The likelihood approach to compare populations : a study on DNA evidence and pitfalls of intuitions

The paper follows on from earlier work [Taroni F and Aitken CGG. Probabilistic reasoning in the law, Part 1: assessment of probabilities and explanation of the value of DNA evidence. Science & Justice 1998; 38: 165-177]. Different explanations of the value of DNA evidence were presented to students from two schools of forensic science and to members of fifteen laboratories all around the world. The responses were divided into two groups; those which came from a school or laboratory identified as Bayesian and those which came from a school or laboratory identified as non-Bayesian. The paper analyses these responses using a likelihood approach. This approach is more consistent with a Bayesian analysis than one based on a frequentist approach, as was reported by Taroni F and Aitken CGG. [Probabilistic reasoning in the law, Part 1: assessment of probabilities and explanation of the value of DNA evidence] in Science & Justice 1998.

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods [Authors]

Uncertainty about the true source. A note on the likelihood ratio at the activity level.

This paper focuses on likelihood ratio based evaluations of fibre evidence in cases in which there is uncertainty about whether or not the reference item available for analysis - that is, an item typically taken from the suspect or seized at his home - is the item actually worn at the time of the offence. A likelihood ratio approach is proposed that, for situations in which certain categorical assumptions can be made about additionally introduced parameters, converges to formula described in existing literature. The properties of the proposed likelihood ratio approach are analysed through sensitivity analyses and discussed with respect to possible argumentative implications that arise in practice.

Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

On the use of the likelihood ratio for forensic evaluation: response to Fenton et al.

This letter to the Editor comments on the article When 'neutral' evidence still has probative value (with implications from the Barry George Case) by N. Fenton et al. [[1], 2014].

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Maximum-likelihood estimation of specific differential phase and attenuation in rain

Precise estimation of propagation parameters inprecipitation media is of interest to improve the performanceof communications systems and in remote sensing applications.In this paper, we present maximum-likelihood estimators ofspecific attenuation and specific differential phase in rain. Themodel used for obtaining the cited estimators assumes coherentpropagation, reflection symmetry of the medium, and Gaussianstatistics of the scattering matrix measurements. No assumptionsabout the microphysical properties of the medium are needed.The performance of the estimators is evaluated through simulateddata. Results show negligible estimators bias and variances closeto Cramer–Rao bounds.

A new maximum likelihood method for luminosity calibrations

A new statistical parallax method using the Maximum Likelihood principle is presented, allowing the simultaneous determination of a luminosity calibration, kinematic characteristics and spatial distribution of a given sample. This method has been developed for the exploitation of the Hipparcos data and presents several improvements with respect to the previous ones: the effects of the selection of the sample, the observational errors, the galactic rotation and the interstellar absorption are taken into account as an intrinsic part of the formulation (as opposed to external corrections). Furthermore, the method is able to identify and characterize physically distinct groups in inhomogeneous samples, thus avoiding biases due to unidentified components. Moreover, the implementation used by the authors is based on the extensive use of numerical methods, so avoiding the need for simplification of the equations and thus the bias they could introduce. Several examples of application using simulated samples are presented, to be followed by applications to real samples in forthcoming articles.

Statistical analysis of maximum likelihood estimator images of human brain FDG PET studies

The work presented evaluates the statistical characteristics of regional bias and expected error in reconstructions of real positron emission tomography (PET) data of human brain fluoro-deoxiglucose (FDG) studies carried out by the maximum likelihood estimator (MLE) method with a robust stopping rule, and compares them with the results of filtered backprojection (FBP) reconstructions and with the method of sieves. The task of evaluating radioisotope uptake in regions-of-interest (ROIs) is investigated. An assessment of bias and variance in uptake measurements is carried out with simulated data. Then, by using three different transition matrices with different degrees of accuracy and a components of variance model for statistical analysis, it is shown that the characteristics obtained from real human FDG brain data are consistent with the results of the simulation studies.

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.

Fertilization response likelihood for the interpretation of leaf analyses

Leaf analysis is the chemical evaluation of the nutritional status where the nutrient concentrations found in the tissue reflect the nutritional status of the plants. Thus, a correct interpretation of the results of leaf analysis is fundamental for an effective use of this tool. The purpose of this study was to propose and compare the method of Fertilization Response Likelihood (FRL) for interpretation of leaf analysis with that of the Diagnosis and Recommendation Integrated System (DRIS). The database consisted of 157 analyses of the N, P, K, Ca, Mg, S, Cu, Fe, Mn, Zn, and B concentrations in coffee leaves, which were divided into two groups: low yield (< 30 bags ha-1) and high yield (> 30 bags ha-1). The DRIS indices were calculated using the method proposed by Jones (1981). The fertilization response likelihood was computed based on the approximation of normal distribution. It was found that the Fertilization Response Likelihood (FRL) allowed an evaluation of the nutritional status of coffee trees, coinciding with the DRIS-based diagnoses in 84.96 % of the crops.

Comparison between artificial neural networks and maximum likelihood classification in digital soil mapping

Soil surveys are the main source of spatial information on soils and have a range of different applications, mainly in agriculture. The continuity of this activity has however been severely compromised, mainly due to a lack of governmental funding. The purpose of this study was to evaluate the feasibility of two different classifiers (artificial neural networks and a maximum likelihood algorithm) in the prediction of soil classes in the northwest of the state of Rio de Janeiro. Terrain attributes such as elevation, slope, aspect, plan curvature and compound topographic index (CTI) and indices of clay minerals, iron oxide and Normalized Difference Vegetation Index (NDVI), derived from Landsat 7 ETM+ sensor imagery, were used as discriminating variables. The two classifiers were trained and validated for each soil class using 300 and 150 samples respectively, representing the characteristics of these classes in terms of the discriminating variables. According to the statistical tests, the accuracy of the classifier based on artificial neural networks (ANNs) was greater than of the classic Maximum Likelihood Classifier (MLC). Comparing the results with 126 points of reference showed that the resulting ANN map (73.81 %) was superior to the MLC map (57.94 %). The main errors when using the two classifiers were caused by: a) the geological heterogeneity of the area coupled with problems related to the geological map; b) the depth of lithic contact and/or rock exposure, and c) problems with the environmental correlation model used due to the polygenetic nature of the soils. This study confirms that the use of terrain attributes together with remote sensing data by an ANN approach can be a tool to facilitate soil mapping in Brazil, primarily due to the availability of low-cost remote sensing data and the ease by which terrain attributes can be obtained.