Related papers: Lessons about likelihood functions from nuclear ph…
Overdispersed count data are modelled with likelihood and non-likelihood approaches. Likelihood approaches include the Poisson mixtures with three distributions, the gamma, the lognormal, and the inverse Gaussian distributions.…
Negative probabilities arise primarily in physics, statistical quantum mechanics and quantum computing. Negative probabilities arise as mixing distributions of unobserved latent variables in Bayesian modeling. Our goal is to provide a link…
The log-normal distribution is one of the most common distributions used for modeling skewed and positive data. It frequently arises in many disciplines of science, specially in the biological and medical sciences. The statistical analysis…
A canonical structure compatible with the action of the Lorentz group can be obtained considering the energy and time as conjugate variables of an extended phase space. Scalar probability waves, describing free relativistic particles, are…
The classical likelihood ratio test (LRT) based on the asymptotic chi-squared distribution of the log likelihood is one of the fundamental tools of statistical inference. A recent universal LRT approach based on sample splitting provides…
The Luria-Delbr\"uck distribution is a classical model of mutations in cell kinetics. It is obtained as a limit when the probability of mutation tends to zero and the number of divisions to infinity. It can be interpreted as a compound…
Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required.…
We consider Gaussian signals, i.e. random functions $u(t)$ ($t/L \in [0,1]$) with independent Gaussian Fourier modes of variance $\sim 1/q^{\alpha}$, and compute their statistical properties in small windows $[x, x+\delta]$. We determine…
We demonstrate that the Bayesian evidence can be used to find a good approximation of the ground truth likelihood function of a dataset, a goal of the likelihood-free inference (LFI) paradigm. As a concrete example, we use forward modelled…
The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback--Leibler from the normal and half normal distributions are approximated using…
Organisms and algorithms learn probability distributions from previous observations, either over evolutionary time or on the fly. In the absence of regularities, estimating the underlying distribution from data would require observing each…
The projected normal distribution, also known as the angular Gaussian distribution, is obtained by dividing a multivariate normal random variable $\mathbf{x}$ by its norm $\sqrt{\mathbf{x}^T \mathbf{x}}$. The resulting random variable…
With a rapid increase in volume and complexity of data sets, there is a need for methods that can extract useful information, for example the relationship between two data sets measured for the same persons. The Partial Least Squares (PLS)…
Determining whether the flux distribution of an Astrophysical source is a Gaussian or a log-normal, provides key insight into the nature of its variability. For lightcurves of moderate length ($< 10^3$), a useful first analysis is to test…
Many common loss functions such as mean-squared-error, cross-entropy, and reconstruction loss are unnecessarily rigid. Under a probabilistic interpretation, these common losses correspond to distributions with fixed shapes and scales. We…
Given N data points drawn from a chi-square distribution, we use Bayesian inference to determine most likely values and N-dependent confidence intervals for the width sigma and the number k of degrees of freedom of that distribution. Using…
A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted…
Several authors have explained that the likelihood ratio measures the strength of the evidence represented by observations in statistical problems. This idea works fine when the goal is to evaluate the strength of the available evidence for…
The statistical analysis of cosmological data often assumes a Gaussian sampling distribution and relies on covariance matrices estimated from simulations. In this setting, the likelihood function of the data is not Gaussian but is instead a…
Metrics for rigorously defining a distance between two events have been used to study the properties of the dataspace manifold of particle collider physics. The probability distribution of pairwise distances on this dataspace is unique with…