An informal and elementary introduction to probability scoring and forecast verification and improvement, slightly extended from Significance 22:3(2025)16, which might be useful for less mathematical readers as a prologue to the classic review by Gneiting and Raftery [Strictly proper scoring rules, prediction, and estimation, Journal of the American Statistical Association 102 (2007): 359].
The ability to read, write, and speak mathematics is critical to students becoming comfortable with statistical models and skills. Faster development of those skills may act as encouragement to further engage with the discipline. Vocabulary has been the focus of scholarship in existing literature on the linguistics of mathematics and statistics but there are structures such as grammar that go beyond the content of words and symbols. Here I introduce ideas for grammar structures through a sequence of examples.
In this paper, we state and prove a simple geometric interpretation of the degrees of freedom of a chi-square distribution. The general geometric idea goes back at least to Fisher in the 1920s, but the exact result does not appear to have been explicitly stated or proved prior to this paper.
This protocol is meant to select the samples from 1 source plate that has been growing in different conditions and has different values for these different conditions, one condition gives higher values and the other lower values. From this selected samples this protocol create final plates with 2 types of reactives The instructions for running this protocol are a python script for an Opentrons 2 robot and need a csv file with several variables setted making the protocol modular to reactives, volumes of transfer, type of plates, etc. In our lab we are selecting samples by their after night OD in 2 different antibiotics and create PCR plates (with water) and stock plates (with glycerol), but it can be used with other selection characteristics as expression of GFP and with other types of reactives.
This tutorial reviews the main steps of the principal component analysis of a multivariate data set and its subsequent dimensional reduction on the grounds of identified dominant principal components. The underlying computations are demonstrated and performed by means of a script written in the statistical software package R.
We propose a power-law $m$-uniform random hypergraph on $n$ vertexes. In this hypergraph, each vertex is independently assigned a random weight from a power-law distribution with exponent $α\in(0,\infty)$ and the hyperedge probabilities are defined as functions of the random weights. We characterize the number of hyperedge and the number of loose 2-cycle. There is a phase transition phenomenon for the number of hyperedge at $α=1$. Interestingly, for the number of loose 2-cycle, phase transition occurs at both $α=1$ and $α=2$.
In this note we discuss the mathematical tools to define trend indicators which are used to describe market trends. We explain the relation between averages and moving averages on the one hand and the so called exponential moving average (EMA) on the other hand. We present a lot of examples and give the definition of the most frequently used trend indicator, the MACD, and discuss its properties.
A concrete, stylized example illustrates that inferences may be degraded, rather than improved, by incorporating supplementary data via a joint likelihood. In the example, the likelihood is assumed to be correctly specified, as is the prior over the parameter of interest; all that is necessary for the joint modeling approach to suffer is misspecification of the prior over a nuisance parameter.
We provide accessible insight into the current 'replication crisis' in 'statistical science', by revisiting the old metaphor of 'court trial as hypothesis test'. Inter alia, we define and diagnose harmful statistical witch-hunting both in justice and science, which extends to the replication crisis itself, where a hunt on p-values is currently underway.
The von Mises distribution is one of the most important distribution in statistics to deal with circular data. In this paper we will consider some basic properties and characterizations of the sine skewed von Mises distribution.
An explanation to Lord's paradox using ordinary least square regression models is given. It is not a paradox at all, if the regression parameters are interpreted as predictive or as causal with stricter conditions and be aware of laws of averages. We use derivation of a super-model from a given sub-model, when its residuals can be modelled with other potential predictors as a solution.
A class of derivatives is defined for the pseudo determinant $Det(A)$ of a Hermitian matrix $A$. This class is shown to be non-empty and to have a unique, canonical member $\mathbf{\nabla Det}(A)=Det(A)A^+$, where $A^+$ is the Moore-Penrose pseudo inverse. The classic identity for the gradient of the determinant is thus reproduced. Examples are provided, including the maximum likelihood problem for the rank-deficient covariance matrix of the degenerate multivariate Gaussian distribution.
The fiducial argument of Fisher (1973) has been described as his biggest blunder, but the recent review of Hannig et al. (2016) demonstrates the current and increasing interest in this brilliant idea. This short note analyses an example introduced by Seidenfeld (1992) where the fiducial distribution is restricted to a string. Keywords and phrases: Bayesian and fiducial inference, Restrictions on parameters, Uncertainty quantification, Epistemic probability, Statistics on a manifold.
We show that the recently proposed (enhanced) PUMA estimator for array processing minimizes the same criterion function as the well-established MODE estimator. (PUMA = principal-singular-vector utilization for modal analysis, MODE = method of direction estimation.)
Convoulutional Neural Networks (CNNs) exhibit extraordinary performance on a variety of machine learning tasks. However, their mathematical properties and behavior are quite poorly understood. There is some work, in the form of a framework, for analyzing the operations that they perform. The goal of this project is to present key results from this theory, and provide intuition for why CNNs work.
The author revisits the Blue Bus Problem, a famous thought-experiment in law involving probabilistic proof, and presents simple Bayesian solutions to different versions of the blue bus hypothetical. In addition, the author expresses his solutions in standard and visual formats, i.e. in terms of probabilities and natural frequencies.
Examination of precinct level data in US presidential elections reveals a correlation of large precincts and increased fraction of Republican votes. The large precinct bias is analyzed with respect to voter heterogeneity and voter inconvenience as precinct size increases. The analysis shows that voter inconvenience is a significant factor in election outcomes in certain states, and may significantly disadvantage Democratic candidates.
In this paper, we consider the exponential functional \(A_{\infty}=\int_0^\infty e^{-ξ_s}ds\) of a L{é}vy process \(ξ_s\) and aim to estimate the characteristics of \(ξ_{s}\) from the distribution of \(A_{\infty}\). We present a new approach, which allows to statistically infer on the L{é}vy triplet of \(ξ_{t}\), and study the theoretical properties of the proposed estimators. The suggested algorithms are illustrated with numerical simulations.