"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1875)
"It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence." (Sir Francis Galton, "Natural Inheritance", 1889)
"Physical research by experimental methods is both a broadening and a narrowing field. There are many gaps yet to be filled, data to be accumulated, measurements to be made with great precision, but the limits within which we must work are becoming, at the same time, more and more defined." (Elihu Thomson, "Annual Report of the Board of Regents of the Smithsonian Institution", 1899)
"Statistics may, for instance, be called the science of counting. Counting appears at first sight to be a very simple operation, which any one can perform or which can be done automatically; but, as a matter of fact, when we come to large numbers, e.g., the population of the United Kingdom, counting is by no means easy, or within the power of an individual; limits of time and place alone prevent it being so carried out, and in no way can absolute accuracy be obtained when the numbers surpass certain limits." (Sir Arthur L Bowley, "Elements of Statistics", 1901)
"The usefulness of the models in constructing a testable theory of the process is severely limited by the quickly increasing number of parameters which must be estimated in order to compare the predictions of the models with empirical results" (Anatol Rapoport, "Prisoner's Dilemma: A study in conflict and cooperation", 1965)
"A real change of theory is not a change of equations - it is a change of mathematical structure, and only fragments of competing theories, often not very important ones conceptually, admit comparison with each other within a limited range of phenomena." (Yuri I Manin, "Mathematics and Physics", 1981)
"Models are often used to decide issues in situations marked by uncertainty. However statistical differences from data depend on assumptions about the process which generated these data. If the assumptions do not hold, the inferences may not be reliable either. This limitation is often ignored by applied workers who fail to identify crucial assumptions or subject them to any kind of empirical testing. In such circumstances, using statistical procedures may only compound the uncertainty." (David A Greedman & William C Navidi, "Regression Models for Adjusting the 1980 Census", Statistical Science Vol. 1 (1), 1986)
"[…] an honest exploratory study should indicate how many comparisons were made […] most experts agree that large numbers of comparisons will produce apparently statistically significant findings that are actually due to chance. The data torturer will act as if every positive result confirmed a major hypothesis. The honest investigator will limit the study to focused questions, all of which make biologic sense. The cautious reader should look at the number of ‘significant’ results in the context of how many comparisons were made." (James L Mills, "Data torturing", New England Journal of Medicine, 1993)
"In spite of the insurmountable computational limits, we continue to pursue the many problems that possess the characteristics of organized complexity. These problems are too important for our well being to give up on them. The main challenge in pursuing these problems narrows down fundamentally to one question: how to deal with systems and associated problems whose complexities are beyond our information processing limits? That is, how can we deal with these problems if no computational power alone is sufficient?" (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"The larger, more detailed and complex the model - the less abstract the abstraction – the smaller the number of people capable of understanding it and the longer it takes for its weaknesses and limitations to be found out." (John Adams, "Risk", 1995)
"[...] the NFL theorems mean that if an algorithm does particularly well on average for one class of problems then it must do worse on average over the remaining problems. In particular, if an algorithm performs better than random search on some class of problems then in must perform worse than random search on the remaining problems. Thus comparisons reporting the performance of a particular algorithm with a particular parameter setting on a few sample problems are of limited utility. While such results do indicate behavior on the narrow range of problems considered, one should be very wary of trying to generalize those results to other problems." (David H Wolpert & William G Macready, "No free lunch theorems for optimization", IEEE Transactions on Evolutionary Computation 1 (1), 1997)
"No comparison between two values can be global. A simple comparison between the current figure and some previous value and convey the behavior of any time series. […] While it is simple and easy to compare one number with another number, such comparisons are limited and weak. They are limited because of the amount of data used, and they are weak because both of the numbers are subject to the variation that is inevitably present in weak world data. Since both the current value and the earlier value are subject to this variation, it will always be difficult to determine just how much of the difference between the values is due to variation in the numbers, and how much, if any, of the difference is due to real changes in the process." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)
"[…] an obvious difference between our best classifiers and human learning is the number of examples required in tasks such as object detection. […] the difficulty of a learning task depends on the size of the required hypothesis space. This complexity determines in turn how many training examples are needed to achieve a given level of generalization error. Thus the complexity of the hypothesis space sets the speed limit and the sample complexity for learning." (Tomaso Poggio & Steve Smale, "The Mathematics of Learning: Dealing with Data", Notices of the AMS, 2003)
"Every number has its limitations; every number is a product of choices that inevitably involve compromise. Statistics are intended to help us summarize, to get an overview of part of the world’s complexity. But some information is always sacrificed in the process of choosing what will be counted and how. Something is, in short, always missing. In evaluating statistics, we should not forget what has been lost, if only because this helps us understand what we still have."
"Statistics depend on collecting information. If questions go unasked, or if they are asked in ways that limit responses, or if measures count some cases but exclude others, information goes ungathered, and missing numbers result. Nevertheless, choices regarding which data to collect and how to go about collecting the information are inevitable." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)
"The Bayesian approach is based on the following postulates: (B1) Probability describes degree of belief, not limiting frequency. As such, we can make probability statements about lots of things, not just data which are subject to random variation. […] (B2) We can make probability statements about parameters, even though they are fixed constants. (B3) We make inferences about a parameter θ by producing a probability distribution for θ. Inferences, such as point estimates and interval estimates, may then be extracted from this distribution."
"A population that grows logistically, initially increases exponentially; then the growth lows down and eventually approaches an upper bound or limit. The most well-known form of the model is the logistic differential equation." (Linda J S Allen, "An Introduction to Mathematical Biology", 2007)
"Humans have difficulty perceiving variables accurately […]. However, in general, they tend to have inaccurate perceptions of system states, including past, current, and future states. This is due, in part, to limited ‘mental models’ of the phenomena of interest in terms of both how things work and how to influence things. Consequently, people have difficulty determining the full implications of what is known, as well as considering future contingencies for potential systems states and the long-term value of addressing these contingencies. " (William B. Rouse, "People and Organizations: Explorations of Human-Centered Design", 2007)
"The methodology of feedback design is borrowed from cybernetics (control theory). It is based upon methods of controlled system model’s building, methods of system states and parameters estimation (identification), and methods of feedback synthesis. The models of controlled system used in cybernetics differ from conventional models of physics and mechanics in that they have explicitly specified inputs and outputs. Unlike conventional physics results, often formulated as conservation laws, the results of cybernetical physics are formulated in the form of transformation laws, establishing the possibilities and limits of changing properties of a physical system by means of control." (Alexander L Fradkov, "Cybernetical Physics: From Control of Chaos to Quantum Control", 2007)
"All graphics present data and allow a certain degree of exploration of those same data. Some graphics are almost all presentation, so they allow just a limited amount of exploration; hence we can say they are more infographics than visualization, whereas others are mostly about letting readers play with what is being shown, tilting more to the visualization side of our linear scale. But every infographic and every visualization has a presentation and an exploration component: they present, but they also facilitate the analysis of what they show, to different degrees." (Alberto Cairo, "The Functional Art", 2011)
"There are limits on the data we can gather and the kinds of experiments we can perform."(Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)
"Learning theory claims that a machine learning algorithm can generalize well from a finite training set of examples. This seems to contradict some basic principles of logic. Inductive reasoning, or inferring general rules from a limited set of examples, is not logically valid. To logically infer a rule describing every member of a set, one must have information about every member of that set." (Ian Goodfellow et al, "Deep Learning", 2015)
"Science’s predictions are more trustworthy, but they are limited to what we can systematically observe and tractably model. Big data and machine learning greatly expand that scope. Some everyday things can be predicted by the unaided mind, from catching a ball to carrying on a conversation. Some things, try as we might, are just unpredictable. For the vast middle ground between the two, there’s machine learning." (Pedro Domingos, "The Master Algorithm", 2015)
"To make progress, every field of science needs to have data commensurate with the complexity of the phenomena it studies. [...] With big data and machine learning, you can understand much more complex phenomena than before. In most fields, scientists have traditionally used only very limited kinds of models, like linear regression, where the curve you fit to the data is always a straight line. Unfortunately, most phenomena in the world are nonlinear. [...] Machine learning opens up a vast new world of nonlinear models." (Pedro Domingos, "The Master Algorithm", 2015)
"Repeated observations of the same phenomenon do not always produce the same results, due to random noise or error. Sampling errors result when our observations capture unrepresentative circumstances, like measuring rush hour traffic on weekends as well as during the work week. Measurement errors reflect the limits of precision inherent in any sensing device. The notion of signal to noise ratio captures the degree to which a series of observations reflects a quantity of interest as opposed to data variance. As data scientists, we care about changes in the signal instead of the noise, and such variance often makes this problem surprisingly difficult." (Steven S Skiena, "The Data Science Design Manual", 2017)
"Regularization is particularly important when the amount of available data is limited. A neat biological interpretation of regularization is that it corresponds to gradual forgetting, as a result of which 'less important' (i.e., noisy) patterns are removed. In general, it is often advisable to use more complex models with regularization rather than simpler models without regularization." (Charu C Aggarwal, "Neural Networks and Deep Learning: A Textbook", 2018)
"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)
"Unless we’re collecting data ourselves, there’s a limit to how much we can do to combat the problem of missing data. But we can and should remember to ask who or what might be missing from the data we’re being told about. Some missing numbers are obvious […]. Other omissions show up only when we take a close look at the claim in question." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)
"Despite their predictive power, most analytics and data science practices ignore relationships because it has been historically challenging to process them at scale." (Jesús Barrasa et al, "Knowledge Graphs: Data in Context for Responsive Businesses", 2021)
"Visualisation is fundamentally limited by the number of pixels you can pump to a screen. If you have big data, you have way more data than pixels, so you have to summarise your data. Statistics gives you lots of really good tools for this." (Hadley Wickham)
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