How to assign a likelihood ratio in a footwear mark case: an analysis and discussion in the light of R v T

This article analyses and discusses issues that pertain to the choice of relevant databases for assigning values to the components of evaluative likelihood ratio procedures at source level. Although several formal likelihood ratio developments currently exist, both case practitioners and recipients of expert information (such as judiciary) may be reluctant to consider them as a framework for evaluating scientific evidence in context. The recent ruling R v T and ensuing discussions in many forums provide illustrative examples for this. In particular, it is often felt that likelihood ratio-based reasoning amounts to an application that requires extensive quantitative information along with means for dealing with technicalities related to the algebraic formulation of these approaches. With regard to this objection, this article proposes two distinct discussions. In a first part, it is argued that, from a methodological point of view, there are additional levels of qualitative evaluation that are worth considering prior to focusing on particular numerical probability assignments. Analyses will be proposed that intend to show that, under certain assumptions, relative numerical values, as opposed to absolute values, may be sufficient to characterize a likelihood ratio for practical and pragmatic purposes. The feasibility of such qualitative considerations points out that the availability of hard numerical data is not a necessary requirement for implementing a likelihood ratio approach in practice. It is further argued that, even if numerical evaluations can be made, qualitative considerations may be valuable because they can further the understanding of the logical underpinnings of an assessment. In a second part, the article will draw a parallel to R v T by concentrating on a practical footwear mark case received at the authors' institute. This case will serve the purpose of exemplifying the possible usage of data from various sources in casework and help to discuss the difficulty associated with reconciling the depth of theoretical likelihood ratio developments and limitations in the degree to which these developments can actually be applied in practice.

Transformation kernel density estimation of actuarial loss functions

[cat] Es presenta un estimador nucli transformat que és adequat per a distribucions de cua pesada. Utilitzant una transformació basada en la distribució de probabilitat Beta l’elecció del paràmetre de finestra és molt directa. Es presenta una aplicació a dades d’assegurances i es mostra com calcular el Valor en Risc.

Invariant decomposition of the retarded electromagnetic field.

The integral representation of the electromagnetic two-form, defined on Minkowski space-time, is studied from a new point of view. The aim of the paper is to obtain an invariant criteria in order to define the radiative field. This criteria generalizes the well-known structureless charge case. We begin with the curvature two-form, because its field equations incorporate the motion of the sources. The gauge theory methods (connection one-forms) are not suited because their field equations do not incorporate the motion of the sources. We obtain an integral solution of the Maxwell equations in the case of a flow of charges in irrotational motion. This solution induces us to propose a new method of solving the problem of the nature of the retarded radiative field. This method is based on a projection tensor operator which, being local, is suited to being implemented on general relativity. We propose the field equations for the pair {electromagnetic field, projection tensor J. These field equations are an algebraic differential first-order system of oneforms, which verifies automatically the integrability conditions.

Consideraciones sobre cuerpos ordenados y cuerpos reales maximales

El presente trabajo se puede considerar como prolongación del trabajo (I) del autor, se amplía un resultado de este último que era válido sólo en el caso metrizable, y aquí se estudia el caso general; se hace un estudio de los cuerpos reales maximales y completos, y, finalmente se da un apéndice sobre cuerpos reales contenidos en un cuerpo algebraicamente cerrado y de característica 0

Short-time dynamics of colloidal suspensions in confined geometries

We analyze the short-time dynamical behavior of a colloidal suspension in a confined geometry. We analyze the relevant dynamical response of the solvent, and derive the temporal behavior of the velocity autocorrelation function, which exhibits an asymptotic negative algebraic decay. We are able to compare quantitatively with theoretical expressions, and analyze the effects of confinement on the diffusive behavior of the suspension.

Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Interaction of Spiral Waves in the Complex Ginzburg-Landau Equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered, and laws of motion for the centers are derived. The direction of the motion changes from along the line of centers to perpendicular to the line of centers as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wave number and frequency are also determined, which evolve slowly as the spirals move

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Bayes linear spaces

Linear spaces consisting of σ-finite probability measures and infinite measures (improper priors and likelihood functions) are defined. The commutative group operation, called perturbation, is the updating given by Bayes theorem; the inverse operation is the Radon-Nikodym derivative. Bayes spaces of measures are sets of classes of proportional measures. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. For example, exponential families appear as affine subspaces with their sufficient statistics as a basis. Bayesian statistics, in particular some well-known properties of conjugated priors and likelihood functions, are revisited and slightly extended

Site quality curves for birch stands in north-western Spain

Abstract

Bayesian Networks and the Value of the Evidence for the Forensic Two-Trace Transfer Problem

Forensic scientists face increasingly complex inference problems for evaluating likelihood ratios (LRs) for an appropriate pair of propositions. Up to now, scientists and statisticians have derived LR formulae using an algebraic approach. However, this approach reaches its limits when addressing cases with an increasing number of variables and dependence relationships between these variables. In this study, we suggest using a graphical approach, based on the construction of Bayesian networks (BNs). We first construct a BN that captures the problem, and then deduce the expression for calculating the LR from this model to compare it with existing LR formulae. We illustrate this idea by applying it to the evaluation of an activity level LR in the context of the two-trace transfer problem. Our approach allows us to relax assumptions made in previous LR developments, produce a new LR formula for the two-trace transfer problem and generalize this scenario to n traces.

Oscillations, phase-of-firing coding, and spike timing-dependent plasticity: an efficient learning scheme

Recent experiments have established that information can be encoded in the spike times of neurons relative to the phase of a background oscillation in the local field potential—a phenomenon referred to as “phase-of-firing coding” (PoFC). These firing phase preferences could result from combining an oscillation in the input current with a stimulus-dependent static component that would produce the variations in preferred phase, but it remains unclear whether these phases are an epiphenomenon or really affect neuronal interactions—only then could they have a functional role. Here we show that PoFC has a major impact on downstream learning and decoding with the now well established spike timing-dependent plasticity (STDP). To be precise, we demonstrate with simulations how a single neuron equipped with STDP robustly detects a pattern of input currents automatically encoded in the phases of a subset of its afferents, and repeating at random intervals. Remarkably, learning is possible even when only a small fraction of the afferents (~10%) exhibits PoFC. The ability of STDP to detect repeating patterns had been noted before in continuous activity, but it turns out that oscillations greatly facilitate learning. A benchmark with more conventional rate-based codes demonstrates the superiority of oscillations and PoFC for both STDP-based learning and the speed of decoding: the oscillation partially formats the input spike times, so that they mainly depend on the current input currents, and can be efficiently learned by STDP and then recognized in just one oscillation cycle. This suggests a major functional role for oscillatory brain activity that has been widely reported experimentally.