenize, chunk, embed, and version this content; store it in vector indexes for similarity search; and apply filters for personally identifiable information, toxicity, and licensing constraints. Instead of ETL jobs, teams run continuous ingestion and reembedding loops so that RAG systems stay fresh, and they log every prompt–response pair so that the inputs and outputs can be evaluated and improve the future performance of this system. Data quality in this context is judged by grounding, factuality, and bias metrics - attributes that require automated red-teaming and humanin-the-loop (HITL) review rather than the data structure violation checks of the past." (Abi Aryan, "LLMOps: Managing Large Language Models in Production", 2025)

"Vector databases are designed to store and index highdimensional embeddings - dense numeric vectors that capture the semantic meaning of text, images, audio, or other content. Instead of looking for exact matches, they use approximate nearest neighbor (ANN) algorithms to return the items whose vectors lie closest to a query vector in that multidimensional space. This makes them the engine behind semantic search, recommendation systems, image-or-audio similarity matching, and retrieval augmented generation (RAG) pipelines that supply LLM prompts with relevant context in milliseconds." (Abi Aryan, "LLMOps: Managing Large Language Models in Production", 2025)

"GenAI is a type of AI that can generate new content, such as text, images, music, or video. It is a class of AI models that is designed to generate new data or content that resembles existing data. Traditional AI models tend to be primarily focused on classification, regression, or decision making, but GenAI models learn the underlying patterns and structure of the input data and use this knowledge to create new, synthetic instances that are similar to the original data." (Bennie Haelen, "ML and Generative AI in the Data Lakehouse Building and Deploying AI Applications at Scale", 2026)

"The foundational form of RAG, often called naive RAG, follows a straightforward pattern. A pipeline retrieves supporting context from external sources such as enterprise documents, knowledge bases, or structured datasets and appends that information to the model’s prompt before inference. In the most common implementation, each document is converted into an embedding, a numerical representation of its semantic meaning, using either the same foundation model or a specialized embedding model. When a user submits a query, the system performs a vector similarity search to find documents whose embeddings most closely match the query’s vector representation, and the retrieved content is concatenated with the user query before being passed to the language model." (Bennie Haelen, "ML and Generative AI in the Data Lakehouse Building and Deploying AI Applications at Scale", 2026) 

🔭Data Science: Decision-Making (Just the Quotes)

"There are, indeed, plenty of ways in which statistics can help in the process of decision-taking. But exaggerated claims for the role they can play merely serve to confuse rather than clarify issues of public policy, and lead those responsible for action to oscillate between over-confidence and over-scepticism in using them." (Ely Devons, "Essays in Economics", 1961)

"Statistics is a body of methods and theory applied to numerical evidence in making decisions in the face of uncertainty." (Lawrence Lapin, "Statistics for Modern Business Decisions", 1973)

"A model for simulating dynamic system behavior requires formal policy descriptions to specify how individual decisions are to be made. Flows of information are continuously converted into decisions and actions. No plea about the inadequacy of our understanding of the decision-making processes can excuse us from estimating decision-making criteria. To omit a decision point is to deny its presence - a mistake of far greater magnitude than any errors in our best estimate of the process." (Jay W Forrester, "Policies, decisions and information sources for modeling", 1994)

"The science of statistics may be described as exploring, analyzing and summarizing data; designing or choosing appropriate ways of collecting data and extracting information from them; and communicating that information. Statistics also involves constructing and testing models for describing chance phenomena. These models can be used as a basis for making inferences and drawing conclusions and, finally, perhaps for making decisions." (Fergus Daly et al, "Elements of Statistics", 1995)

"So we pour in data from the past to fuel the decision-making mechanisms created by our models, be they linear or nonlinear. But therein lies the logician's trap: past data from real life constitute a sequence of events rather than a set of independent observations, which is what the laws of probability demand. [...] It is in those outliers and imperfections that the wildness lurks." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"Under conditions of uncertainty, both rationality and measurement are essential to decision-making. Rational people process information objectively: whatever errors they make in forecasting the future are random errors rather than the result of a stubborn bias toward either optimism or pessimism. They respond to new information on the basis of a clearly defined set of preferences. They know what they want, and they use the information in ways that support their preferences." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"Delay time, the time between causes and their impacts, can highly influence systems. Yet the concept of delayed effect is often missed in our impatient society, and when it is recognized, it’s almost always underestimated. Such oversight and devaluation can lead to poor decision making as well as poor problem solving, for decisions often have consequences that don’t show up until years later. Fortunately, mind mapping, fishbone diagrams, and creativity/brainstorming tools can be quite useful here." (Stephen G Haines, "The Manager's Pocket Guide to Strategic and Business Planning", 1998)

"Just as dynamics arise from feedback, so too all learning depends on feedback. We make decisions that alter the real world; we gather information feedback about the real world, and using the new information we revise our understanding of the world and the decisions we make to bring our perception of the state of the system closer to our goals." (John D Sterman, "Business dynamics: Systems thinking and modeling for a complex world", 2000)

"Balanced Scorecard theory is flawed because it presents managers with a scorecard which gives no score – that is no single-valued measure how they have performed. Thus managers evaluated with such a system […] have no way to make principled or purposeful decisions." (Michael Jensen,"Value Maximization, Stakeholder Theory, and the Corporate Objective Function", Journal of Applied Corporate Finance, 2001)

"Compared to traditional statistical studies, which are often hindsight, the field of data mining finds patterns and classifications that look toward and even predict the future. In summary, data mining can (1) provide a more complete understanding of data by finding patterns previously not seen and (2) make models that predict, thus enabling people to make better decisions, take action, and therefore mold future events." (Robert Nisbet et al, "Handbook of statistical analysis and data mining applications", 2009)

"What is so unconventional about the statistical way of thinking? First, statisticians do not care much for the popular concept of the statistical average; instead, they fixate on any deviation from the average. They worry about how large these variations are, how frequently they occur, and why they exist. [...] Second, variability does not need to be explained by reasonable causes, despite our natural desire for a rational explanation of everything; statisticians are frequently just as happy to pore over patterns of correlation. [...] Third, statisticians are constantly looking out for missed nuances: a statistical average for all groups may well hide vital differences that exist between these groups. Ignoring group differences when they are present frequently portends inequitable treatment. [...] Fourth, decisions based on statistics can be calibrated to strike a balance between two types of errors. Predictably, decision makers have an incentive to focus exclusively on minimizing any mistake that could bring about public humiliation, but statisticians point out that because of this bias, their decisions will aggravate other errors, which are unnoticed but serious. [...] Finally, statisticians follow a specific protocol known as statistical testing when deciding whether the evidence fits the crime, so to speak. Unlike some of us, they don’t believe in miracles. In other words, if the most unusual coincidence must be contrived to explain the inexplicable, they prefer leaving the crime unsolved." (Kaiser Fung, "Numbers Rule the World", 2010)

"Statistics is the scientific discipline that provides methods to help us make sense of data. […] The field of statistics teaches us how to make intelligent judgments and informed decisions in the presence of uncertainty and variation." (Roxy Peck & Jay L Devore, "Statistics: The Exploration and Analysis of Data" 7th Ed, 2012)

"There is convincing evidence that data-driven decision-making and big data technologies substantially improve business performance. Data science supports data-driven decision-making - and sometimes conducts such decision-making automatically - and depends upon technologies for 'big data' storage and engineering, but its principles are separate." (Foster Provost & Tom Fawcett, "Data Science for Business", 2013)

"A study that leaves out data is waving a big red flag. A decision to include or exclude data sometimes makes all the difference in the world. This decision should be based on the relevance and quality of the data, not on whether the data support or undermine a conclusion that is expected or desired." (Gary Smith, "Standard Deviations", 2014)

"It is important to remember that predictive data analytics models built using machine learning techniques are tools that we can use to help make better decisions within an organization and are not an end in themselves. It is paramount that, when tasked with creating a predictive model, we fully understand the business problem that this model is being constructed to address and ensure that it does address it." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, worked examples, and case studies", 2015)

"Big data is, in a nutshell, large amounts of data that can be gathered up and analyzed to determine whether any patterns emerge and to make better decisions." (Daniel Covington, Analytics: Data Science, Data Analysis and Predictive Analytics for Business, 2016)

"Machine learning is often associated with the automation of decision making, but in practice, the process of constructing a predictive model generally requires a human in the loop. While computers are good at fast, accurate numerical computation, humans are instinctively and instantly able to identify patterns. The bridge between these two necessary skill sets lies in visualization - the precise and accurate rendering of data by a computer in visual terms and the immediate assignation of meaning to that data by humans." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

"The goal of data science is to improve decision making by basing decisions on insights extracted from large data sets. As a field of activity, data science encompasses a set of principles, problem definitions, algorithms, and processes for extracting nonobvious and useful patterns from large data sets. It is closely related to the fields of data mining and machine learning, but it is broader in scope." (John D Kelleher & Brendan Tierney, "Data Science", 2018)

"Ideally, a decision maker or a forecaster will combine the outside view and the inside view - or, similarly, statistics plus personal experience. But it’s much better to start with the statistical view, the outside view, and then modify it in the light of personal experience than it is to go the other way around. If you start with the inside view you have no real frame of reference, no sense of scale - and can easily come up with a probability that is ten times too large, or ten times too small." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)

"Our machines are helpers, not decision makers. Their insights are not the final word in the discussion, merely the work of our most nimble observers who can ramp up time spent on analysis by factors that our counterparts even a generation ago would have a hard time believing." (Kate Strachnyi, "ColorWise: A Data Storyteller’s Guide to the Intentional Use of Color", 2023)

🔭Data Science: Misconceptions (Just the Quotes)

"Science does not begin with facts; one of its tasks is to uncover the facts by removing misconceptions." (Lancelot L Whyte, "Accent on Form", 1954)

"A common misconception is that an effect exists only if it is statistically significant and that it does not exist if it is not [statistically significant]." (Jonas Ranstam, "A common misconception about p-value and its consequences", Acta Orthopaedica Scandinavica 67, 1996)

"The standard deviation (often SD) is a measure of variability. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. For data with a normal distribution, about 95% of individu als will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. About 95% of observa tions of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. We may choose a different summary statistic, how ever, when data have a skewed distribution." (Douglas G Altman & J Martin Bland, "Statistics Notes: Standard Deviations And Standard Errors", British Medical Journal Vol. 331 (7521) 2005)

"[...] the term statistical misconception refers to any of several widely held but incorrect notions about statistical concepts, about procedures for analyzing data and about the meaning of results produced by such analyses. To illustrate, many people think that (1) normal curves are bell shaped, (2) a correlation coeffi cient should never be used to address questios of causality, and (3) the level of signifi cance dictates the probability of a Type I error. Some people, of course, have only one or two (rather than all three) of these misconceptions, and a few individuals realize that all three of those beliefs are false."(Schuyler W Huck, "Statistical Misconceptions", 2008)

"Science would be better understood if we called theories ‘misconceptions’ from the outset, instead of only after we have discovered their successors." (David Deutsch, "Beginning of Infinity", 2011)

"A popular misconception holds that the era of Big Data means the end of a need for sampling. In fact, the proliferation of data of varying quality and relevance reinforces the need for sampling as a tool to work efficiently with a variety of data, and minimize bias. Even in a Big Data project, predictive models are typically developed and piloted with samples." (Peter C Bruce & Andrew G Bruce, "Statistics for Data Scientists: 50 Essential Concepts", 2016)

"An oft-repeated rule of thumb in any sort of statistical model fitting is 'you can't fit a model with more parameters than data points'. This idea appears to be as wide-spread as it is incorrect. On the contrary, if you construct your models carefully, you can fit models with more parameters than datapoints [...]. A model with more parameters than datapoints is known as an under-determined system, and it's a common misperception that such a model cannot be solved in any circumstance. [...] this misconception, which I like to call the 'model complexity myth' [...] is not true in general, it is true in the specific case of simple linear models, which perhaps explains why the myth is so pervasive." (Jake Vanderplas", "The Model Complexity Myth", 2015)

🔭Data Science: Boosting (Just the Quotes)

"In bagging, generating complementary base-learners is left to chance and to the unstability of the learning method. In boosting, we actively try to generate complementary base-learners by training the next learner boosting on the mistakes of the previous learners." (Ethem Alpaydin, "Introduction to Machine Learning" 2nd Ed, 2010)

"The key idea behind boosting techniques is to use a weak learning algorithm to build a strong learner, that is, an accurate PAC-learning algorithm. To do so, boosting techniques use an ensemble method: they combine different base classifiers returned by a weak learner to create a more accurate predictor." (Mehryar Mohri et al, "Foundations of Machine Learning", 2012)

"Decision trees are also discriminative models. Decision trees are induced by recursively partitioning the feature space into regions belonging to the different classes, and consequently they define a decision boundary by aggregating the neighboring regions belonging to the same class. Decision tree model ensembles based on bagging and boosting are also discriminative models." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"Boosting defines an objective function to measure the performance of a model given a certain set of parameters. The objective function contains two parts: regularization and training loss, both of which add to one another. The training loss measures how predictive our model is on the training data. The most commonly used training loss function includes mean squared error and logistic regression. The regularization term controls the complexity of the model, which helps avoid overfitting." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"Boosting is a non-linear flexible regression technique that helps increase the accuracy of trees by assigning more weights to wrong predictions. The reason for inducing more weight is so the model can emphasize more on these wrongly predicted samples and tune itself to increase accuracy. The gradient boosting method solves the inherent problem in boosting trees (i.e., low speed and human interpretability). The algorithm supports parallelism by specifying the number of threads." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"Gradient boosted decision trees are among the most powerful and widely used models for supervised learning. Their main drawback is that they require careful tuning of the parameters and may take a long time to train. Similarly to other tree-based models, the algorithm works well without scaling and on a mixture of binary and continuous features. As with other tree-based models, it also often does not work well on high-dimensional sparse data." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"In Boosting, the selection of samples is done by giving more and more weight to hard-to-classify observations. Gradient boosting classification produces a prediction model in the form of an ensemble of weak predictive models, usually decision trees. It generalizes the model by optimizing for the arbitrary differentiable loss function. At each stage, regression trees fit on the negative gradient of binomial or multinomial deviance loss function." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"The danger of overfitting is particularly severe when the training data is not a perfect gold standard. Human class annotations are often subjective and inconsistent, leading boosting to amplify the noise at the expense of the signal. The best boosting algorithms will deal with overfitting though regularization. The goal will be to minimize the number of non-zero coefficients, and avoid large coefficients that place too much faith in any one classifier in the ensemble." (Steven S Skiena, "The Data Science Design Manual", 2017)

"The no free lunch theorems set limits on the range of optimality of any method. That is, each methodology has a ‘catchment area’ where it is optimal or nearly so. Often, intuitively, if the optimality is particularly strong then the effectiveness of the methodology falls off more quickly outside its catchment area than if its optimality were not so strong. Boosting is a case in point: it seems so well suited to binary classification that efforts to date to extend it to give effective classification (or regression) more generally have not been very successful. Overall, it remains to characterize the catchment areas where each class of predictors performs optimally, performs generally well, or breaks down." (Bertrand S Clarke & Jennifer L. Clarke, "Predictive Statistics: Analysis and Inference beyond Models", 2018)

"A recurring theme in machine learning is combining predictions across multiple models. There are techniques called bagging and boosting which seek to tweak the data and fit many estimates to it. Averaging across these can give a better prediction than any one model on its own. But here a serious problem arises: it is then very hard to explain what the model is (often referred to as a 'black box'). It is now a mixture of many, perhaps a thousand or more, models." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)