"If the number of experiments be very large, we may have precise information as to the value of the mean, but if our sample be small, we have two sources of uncertainty: (I) owing to the 'error of random sampling' the mean of our series of experiments deviates more or less widely from the mean of the population, and (2) the sample is not sufficiently large to determine what is the law of distribution of individuals." (William S Gosset, "The Probable Error of a Mean", Biometrika, 1908)
"The making of decisions, as everyone knows from personal experience, is a burdensome task. Offsetting the exhilaration that may result from correct and successful decision and the relief that follows the termination of a struggle to determine issues is the depression that comes from failure, or error of decision, and the frustration which ensues from uncertainty." (Chester I Barnard, "The Functions of the Executive", 1938)
"Uncertainty is introduced, however, by the impossibility of making generalizations, most of the time, which happens to all members of a class. Even scientific truth is a matter of probability and the degree of probability stops somewhere short of certainty." (Wayne C Minnick, "The Art of Persuasion", 1957)
"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)
"The most dominant decision type [that will have to be made in an organic organization] will be decisions under uncertainty." (Henry L Tosi & Stephen J Carroll, "Management", 1976)
"The greater the uncertainty, the greater the amount of decision making and information processing. It is hypothesized that organizations have limited capacities to process information and adopt different organizing modes to deal with task uncertainty. Therefore, variations in organizing modes are actually variations in the capacity of organizations to process information and make decisions about events which cannot be anticipated in advance." (John K Galbraith, "Organization Design", 1977)
"Probability is the mathematics of uncertainty. Not only do we constantly face situations in which there is neither adequate data nor an adequate theory, but many modem theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential. Statistics is the reverse of probability (glibly speaking). In probability you go from the model of the situation to what you expect to see; in statistics you have the observations and you wish to estimate features of the underlying model." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"Probability plays a central role in many fields, from quantum mechanics to information theory, and even older fields use probability now that the presence of 'noise' is officially admitted. The newer aspects of many fields start with the admission of uncertainty." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"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)
"The mathematical theories generally called 'mathematical theories of chance' actually ignore chance, uncertainty and probability. The models they consider are purely deterministic, and the quantities they study are, in the final analysis, no more than the mathematical frequencies of particular configurations, among all equally possible configurations, the calculation of which is based on combinatorial analysis. In reality, no axiomatic definition of chance is conceivable." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)
"The worst, i.e., most dangerous, feature of 'accepting the null hypothesis' is the giving up of explicit uncertainty. [...] Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can." (John Tukey, "The Philosophy of Multiple Comparisons", Statistical Science Vol. 6 (1), 1991)
"In nonlinear systems - and the economy is most certainly nonlinear - chaos theory tells you that the slightest uncertainty in your knowledge of the initial conditions will often grow inexorably. After a while, your predictions are nonsense." (M Mitchell Waldrop, "Complexity: The Emerging Science at the Edge of Order and Chaos", 1992)
"Statistics as a science is to quantify uncertainty, not unknown." (Chamont Wang, "Sense and Nonsense of Statistical Inference: Controversy, Misuse, and Subtlety", 1993)
"There is a new science of complexity which says that the link between cause and effect is increasingly difficult to trace; that change (planned or otherwise) unfolds in non-linear ways; that paradoxes and contradictions abound; and that creative solutions arise out of diversity, uncertainty and chaos." (Andy P Hargreaves & Michael Fullan, "What’s Worth Fighting for Out There?", 1998)
"Information entropy has its own special interpretation and is defined as the degree of unexpectedness in a message. The more unexpected words or phrases, the higher the entropy. It may be calculated with the regular binary logarithm on the number of existing alternatives in a given repertoire. A repertoire of 16 alternatives therefore gives a maximum entropy of 4 bits. Maximum entropy presupposes that all probabilities are equal and independent of each other. Minimum entropy exists when only one possibility is expected to be chosen. When uncertainty, variety or entropy decreases it is thus reasonable to speak of a corresponding increase in information." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)
"Any scientific data without (a stated) uncertainty is of no avail. Therefore the analysis and description of uncertainty are almost as important as those of the data value itself . It should be clear that the uncertainty itself also has an uncertainty – due to its nature as a scientific quantity – and so on. The uncertainty of an uncertainty is generally not determined.
"As uncertainties of scientific data values are nearly as important as the data values themselves, it is usually not acceptable that a best estimate is only accompanied by an estimated uncertainty. Therefore, only the size of nondominant uncertainties should be estimated. For estimating the size of a nondominant uncertainty we need to find its upper limit, i.e., we want to be as sure as possible that the uncertainty does not exceed a certain value.
"Before best estimates are extracted from data sets by way of a regression analysis, the uncertainties of the individual data values must be determined.In this case care must be taken to recognize which uncertainty components are common to all the values, i.e., those that are correlated (systematic).
"Due to the theory that underlies uncertainties an infinite number of data values would be necessary to determine the true value of any quantity. In reality the number of available data values will be relatively small and thus this requirement can never be fully met; all one can get is the best estimate of the true value."
"For linear dependences the main information usually lies in the slope. It is obvious that those points that lie far apart have the strongest influence on the slope if all points have the same uncertainty. In this context we speak of the strong leverage of distant points; when determining the parameter 'slope' these distant points carry more effective weight. Naturally, this weight is distinct from the 'statistical' weight usually used in regression analysis.
"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.
"In many cases systematic errors are interpreted as the systematic difference between nature (which is being questioned by the experimenter in his experiment) and the model (which is used to describe nature). If the model used is not good enough, but the measurement result is interpreted using this model, the final result (the interpretation) will be wrong because it is biased, i.e., it has a systematic deviation (not uncertainty). If we do not use the best model (the best theory) available for the description of a certain phenomenon this procedure is just wrong. It has nothing to do with an uncertainty.
"It is also inevitable for any model or theory to have an uncertainty (a difference between model and reality). Such uncertainties apply both to the numerical parameters of the model and to the inadequacy of the model as well. Because it is much harder to get a grip on these types of uncertainties, they are disregarded, usually.
"It is important that uncertainty components that are independent of each other are added quadratically. This is also true for correlated uncertainty components, provided they are independent of each other, i.e., as long as there is no correlation between the components.
"It is the nature of an uncertainty that it is not known and can never be known, whether the best estimate is greater or less than the true value.
"Outliers or flyers are those data points in a set that do not quite fit within the rest of the data, that agree with the model in use. The uncertainty of such an outlier is seemingly too small. The discrepancy between outliers and the model should be subject to thorough examination and should be given much thought. Isolated data points, i.e., data points that are at some distance from the bulk of the data are not outliers if their values are in agreement with the model in use.
"The fact that the same uncertainty (e.g., scale uncertainty) is uncorrelated if we are dealing with only one measurement, but correlated (i.e., systematic) if we look at more than one measurement using the same instrument shows that both types of uncertainties are of the same nature. Of course, an uncertainty keeps its characteristics (e.g., Poisson distributed), independent of the fact whether it occurs only once or more often.
"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.
"In fact, H [entropy] measures the amount of uncertainty that exists in the phenomenon. If there were only one event, its probability would be equal to 1, and H would be equal to 0 - that is, there is no uncertainty about what will happen in a phenomenon with a single event because we always know what is going to occur. The more events that a phenomenon possesses, the more uncertainty there is about the state of the phenomenon. In other words, the more entropy, the more information." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)
"Data always vary randomly because the object of our inquiries, nature itself, is also random. We can analyze and predict events in nature with an increasing amount of precision and accuracy, thanks to improvements in our techniques and instruments, but a certain amount of random variation, which gives rise to uncertainty, is inevitable." (Alberto Cairo, "The Functional Art", 2011)
"The storytelling mind is allergic to uncertainty, randomness, and coincidence. It is addicted to meaning. If the storytelling mind cannot find meaningful patterns in the world, it will try to impose them. In short, the storytelling mind is a factory that churns out true stories when it can, but will manufacture lies when it can't." (Jonathan Gottschall, "The Storytelling Animal: How Stories Make Us Human", 2012)
"The data is a simplification - an abstraction - of the real world. So when you visualize data, you visualize an abstraction of the world, or at least some tiny facet of it. Visualization is an abstraction of data, so in the end, you end up with an abstraction of an abstraction, which creates an interesting challenge. […] Just like what it represents, data can be complex with variability and uncertainty, but consider it all in the right context, and it starts to make sense."
"Without precise predictability, control is impotent and almost meaningless. In other words, the lesser the predictability, the harder the entity or system is to control, and vice versa. If our universe actually operated on linear causality, with no surprises, uncertainty, or abrupt changes, all future events would be absolutely predictable in a sort of waveless orderliness." (Lawrence K Samuels, "Defense of Chaos", 2013)
"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)
"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."
"The elements of this cloud of uncertainty (the set of all possible errors) can be described in terms of probability. The center of the cloud is the number zero, and elements of the cloud that are close to zero are more probable than elements that are far away from that center. We can be more precise in this definition by defining the cloud of uncertainty in terms of a mathematical function, called the probability distribution." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)
"Uncertainty is an adversary of coldly logical algorithms, and being aware of how those algorithms might break down in unusual circumstances expedites the process of fixing problems when they occur - and they will occur. A data scientist’s main responsibility is to try to imagine all of the possibilities, address the ones that matter, and reevaluate them all as successes and failures happen." (Brian Godsey, "Think Like a Data Scientist", 2017)
"Bootstrapping provides an intuitive, computer-intensive way of assessing the uncertainty in our estimates, without making strong assumptions and without using probability theory. But the technique is not feasible when it comes to, say, working out the margins of error on unemployment surveys of 100,000 people. Although bootstrapping is a simple, brilliant and extraordinarily effective idea, it is just too clumsy to bootstrap such large quantities of data, especially when a convenient theory exists that can generate formulae for the width of uncertainty intervals." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)
"Entropy is a measure of amount of uncertainty or disorder present in the system within the possible probability distribution. The entropy and amount of unpredictability are directly proportional to each other." (G Suseela & Y Asnath V Phamila, "Security Framework for Smart Visual Sensor Networks", 2019)
"Estimates based on data are often uncertain. If the data were intended to tell us something about a wider population (like a poll of voting intentions before an election), or about the future, then we need to acknowledge that uncertainty. This is a double challenge for data visualization: it has to be calculated in some meaningful way and then shown on top of the data or statistics without making it all too cluttered." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)
"Uncertainty confuses many people because they have the unreasonable expectation that science and statistics will unearth precise truths, when all they can yield is imperfect estimates that can always be subject to changes and updates." (Alberto Cairo, "How Charts Lie", 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)
