An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

Progression of auditory discrimination based on neural decoding predicts awakening from coma.

Auditory evoked potentials are informative of intact cortical functions of comatose patients. The integrity of auditory functions evaluated using mismatch negativity paradigms has been associated with their chances of survival. However, because auditory discrimination is assessed at various delays after coma onset, it is still unclear whether this impairment depends on the time of the recording. We hypothesized that impairment in auditory discrimination capabilities is indicative of coma progression, rather than of the comatose state itself and that rudimentary auditory discrimination remains intact during acute stages of coma. We studied 30 post-anoxic comatose patients resuscitated from cardiac arrest and five healthy, age-matched controls. Using a mismatch negativity paradigm, we performed two electroencephalography recordings with a standard 19-channel clinical montage: the first within 24 h after coma onset and under mild therapeutic hypothermia, and the second after 1 day and under normothermic conditions. We analysed electroencephalography responses based on a multivariate decoding algorithm that automatically quantifies neural discrimination at the single patient level. Results showed high average decoding accuracy in discriminating sounds both for control subjects and comatose patients. Importantly, accurate decoding was largely independent of patients' chance of survival. However, the progression of auditory discrimination between the first and second recordings was informative of a patient's chance of survival. A deterioration of auditory discrimination was observed in all non-survivors (equivalent to 100% positive predictive value for survivors). We show, for the first time, evidence of intact auditory processing even in comatose patients who do not survive and that progression of sound discrimination over time is informative of a patient's chance of survival. Tracking auditory discrimination in comatose patients could provide new insight to the chance of awakening in a quantitative and automatic fashion during early stages of coma.

Decoding decisions in healthy subjects and auditory discrimination in comatose patients through single-trial topographic EEG analyses

Neuroimaging studies analyzing neurophysiological signals are typically based on comparing averages of peri-stimulus epochs across experimental conditions. This approach can however be problematic in the case of high-level cognitive tasks, where response variability across trials is expected to be high and in cases where subjects cannot be considered part of a group. The main goal of this thesis has been to address this issue by developing a novel approach for analyzing electroencephalography (EEG) responses at the single-trial level. This approach takes advantage of the spatial distribution of the electric field on the scalp (topography) and exploits repetitions across trials for quantifying the degree of discrimination between experimental conditions through a classification scheme. In the first part of this thesis, I developed and validated this new method (Tzovara et al., 2012a,b). Its general applicability was demonstrated with three separate datasets, two in the visual modality and one in the auditory. This development allowed then to target two new lines of research, one in basic and one in clinical neuroscience, which represent the second and third part of this thesis respectively. For the second part of this thesis (Tzovara et al., 2012c), I employed the developed method for assessing the timing of exploratory decision-making. Using single-trial topographic EEG activity during presentation of a choice's payoff, I could predict the subjects' subsequent decisions. This prediction was due to a topographic difference which appeared on average at ~516ms after the presentation of payoff and was subject-specific. These results exploit for the first time the temporal correlates of individual subjects' decisions and additionally show that the underlying neural generators start differentiating their responses already ~880ms before the button press. Finally, in the third part of this project, I focused on a clinical study with the goal of assessing the degree of intact neural functions in comatose patients. Auditory EEG responses were assessed through a classical mismatch negativity paradigm, during the very early phase of coma, which is currently under-investigated. By taking advantage of the decoding method developed in the first part of the thesis, I could quantify the degree of auditory discrimination at the single patient level (Tzovara et al., in press). Our results showed for the first time that even patients who do not survive the coma can discriminate sounds at the neural level, during the first hours after coma onset. Importantly, an improvement in auditory discrimination during the first 48hours of coma was predictive of awakening and survival, with 100% positive predictive value. - L'analyse des signaux électrophysiologiques en neuroimagerie se base typiquement sur la comparaison des réponses neurophysiologiques à différentes conditions expérimentales qui sont moyennées après plusieurs répétitions d'une tâche. Pourtant, cette approche peut être problématique dans le cas des fonctions cognitives de haut niveau, où la variabilité des réponses entre les essais peut être très élevéeou dans le cas où des sujets individuels ne peuvent pas être considérés comme partie d'un groupe. Le but principal de cette thèse est d'investiguer cette problématique en développant une nouvelle approche pour l'analyse des réponses d'électroencephalographie (EEG) au niveau de chaque essai. Cette approche se base sur la modélisation de la distribution du champ électrique sur le crâne (topographie) et profite des répétitions parmi les essais afin de quantifier, à l'aide d'un schéma de classification, le degré de discrimination entre des conditions expérimentales. Dans la première partie de cette thèse, j'ai développé et validé cette nouvelle méthode (Tzovara et al., 2012a,b). Son applicabilité générale a été démontrée avec trois ensembles de données, deux dans le domaine visuel et un dans l'auditif. Ce développement a permis de cibler deux nouvelles lignes de recherche, la première dans le domaine des neurosciences cognitives et l'autre dans le domaine des neurosciences cliniques, représentant respectivement la deuxième et troisième partie de ce projet. En particulier, pour la partie cognitive, j'ai appliqué cette méthode pour évaluer l'information temporelle de la prise des décisions (Tzovara et al., 2012c). En se basant sur l'activité topographique de l'EEG au niveau de chaque essai pendant la présentation de la récompense liée à un choix, on a pu prédire les décisions suivantes des sujets (en termes d'exploration/exploitation). Cette prédiction s'appuie sur une différence topographique qui apparaît en moyenne ~516ms après la présentation de la récompense. Ces résultats exploitent pour la première fois, les corrélés temporels des décisions au niveau de chaque sujet séparément et montrent que les générateurs neuronaux de ces décisions commencent à différentier leurs réponses déjà depuis ~880ms avant que les sujets appuient sur le bouton. Finalement, pour la dernière partie de ce projet, je me suis focalisée sur une étude Clinique afin d'évaluer le degré des fonctions neuronales intactes chez les patients comateux. Des réponses EEG auditives ont été examinées avec un paradigme classique de mismatch negativity, pendant la phase précoce du coma qui est actuellement sous-investiguée. En utilisant la méthode de décodage développée dans la première partie de la thèse, j'ai pu quantifier le degré de discrimination auditive au niveau de chaque patient (Tzovara et al., in press). Nos résultats montrent pour la première fois que même des patients comateux qui ne vont pas survivre peuvent discriminer des sons au niveau neuronal, lors de la phase aigue du coma. De plus, une amélioration dans la discrimination auditive pendant les premières 48heures du coma a été observée seulement chez des patients qui se sont réveillés par la suite (100% de valeur prédictive pour un réveil).

Decoding sequence learning from single-trial intracranial EEG in humans.

We propose and validate a multivariate classification algorithm for characterizing changes in human intracranial electroencephalographic data (iEEG) after learning motor sequences. The algorithm is based on a Hidden Markov Model (HMM) that captures spatio-temporal properties of the iEEG at the level of single trials. Continuous intracranial iEEG was acquired during two sessions (one before and one after a night of sleep) in two patients with depth electrodes implanted in several brain areas. They performed a visuomotor sequence (serial reaction time task, SRTT) using the fingers of their non-dominant hand. Our results show that the decoding algorithm correctly classified single iEEG trials from the trained sequence as belonging to either the initial training phase (day 1, before sleep) or a later consolidated phase (day 2, after sleep), whereas it failed to do so for trials belonging to a control condition (pseudo-random sequence). Accurate single-trial classification was achieved by taking advantage of the distributed pattern of neural activity. However, across all the contacts the hippocampus contributed most significantly to the classification accuracy for both patients, and one fronto-striatal contact for one patient. Together, these human intracranial findings demonstrate that a multivariate decoding approach can detect learning-related changes at the level of single-trial iEEG. Because it allows an unbiased identification of brain sites contributing to a behavioral effect (or experimental condition) at the level of single subject, this approach could be usefully applied to assess the neural correlates of other complex cognitive functions in patients implanted with multiple electrodes.

Decoding stimulus-related information from single-trial EEG responses based on voltage topographies

Neuroimaging studies typically compare experimental conditions using average brain responses, thereby overlooking the stimulus-related information conveyed by distributed spatio-temporal patterns of single-trial responses. Here, we take advantage of this rich information at a single-trial level to decode stimulus-related signals in two event-related potential (ERP) studies. Our method models the statistical distribution of the voltage topographies with a Gaussian Mixture Model (GMM), which reduces the dataset to a number of representative voltage topographies. The degree of presence of these topographies across trials at specific latencies is then used to classify experimental conditions. We tested the algorithm using a cross-validation procedure in two independent EEG datasets. In the first ERP study, we classified left- versus right-hemifield checkerboard stimuli for upper and lower visual hemifields. In a second ERP study, when functional differences cannot be assumed, we classified initial versus repeated presentations of visual objects. With minimal a priori information, the GMM model provides neurophysiologically interpretable features - vis à vis voltage topographies - as well as dynamic information about brain function. This method can in principle be applied to any ERP dataset testing the functional relevance of specific time periods for stimulus processing, the predictability of subject's behavior and cognitive states, and the discrimination between healthy and clinical populations.

A game theoretical approach to the algebraic counterpart of the Wagner hierarchy

La hiérarchie de Wagner constitue à ce jour la plus fine classification des langages ω-réguliers. Par ailleurs, l'approche algébrique de la théorie de langages formels montre que ces ensembles ω-réguliers correspondent précisément aux langages reconnaissables par des ω-semigroupes finis pointés. Ce travail s'inscrit dans ce contexte en fournissant une description complète de la contrepartie algébrique de la hiérarchie de Wagner, et ce par le biais de la théorie descriptive des jeux de Wadge. Plus précisément, nous montrons d'abord que le degré de Wagner d'un langage ω-régulier est effectivement un invariant syntaxique. Nous définissons ensuite une relation de réduction entre ω-semigroupes pointés par le biais d'un jeu infini de type Wadge. La collection de ces structures algébriques ordonnée par cette relation apparaît alors comme étant isomorphe à la hiérarchie de Wagner, soit un quasi bon ordre décidable de largeur 2 et de hauteur ω. Nous exposons par la suite une procédure de décidabilité de cette hiérarchie algébrique : on décrit une représentation graphique des ω-semigroupes finis pointés, puis un algorithme sur ces structures graphiques qui calcule le degré de Wagner de n'importe quel élément. Ainsi le degré de Wagner de tout langage ω-régulier peut être calculé de manière effective directement sur son image syntaxique. Nous montrons ensuite comment construire directement et inductivement une structure de n''importe quel degré. Nous terminons par une description détaillée des invariants algébriques qui caractérisent tous les degrés de cette hiérarchie. Abstract The Wagner hierarchy is known so far to be the most refined topological classification of ω-rational languages. Also, the algebraic study of formal languages shows that these ω-rational sets correspond precisely to the languages recognizable by finite pointed ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on finite pointed ω-semigroups by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height $\omega^\omega$. We also describe a decidability procedure of this hierarchy: we introduce a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of every ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every Wagner degree of this hierarchy.

Decoding auditory EEG responses in healthy and clinical populations: A comparative study.

BACKGROUND: Analyses of brain responses to external stimuli are typically based on the means computed across conditions. However in many cognitive and clinical applications, taking into account their variability across trials has turned out to be statistically more sensitive than comparing their means. NEW METHOD: In this study we present a novel implementation of a single-trial topographic analysis (STTA) for discriminating auditory evoked potentials at predefined time-windows. This analysis has been previously introduced for extracting spatio-temporal features at the level of the whole neural response. Adapting the STTA on specific time windows is an essential step for comparing its performance to other time-window based algorithms. RESULTS: We analyzed responses to standard vs. deviant sounds and showed that the new implementation of the STTA gives above-chance decoding results in all subjects (in comparison to 7 out of 11 with the original method). In comatose patients, the improvement of the decoding performance was even more pronounced than in healthy controls and doubled the number of significant results. COMPARISON WITH EXISTING METHOD(S): We compared the results obtained with the new STTA to those based on a logistic regression in healthy controls and patients. We showed that the first of these two comparisons provided a better performance of the logistic regression; however only the new STTA provided significant results in comatose patients at group level. CONCLUSIONS: Our results provide quantitative evidence that a systematic investigation of the accuracy of established methods in normal and clinical population is an essential step for optimizing decoding performance.

Quantifying the time for accurate EEG decoding of single value-based decisions.

BACKGROUND: Recent neuroimaging studies suggest that value-based decision-making may rely on mechanisms of evidence accumulation. However no studies have explicitly investigated the time when single decisions are taken based on such an accumulation process. NEW METHOD: Here, we outline a novel electroencephalography (EEG) decoding technique which is based on accumulating the probability of appearance of prototypical voltage topographies and can be used for predicting subjects' decisions. We use this approach for studying the time-course of single decisions, during a task where subjects were asked to compare reward vs. loss points for accepting or rejecting offers. RESULTS: We show that based on this new method, we can accurately decode decisions for the majority of the subjects. The typical time-period for accurate decoding was modulated by task difficulty on a trial-by-trial basis. Typical latencies of when decisions are made were detected at ∼500ms for 'easy' vs. ∼700ms for 'hard' decisions, well before subjects' response (∼340ms). Importantly, this decision time correlated with the drift rates of a diffusion model, evaluated independently at the behavioral level. COMPARISON WITH EXISTING METHOD(S): We compare the performance of our algorithm with logistic regression and support vector machine and show that we obtain significant results for a higher number of subjects than with these two approaches. We also carry out analyses at the average event-related potential level, for comparison with previous studies on decision-making. CONCLUSIONS: We present a novel approach for studying the timing of value-based decision-making, by accumulating patterns of topographic EEG activity at single-trial level.

A cellular system for spatial signal decoding in chemical gradients.

Directional cell growth requires that cells read and interpret shallow chemical gradients, but how the gradient directional information is identified remains elusive. We use single-cell analysis and mathematical modeling to define the cellular gradient decoding network in yeast. Our results demonstrate that the spatial information of the gradient signal is read locally within the polarity site complex using double-positive feedback between the GTPase Cdc42 and trafficking of the receptor Ste2. Spatial decoding critically depends on low Cdc42 activity, which is maintained by the MAPK Fus3 through sequestration of the Cdc42 activator Cdc24. Deregulated Cdc42 or Ste2 trafficking prevents gradient decoding and leads to mis-oriented growth. Our work discovers how a conserved set of components assembles a network integrating signal intensity and directionality to decode the spatial information contained in chemical gradients.

Robust discrimination between EEG responses to categories of environmental sounds in early coma.

Humans can recognize categories of environmental sounds, including vocalizations produced by humans and animals and the sounds of man-made objects. Most neuroimaging investigations of environmental sound discrimination have studied subjects while consciously perceiving and often explicitly recognizing the stimuli. Consequently, it remains unclear to what extent auditory object processing occurs independently of task demands and consciousness. Studies in animal models have shown that environmental sound discrimination at a neural level persists even in anesthetized preparations, whereas data from anesthetized humans has thus far provided null results. Here, we studied comatose patients as a model of environmental sound discrimination capacities during unconsciousness. We included 19 comatose patients treated with therapeutic hypothermia (TH) during the first 2 days of coma, while recording nineteen-channel electroencephalography (EEG). At the level of each individual patient, we applied a decoding algorithm to quantify the differential EEG responses to human vs. animal vocalizations as well as to sounds of living vocalizations vs. man-made objects. Discrimination between vocalization types was accurate in 11 patients and discrimination between sounds from living and man-made sources in 10 patients. At the group level, the results were significant only for the comparison between vocalization types. These results lay the groundwork for disentangling truly preferential activations in response to auditory categories, and the contribution of awareness to auditory category discrimination.

MAMOT: hidden Markov modeling tool.

Hidden Markov models (HMMs) are probabilistic models that are well adapted to many tasks in bioinformatics, for example, for predicting the occurrence of specific motifs in biological sequences. MAMOT is a command-line program for Unix-like operating systems, including MacOS X, that we developed to allow scientists to apply HMMs more easily in their research. One can define the architecture and initial parameters of the model in a text file and then use MAMOT for parameter optimization on example data, decoding (like predicting motif occurrence in sequences) and the production of stochastic sequences generated according to the probabilistic model. Two examples for which models are provided are coiled-coil domains in protein sequences and protein binding sites in DNA. A wealth of useful features include the use of pseudocounts, state tying and fixing of selected parameters in learning, and the inclusion of prior probabilities in decoding. AVAILABILITY: MAMOT is implemented in C++, and is distributed under the GNU General Public Licence (GPL). The software, documentation, and example model files can be found at http://bcf.isb-sib.ch/mamot

Binary classification of dyslipidemia from the waist-to-hip ratio and body mass index: a comparison of linear, logistic, and CART models.

BACKGROUND: We sought to improve upon previously published statistical modeling strategies for binary classification of dyslipidemia for general population screening purposes based on the waist-to-hip circumference ratio and body mass index anthropometric measurements. METHODS: Study subjects were participants in WHO-MONICA population-based surveys conducted in two Swiss regions. Outcome variables were based on the total serum cholesterol to high density lipoprotein cholesterol ratio. The other potential predictor variables were gender, age, current cigarette smoking, and hypertension. The models investigated were: (i) linear regression; (ii) logistic classification; (iii) regression trees; (iv) classification trees (iii and iv are collectively known as "CART"). Binary classification performance of the region-specific models was externally validated by classifying the subjects from the other region. RESULTS: Waist-to-hip circumference ratio and body mass index remained modest predictors of dyslipidemia. Correct classification rates for all models were 60-80%, with marked gender differences. Gender-specific models provided only small gains in classification. The external validations provided assurance about the stability of the models. CONCLUSIONS: There were no striking differences between either the algebraic (i, ii) vs. non-algebraic (iii, iv), or the regression (i, iii) vs. classification (ii, iv) modeling approaches. Anticipated advantages of the CART vs. simple additive linear and logistic models were less than expected in this particular application with a relatively small set of predictor variables. CART models may be more useful when considering main effects and interactions between larger sets of predictor variables.

Synchronization of EEG: bivariate and multivariate measures.

Synchronization behavior of electroencephalographic (EEG) signals is important for decoding information processing in the human brain. Modern multichannel EEG allows a transition from traditional measurements of synchronization in pairs of EEG signals to whole-brain synchronization maps. The latter can be based on bivariate measures (BM) via averaging over pair-wise values or, alternatively, on multivariate measures (MM), which directly ascribe a single value to the synchronization in a group. In order to compare BM versus MM, we applied nine different estimators to simulated multivariate time series with known parameters and to real EEGs.We found widespread correlations between BM and MM, which were almost frequency-independent for all the measures except coherence. The analysis of the behavior of synchronization measures in simulated settings with variable coupling strength, connection probability, and parameter mismatch showed that some of them, including S-estimator, S-Renyi, omega, and coherence, aremore sensitive to linear interdependences,while others, like mutual information and phase locking value, are more responsive to nonlinear effects. Onemust consider these properties together with the fact thatMM are computationally less expensive and, therefore, more efficient for the large-scale data sets than BM while choosing a synchronization measure for EEG analysis.

Temporo-Parietal Junction encodes the Embodied Self: Neuro-Robotics, fMRI, and Lesion mapping.

Introduction: Neuroimaging of the self focused on high-level mechanisms such as language, memory or imagery of the self. Recent evidence suggests that low-level mechanisms of multisensory and sensorimotor integration may play a fundamental role in encoding self-location and the first-person perspective (Blanke and Metzinger, 2009). Neurological patients with out-of body experiences (OBE) suffer from abnormal self-location and the first-person perspective due to a damage in the temporo-parietal junction (Blanke et al., 2004). Although self-location and the first-person perspective can be studied experimentally (Lenggenhager et al., 2009), the neural underpinnings of self-location have yet to be investigated. To investigate the brain network involved in self-location and first-person perspective we used visuo-tactile multisensory conflict, magnetic resonance (MR)-compatible robotics, and fMRI in study 1, and lesion analysis in a sample of 9 patients with OBE due to focal brain damage in study 2. Methods: Twenty-two participants saw a video showing either a person's back or an empty room being stroked (visual stimuli) while the MR-compatible robotic device stroked their back (tactile stimulation). Direction and speed of the seen stroking could either correspond (synchronous) or not (asynchronous) to those of the seen stroking. Each run comprised the four conditions according to a 2x2 factorial design with Object (Body, No-Body) and Synchrony (Synchronous, Asynchronous) as main factors. Self-location was estimated using the mental ball dropping (MBD; Lenggenhager et al., 2009). After the fMRI session participants completed a 6-item adapted from the original questionnaire created by Botvinick and Cohen (1998) and based on questions and data obtained by Lenggenhager et al. (2007, 2009). They were also asked to complete a questionnaire to disclose the perspective they adopted during the illusion. Response times (RTs) for the MBD and fMRI data were analyzed with a 3-way mixed model ANOVA with the in-between factor Perspective (up, down) and the two with-in factors Object (body, no-body) and Stroking (synchronous, asynchronous). Quantitative lesion analysis was performed using MRIcron (Rorden et al., 2007). We compared the distributions of brain lesions confirmed by multimodality imaging (Knowlton, 2004) in patients with OBE with those showing complex visual hallucinations involving people or faces, but without any disturbance of self-location and first person perspective. Nine patients with OBE were investigated. The control group comprised 8 patients. Structural imaging data were available for normalization and co-registration in all the patients. Normalization of each patient's lesion into the common MNI (Montreal Neurological Institute) reference space permitted simple, voxel-wise, algebraic comparisons to be made. Results: Even if in the scanner all participants were lying on their back and were facing upwards, analysis of perspective showed that half of the participants had the impression to be looking down at the virtual human body below them, despite any cues about their body position (Down-group). The other participants had the impression to be looking up at the virtual body above them (Up-group). Analysis of Q3 ("How strong was the feeling that the body you saw was you?") indicated stronger self-identification with the virtual body during the synchronous stroking. RTs in the MBD task confirmed these subjective data (significant 3-way interaction between perspective, object and stroking). fMRI results showed eight cortical regions where the BOLD signal was significantly different during at least one of the conditions resulting from the combination of Object and Stroking, relative to baseline: right and left temporo-parietal junction, right EBA, left middle occipito-temporal gyrus, left postcentral gyrus, right medial parietal lobe, bilateral medial occipital lobe (Fig 1). The activation patterns in right and left temporo-parietal junction and right EBA reflected changes in self-location and perspective as revealed by statistical analysis that was performed on the percentage of BOLD change with respect to the baseline. Statistical lesion overlap comparison (using nonparametric voxel based lesion symptom mapping) with respect to the control group revealed the right temporo-parietal junction, centered at the angular gyrus (Talairach coordinates x = 54, y =-52, z = 26; p>0.05, FDR corrected). Conclusions: The present questionnaire and behavioural results show that - despite the noisy and constraining MR environment) our participants had predictable changes in self-location, self-identification, and first-person perspective when robotic tactile stroking was applied synchronously with the robotic visual stroking. fMRI data in healthy participants and lesion data in patients with abnormal self-location and first-person perspective jointly revealed that the temporo-parietal cortex especially in the right hemisphere encodes these conscious experiences. We argue that temporo-parietal activity reflects the experience of the conscious "I" as embodied and localized within bodily space.

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.