It is usually supposed that the central limit theorem explains why various quantities we find in nature are approximately normally distributed—people's heights, examination grades, snowflake sizes, and so on. This sort of explanation is found in many textbooks across the sciences, particularly in biology, economics, and sociology. Contrary to this received wisdom, I argue that in many cases we are not justified in claiming that the central limit theorem explains why a particular quantity is normally distributed, and that in some (...) cases, we are actually wrong. 1 Introduction2 Normal Distributions and the Central Limit Theorem2.1 Normal distributions2.2 The central limit theorem2.3 Terminology3 Explaining Normality3.1 Loaves of bread3.2 Varying variances and probability densities3.3 Tensile strengths and problems with summation3.4 Products of factors and log-normal distributions3.5 Transforming factors and sub-factors3.6 Transformations of quantities3.7 Quantitative genetics3.8 Inference to the best explanation4 Maximum Entropy Explanations5 Conclusion. (shrink)

A pure probabilistic approach to Gibbs' distributions is given, starting from the notion of “finite exchangeable random process.” The differences between bosons, fermions, and classical particles are ascribed to different values of correlation. The relationship between exchangeability and constraints on the energy distribution is investigated.

A criterion is given for characterizing quantum mechanical joint distributions that have the same covariance of position and momentum.

Multisets are ‘sets’ in which elements may occur multiple times. Discrete probability distributions capture states in which elements may occur with probabilities that add up to one. This paper describes how the interaction between multisets and distributions lies at the heart of some basic constructions in probability theory, especially in distributions arising from drawing from an urn with multiple balls and in learning distributions from multiple occurrences of data. Drawing multiple balls from an urn is described (...) uniformly in terms of Kleisli iteration for a monad, covering the four standard distinctions of ordered/unordered draws, with/without replacement. In probabilistic learning the paper distinguishes two forms of likelihood, based on also on iteration, with corresponding forms of learning. Both of these forms occur in the literature, but they are not clearly distinguished, even though they lead to different outcomes. (shrink)

Next SectionBackground International and national agencies play a major role in setting HIV care-and-treatment priorities in low-income-countries. Little is known about priority setting at lower health-system levels. The objective of this article is to explore experiences of HIV priority decisions, at what levels these decisions are made and how they might influence the distribution of health benefits in a high-endemic region in Tanzania. Methods This is a qualitative study using observations, key documents and semistructured focus-group and individual interviews (43) with (...) health workers, patients and administrators at one regional and one district hospital. The analysis was based on an editing analysis style. Results Health workers did not perceive themselves as bedside rationing agents and they reported following national recommendations and felt they have little impact on important priority decisions. Health benefit distributions were largely determined by priority decisions made at a national level. External factors, such as eligibility criteria, inadequate funding and barriers to access seemed to play a major role in the actual distribution of health services and benefits in the region. Patient groups who were actually given high priority were the sickest patients, those living near a facility and those who could afford long journeys and frequent visits. Low-priority groups were those experiencing other co-morbidities or treatment failure, children, asymptomatic patients and the poorest. Conclusions/significance The interaction of priority setting across health-system levels and their impact on distribution of health outcomes in the population has been underestimated. The distributional pattern of health outcomes calls for further normative assessment. (shrink)

This study describes the results of experiments motivated by an attempt to understand spectral processing in the cerebral cortex (DeValois and DeValois, 1988; Pribram, 1971, 1991). This level of inquiry concerns processing within a restricted cortical area rather than that by which spatially separate circuits become synchronized during certain behavioral and experiential processes. We recorded neural responses for 55 locations in the somatosensory (barrel) cortex of the rat to various combinations of spatial frequency (texture) and temporal frequency stimulation of their (...) vibrissae. The recordings obtained from single and multi-unit bursts of spikes were mapped as surface distributions of local dendritic potentials. The distributions showed a variety of patterns that are asymmetric with respect to the spatial and temporal parameters of stimulation, and were, therefore, not simply reflecting whisker flick rate. Next, a simulation of our results showed that these surface distributions of local dendritic potentials can be described by Gabor-like functions much as in the visual system. The results provide support for a model of distributed cortical processing that imposes a physiologically derived frame (the limited extent of a dendritic patch) and an anatomically derived (axonal) sampling of the distributed process. This combination provides a complex Gabor wavelet that encodes phase, which is necessary to processing such details as edges and texture in a scene. The synchronization across cortical areas that make the Gabor wavelet processes within restricted cortical areas available to one another (the binding problem) proceed at a ''higher order'' level of integration. Both levels of distributed processing accomplish computation in the conjoint spacetime and spectral domain. (shrink)

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

There is now considerable evidence that human sentence processing is expectation based: As people read a sentence, they use their statistical experience with their language to generate predictions about upcoming syntactic structure. This study examines how sentence processing is affected by readers' uncertainty about those expectations. In a self-paced reading study, we use lexical subcategorization distributions to factorially manipulate both the strength of expectations and the uncertainty about them. We compare two types of uncertainty: uncertainty about the verb's complement, (...) reflecting the next prediction step; and uncertainty about the full sentence, reflecting an unbounded number of prediction steps. We find that uncertainty about the full structure, but not about the next step, was a significant predictor of processing difficulty: Greater reduction in uncertainty was correlated with increased reading times. We additionally replicated previously observed effects of expectation violation, orthogonal to the effect of uncertainty. This suggests that both surprisal and uncertainty affect human RTs. We discuss the consequences for theories of sentence comprehension. (shrink)

Subjects in experiments use real-life strategies that differ significantly from those assumed by experimenters. First, true randomness is rare in both natural and constructed environments. Second, communication follows conventions which depend on the game-theoretic aspects of situations. Third, in the common rhetorical stance of storytelling, people do not tell about the representative but about unusual, exceptional, and rare cases.

In the philosophy of the analytical tradition, set theory and formal logic are familiar formal tools. I think there is no deep reason why the philosopher’s tool kit should be restricted to just these theories. It might well be the case—to generalize a dictum of Suppes concerning philosophy of science—that the appropriate formal device for doing philosophy is mathematics in general; it may be set theory, algebra, topology, or any other realm of mathematics. In this paper I want to employ (...) elementary topological considerations to shed new light on the intricate problem of the relation of qualities and similarity. Thereby I want to make plausible the general thesis that topology might be a useful device for matters epistemological. (shrink)

This work compares Autometrics with dual penalization techniques such as minimax concave penalty and smoothly clipped absolute deviation under asymmetric error distributions such as exponential, gamma, and Frechet with varying sample sizes as well as predictors. Comprehensive simulations, based on a wide variety of scenarios, reveal that the methods considered show improved performance for increased sample size. In the case of low multicollinearity, these methods show good performance in terms of potency, but in gauge, shrinkage methods collapse, and higher (...) gauge leads to overspecification of the models. High levels of multicollinearity adversely affect the performance of Autometrics. In contrast, shrinkage methods are robust in presence of high multicollinearity in terms of potency, but they tend to select a massive set of irrelevant variables. Moreover, we find that expanding the data mitigates the adverse impact of high multicollinearity on Autometrics rapidly and gradually corrects the gauge of shrinkage methods. For empirical application, we take the gold prices data spanning from 1981 to 2020. While comparing the forecasting performance of all selected methods, we divide the data into two parts: data over 1981–2010 are taken as training data, and those over 2011–2020 are used as testing data. All methods are trained for the training data and then are assessed for performance through the testing data. Based on a root-mean-square error and mean absolute error, Autometrics remain the best in capturing the gold prices trend and producing better forecasts than MCP and SCAD. (shrink)

Jaynes contends that in many statistical problems a seemingly indeterminate probability distribution is made unique by the transformation group of necessarily implied invariance properties, thereby justifying the principle of indifference. To illustrate and substantiate his claims he considers Bertrand's Paradox. These assertions are here refuted and the traditional attitude is vindicated.

Probability distributions play an essential role in modeling and predicting biomedical datasets. To have the best description and accurate prediction of the biomedical datasets, numerous probability distributions have been introduced and implemented. We investigate a novel family of lifetime probability distributions to represent biological datasets in this paper. The proposed family is called a new flexible logarithmic- X family. The suggested NFLog- X family is obtained by applying the T- X method together with the exponential model having (...) the PDF m t = e − t. Based on the NFLog- X approach, a three parameters probability distribution, namely, a new flexible logarithmic-Weibull distribution is introduced. The method of maximum likelihood estimation is adopted for estimating the parameters of the NFLog- X family. In the end, we examine three different biological datasets in order to give a thorough numerical research that illustrates the NFLog-Wei distribution. Comparisons are made between the analytical goodness-of-fit metrics of the suggested distribution. We made comparison with the alpha power transformed Weibull, exponentiated Weibull, Weibull, flexible reduced logarithmic-Weibull, and Marshall–Olkin Weibull distributions. After performing the analyses, we observe that the proposed method outclassed other competitive distributions. (shrink)

Statistical distributions play a prominent role for modeling data in applied fields, particularly in actuarial, financial sciences, and risk management fields. Among the statistical distributions, the heavy-tailed distributions have proven the best choice to use for modeling heavy-tailed financial data. The actuaries are often in search of such types of distributions to provide the best description of the actuarial and financial data. This study presents a new power transformation to introduce a new family of heavy-tailed (...) class='Hi'>distributions useful for modeling heavy-tailed financial data. A submodel, namely, heavy-tailed beta-power transformed Weibull model is considered to demonstrate the adequacy of the proposed method. Some actuarial measures such as value at risk, tail value at risk, tail variance, and tail variance premium are calculated. A brief simulation study based on these measures is provided. Finally, an application to the insurance loss dataset is analyzed, which revealed that the proposed distribution is a superior model among the competitors and could potentially be very adequate in describing and modeling actuarial and financial data. (shrink)

Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach’s paper published in 1913. We reviewed his analysis and found (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent $$\approx 1$$ ≈ 1, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time (...) periods. We further investigate his legacy as reflected in citations and find that important follow-up work, e.g. by Lotka (Elements of physical biology. Williams & Wilkins Company, Baltimore, 1925) and Zipf (Human behavior and the principle of least effort: an introduction to human ecology, Martino Publishing, Manfield Centre, CT (2012), 1949), does give proper reference to his discovery—but others do not. For example, only approximately 20% of city-related works citing Zipf (1949) also cite Auerbach (Petermanns Geogr Mitteilungen 59(74):74–76, 1913). To our best knowledge, Lotka (1925) was the first to describe the power-law rank-size rule as it is analyzed today. Saibante (Metron Rivista Internazionale di Statistica 7(2):53–99, 1928), building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. Zipf’s achievement was to embed these findings in his monumental 1949 book. We suggest that the use of “Auerbach–Lotka–Zipf law” (or “ALZ-law”) is more appropriate than “Zipf’s law for cities”, which also avoids confusion with Zipf’s law for word frequency. We end the treatise with biographical notes on Auerbach. (shrink)

Cladistic analyses are based on the distinction between primitive and derived character states (hypotheses of the polarity of evolutionary transformations) and a complete reliance on only derived character state distributions as bona fide evidence of holophyletic assemblages of taxa. The cladistic premise that only derived character state distributions provide evidence of holophyly is reconsidered and shown to be both unjustified and inconsistent with the desire or methodological prescription of using all the available evidence. Cladistic techniques are here viewed (...) primarily as methods for the ordering of character states so that they may be differentially weighted. The problem of assigning realistic differential weight to primitive and derived character state distributions is briefly discussed. One possible use of primitive character state distributions as evidence of holophyly that is largely free of the problem of weight is proposed. This application is illustrated with an example from the caecilian amphibians (Amphibia: Gymnophiona). (shrink)