Numerical Judgments Research Articles

Spontaneous representation of numerosity zero in a deep neural network for visual object recognition

SummaryConceiving “nothing” as a numerical value zero is considered a sophisticated numerical capability that humans share with cognitively advanced animals. We demonstrate that representation of zero spontaneously emerges in a deep learning neural network without any number training. As a signature of numerical quantity representation, and similar to real neurons from animals, numerosity zero network units show maximum activity to empty sets and a gradual decrease in activity with increasing countable numerosities. This indicates that the network spontaneously ordered numerosity zero as the smallest numerical value along the number line. Removal of empty-set network units caused specific deficits in the network's judgment of numerosity zero, thus reflecting these units' functional relevance. These findings suggest that processing visual information is sufficient for a visual number sense that includes zero to emerge and explains why cognitively advanced animals with whom we share a nonverbal number system exhibit rudiments of numerosity zero.

The effect of grouping on numerosity judgment varies with the autism-spectrum personality traits, consistent with changes in local-global biases.

The effect of grouping on numerosity judgment varies with the autism-spectrum personality traits, consistent with changes in local-global biases.

Response bias in numerosity perception at early judgments and systematic underestimation.

Mental number representation relies on mapping numerosity based on nonsymbolic stimuli to symbolic magnitudes. It is known that mental number representation builds on a logarithmic scale, and thus numerosity decisions result in underestimation. In the current study, we investigated the temporal dynamics of numerosity perception in four experiments by employing the response-deadline SAT procedure. We presented random number of dots and required participants to make a numerosity judgment by comparing the perceived number of dots to 50. Using temporal dynamics in numerosity perception allowed us to observe a response bias at early decisions and a systematic underestimation at late decisions. In all three experiments, providing feedback diminished the magnitude of underestimation, whereas in Experiment 3 the absence of feedback resulted in greater underestimation errors. These results were in accordance with the findings that suggested feedback is necessary for the calibration of the mental number representation.

Individual differences in anchoring susceptibility: Verbal reasoning, autistic tendencies, and narcissism

Hundreds of studies have documented anchoring, whereby people's numerical judgments assimilate to previously considered values. Much less is known, however, about individual differences in susceptibility to anchoring effects. In this study (N = 399), we found that people with stronger verbal reasoning skills and people higher in social autistic tendencies were less influenced by anchors, whereas people higher in narcissism were more influenced by anchors. Although relatively small, these relations contribute to the integration of the judgment and decision making literature on anchoring with personality and individual differences approaches to cognition.

Consequences of underexplored variation in biodiversity indices used for land-use prioritization.

For biodiversity protection to play a persuasive role in land-use planning, conservationists must be able to offer objective systems for ranking which natural areas to protect or convert. Representing biodiversity in spatially explicit indices is challenging because it entails numerous judgments regarding what variables to measure, how to measure them, and how to combine them. Surprisingly few studies have explored this variation. Here, we explore how this variation affects which areas are selected for agricultural conversion by a land-use prioritization model designed to reduce the biodiversity losses associated with agricultural expansion in Zambia. We first explore the similarity between model recommendations generated by three recently published composite indices and a commonly used rarity-weighted species richness metric. We then explore four underlying sources of ecological and methodological variation within these and other approaches, including different terrestrial vertebrate taxonomic groups, different species-richness metrics, different mathematical methods for combining layers, and different spatial resolutions of inputs. The results generated using different biodiversity approaches show very low spatial agreement regarding which areas to convert to agriculture. There is little overlap in areas identified for conversion using previously published indices (mean Jaccard similarity, Jw , between 0.3 and 3.7%), different taxonomic groups (5.0%<mean Jw <13.5%), or different measures of species richness (15.6%<mean Jw <33.7%). Even with shared conservation goals, different methods for combining layers and different input spatial resolutions still produce meaningful, though smaller, differences among areas selected for conversion (40.9%<mean Jw <67.5%). The choice of taxonomic group had the largest effect on conservation priorities, followed by the choice of species richness metric, the choice of combination method, and finally the choice of spatial resolution. These disagreements highlight the challenge of objectively representing biodiversity in land-use planning tools, and present a credibility challenge for conservation scientists seeking to inform policy making. Our results suggest an urgent need for a more consistent and transparent framework for designing the biodiversity indices used in land-use planning, which we propose here.

Older and younger adults\u2019 hindsight bias after positive and negative outcomes

After learning about facts or outcomes of events, people overestimate in hindsight what they knew in foresight. Prior research has shown that this hindsight bias is more pronounced in older than in younger adults. However, this robust finding is based primarily on a specific paradigm that requires generating and recalling numerical judgments to general knowledge questions that deal with emotionally neutral content. As older and younger adults tend to process positive and negative information differently, they might also show differences in hindsight bias after positive and negative outcomes. Furthermore, hindsight bias can manifest itself as a bias in memory for prior given judgments, but also as retrospective impressions of inevitability and foreseeability. Currently, there is no research on age differences in all three manifestations of hindsight bias. In this study, younger (N = 46, 18–30 years) and older adults (N = 45, 64–90 years) listened to everyday-life scenarios that ended positively or negatively, recalled the expectation they previously held about the outcome (to measure the memory component of hindsight bias), and rated each outcome’s foreseeability and inevitability. Compared with younger adults, older adults recalled their prior expectations as closer to the actual outcomes (i.e., they showed a larger memory component of hindsight bias), and this age difference was more pronounced for negative than for positive outcomes. Inevitability and foreseeability impressions, however, did not differ between the age groups. Thus, there are age differences in hindsight bias after positive and negative outcomes, but only with regard to memory for prior judgments.

Benefits of triple acoustic beamforming during speech-on-speech masking and sound localization for bilateral cochlear-implant users.

Bilateral cochlear-implant (CI) users struggle to understand speech in noisy environments despite receiving some spatial-hearing benefits. One potential solution is to provide acoustic beamforming. A headphone-based experiment was conducted to compare speech understanding under natural CI listening conditions and for two non-adaptive beamformers, one single beam and one binaural, called "triple beam," which provides an improved signal-to-noise ratio (beamforming benefit) and usable spatial cues by reintroducing interaural level differences. Speech reception thresholds (SRTs) for speech-on-speech masking were measured with target speech presented in front and two maskers in co-located or narrow/wide separations. Numerosity judgments and sound-localization performance also were measured. Natural spatial cues, single-beam, and triple-beam conditions were compared. For CI listeners, there was a negligible change in SRTs when comparing co-located to separated maskers for natural listening conditions. In contrast, there were 4.9- and 16.9-dB improvements in SRTs for the beamformer and 3.5- and 12.3-dB improvements for triple beam (narrow and wide separations). Similar results were found for normal-hearing listeners presented with vocoded stimuli. Single beam improved speech-on-speech masking performance but yielded poor sound localization. Triple beam improved speech-on-speech masking performance, albeit less than the single beam, and sound localization. Thus, triple beam was the most versatile across multiple spatial-hearing domains.

Grouping-Induced Numerosity Biases Vary with Autistic-Like Personality Traits

Individuals with autism spectrum disorder are thought to have a more local than global perceptual style. We used a novel paradigm to investigate how grouping-induced response biases in numerosity judgments depend on autistic-like personality traits in neurotypical adults. Participants judged the numerosity of clouds of dot-pairs connected by thin lines, known to cause underestimation of numerosity. The underestimation bias correlated strongly with autism-spectrum quotient (r = 0.72, Bayes factor > 100), being weaker for participants with high autistic traits. As connecting dots probably activates global grouping mechanisms, causing dot-pairs to be processed as an integrated whole rather than as individual dots, the results suggest that these grouping mechanisms may be weaker in individuals self-reporting high levels of autistic-like traits.

Spatial-numerical associations from a novel paradigm support the mental number line account.

Multiple tasks have been used to demonstrate the relation between numbers and space. The classic interpretation of these directional spatial-numerical associations (d-SNAs) is that they are the product of a mental number line (MNL), in which numerical magnitude is intrinsically associated with spatial position. The alternative account is that d-SNAs reflect task demands, such as explicit numerical judgements and/or categorical responses. In the novel "Where was The Number?" task, no explicit numerical judgements were made. Participants were simply required to reproduce the location of a numeral within a rectangular space. Using a between-subject design, we found that numbers, but not letters, biased participants' responses along the horizontal dimension, such that larger numbers were placed more rightward than smaller numbers, even when participants completed a concurrent verbal working memory task. These findings are consistent with the MNL account, such that numbers specifically are inherently left-to-right oriented in Western participants.

Nonsymbolic and symbolic representations of null numerosity.

Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1-9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

Fixed-Cost Pooling Strategies

The empirical nature of Information Retrieval (IR) mandates strong experimental practices. A keystone of such experimental practices is the Cranfield evaluation paradigm. Within this paradigm, the collection of relevance judgments has been the subject of intense scientific investigation. This is because, on one hand, consistent, precise, and numerous judgements are keys to reducing evaluation uncertainty and test collection bias; on the other hand, however, relevance judgements are costly to collect. The selection of which documents to judge for relevance, known as pooling method , has therefore a great impact on IR evaluation. In this paper we focus on the bias introduced by the pooling method, known as pool bias , which affects the reusability of test collections, in particular when building test collections with a limited budget. In this paper we formalize and evaluate a set of 22 pooling strategies based on: traditional strategies, voting systems, retrieval fusion methods, evaluation measures, and multi-armed bandit models. To do this we run a large-scale evaluation by considering a set of 9 standard TREC test collections, in which we show that the choice of the pooling strategy has significant effects on the cost needed to obtain an unbiased test collection. We also identify the least biased pooling strategy in terms of pool bias according to three IR evaluation measures: AP, NDCG, and P@10.

Separable effects of the approximate number system, symbolic number knowledge, and number ordering ability on early arithmetic development

There is evidence that early variations in the development of an approximate number system (ANS) and symbolic number understanding are both influences on the later development of formal arithmetic skills. We report a large-scale (N = 552) longitudinal study of the predictors of arithmetic spanning a critical developmental period (the first 3 years of formal education). Variations in early knowledge of symbolic representations of number and the ordinal associations between them are direct predictors of later arithmetic skills. The development of number ordering ability is in turn predicted by earlier variations in arithmetic, the ANS (numerosity judgments), and rapid automatized naming (RAN). These findings have important implications for theories of numerical and arithmetical development and potentially for the teaching of these skills.

A two-stage ranking method to minimize ordinal violation for pairwise comparisons

Pairwise comparison is a powerful tool in intelligent decision making systems. Items are compared using numerical judgments that estimate the item weight ratios, which are provided by decision makers or transformed by objective data. The reliable assignment of numerical judgments is important, because judgment variety leads to significantly different results. However, it is difficult to provide exact ratios for decision makers owing to the limitations of knowledge. Although objective data provides an estimation of the ratios, a series of artificially defined rules is required to transform data into numerical judgments; however, these rules are subjective and arbitrary. Conversely, the dominance relationships between the items are obvious and reliable. Therefore, this study proposes a two-stage ranking method to minimize the ordinal violation that indicates the degree of conflict between the ranking result and the dominance. First, a 0–1 integer programming is designed and solved. Then, the second stage focuses on the topological sorting of nodes in a graph constructed using the optimal solution. To validate the effectiveness of the proposed method, we perform two experiments: a numerical example provided by participants and a real-world application involving ranking top tennis players. The results show that the proposed method not only avoids subjectivity in judgments, but also obtains the ranking that has the minimum ordinal violation among the compared methods.

Judgment errors in naturalistic numerical estimation

People estimate numerical quantities (such as the calories of foods) on a day-to-day basis. Although these estimates influence behavior and determine wellbeing, they are prone to two important types of errors. Scaling errors occur when people make mistakes reporting their beliefs about a particular numerical quantity (e.g. by inflating small numbers). Belief errors occur when people make mistakes using their knowledge of the judgment target to form their beliefs about the numerical quantity (e.g. by overweighting certain cues). In this paper, we quantitatively model numerical estimates, and in turn, scaling and belief errors, in everyday judgment tasks. Our approach is unique in using insights from semantic memory research to specify knowledge for naturalistic judgment targets, allowing our models to formally describe nuanced errors in belief not considered in prior research. In Studies 1 and 2, we find that belief error models predict participant estimates and errors with very high out-of-sample accuracy rates, significantly outperforming the predictions of scaling error models. In fact, the best-fitting belief error models can closely mimic the inverse-S shaped patterns captured by scaling error models, suggesting that the types of responses previously attributed to scaling errors can be seen as errors of belief. In Studies 3 to 8, we find that belief error models are also able to predict people's responses in semantic judgment, free association, and verbal protocol tasks related to numerical judgment, and thus provide a good account of the cognitive underpinnings of judgment.

When "bad" is good: How evaluative judgments eliminate the standard anchoring effect.

Early accounts of judgmental anchoring attribute the effect to a deliberate, but insufficient, adjustment process; more recent theories point to automatic, priming-based processes as the underlying cause. In this article we introduce a novel anchor assessment manipulation and a decompositional analysis of the standard anchoring effect to determine the extent to which anchoring is driven by automatic versus deliberate processes. Prior to providing a target estimate, participants indicated whether the target was greater or less than the anchor, or whether the anchor would make a good or bad target estimate. Contrary to predictions of priming-based accounts, the decomposition of the anchoring effect revealed that participants generally provided estimates consistent with their prior assessment; in particular, anchoring was eliminated when participants considered the anchor to be a bad target estimate. These findings challenge the view of anchoring as an inevitable bias of numerical judgment and indicate that people have significant control over how they manage numerical information in judgments under uncertainty. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Factors influencing bilinguals’ speed and accuracy of number judgments across languages: A meta-analytic review

• For bilinguals, number judgments are overall faster and more accurate in the first language. • Early schooling in the first language predicts a first language advantage in number speed and accuracy. • Low or medium second language proficiency predicts a first language advantage in number task performance. • Late onset of bilingualism predicts a first language advantage in number task performance. • The findings support models of number representation that posit language experience-based format effects. Are bilinguals faster and/or better at performing numerical judgments in one language than in their other language? What factors moderate language differences in number processing ease? To address these issues, we conducted a meta-analysis of 38 studies (based on 1861 participants) in which bilinguals’ speeded performance and accuracy on number tasks was compared across languages within subjects. Analysis of 63 independent effect sizes for reaction time data and 28 for accuracy data showed an overall language difference favoring the first language. This effect was particularly observed in reaction time studies for number tasks involving arithmetic judgments or number naming. The L1 effect was characteristic of bilinguals with low or medium proficiency in the second language, those who were schooled in the first language, and those whose stated preference for arithmetic was the first language. Further, a first language advantage was most characteristic of late bilinguals (for reaction time). For accuracy, additional moderators of a first language advantage included differences between the languages in decade unit counting system. No other effects were significant. The findings lend support to models of number cognition that posit format-sensitive codes. They further clarify that a privileging of the first language is not uniform but varies by bilingualism onset, proficiency, language of early schooling, and language-specific characteristics.

Torture and inhuman treatment in the context of the relevant practice of the European Court of Human Rights

The article examines the problems of torture and ill-treatment in the relevant practice of the European Court of Human Rights. The Convention for the Protection of Human Rights and Fundamental Freedoms (the Convention) has become an important place in the practice of application of criminal law of Ukraine. After analyzing some practice ECtHR about violation of Article 3 of the Convention, the forms of torture and inhuman treatment were singled out. These include: improper investigation, improper treatment of convicts, improper medical care of convicts, threats of torture, ignoring allegations of threats of torture or ill-treatment, etc.It was found that despite numerous judgments of the European Court of Human Rights against Ukraine, both certain provisions of law and practice of their application continue to contradict the provisions of the Convention, in particular Article 3.It is established that currently the problem of treatment of people by law enforcement agencies and other state representatives is urgent and systemic.The paper proposes that some European experience should be borrowed in providing a real mechanism for the protection of human rights and freedoms. Reforms are needed to ensure the effective participation of victims in criminal proceedings and the use of incentives, as well as the practice of compensating victims. It seems that given the accompanying circumstances, failure to provide the victim with a real opportunity to participate in criminal proceedings, as well as the lack of informing the victims about the progress of the investigation can be considered as inhuman treatment within the meaning of Art. 3 of the Convention.

Examining aging and numerosity using an integrated diffusion model.

Two experiments are presented that use tasks common in research in numerical cognition with young adults and older adults as subjects. In these tasks, one or two arrays of dots are displayed, and subjects decide whether there are more or fewer dots of one kind than another. Results show that older adults, relative to young adults, tend to rely more on the perceptual feature, area, in making numerosity judgments when area is correlated with numerosity. Also, convex hull unexpectedly shows different effects depending on the task (being either correlated with numerosity or anticorrelated). Accuracy and response time (RT) data are interpreted with the integration of the diffusion decision model with models for the representation of numerosity. One model assumes that the representation of the difference depends on the difference between the numerosities and that standard deviations (SDs) increase linearly with numerosity, and the other model assumes a log representation with constant SDs. The representational models have coefficients that are applied to differences between two numerosities to produce drift rates and SDs in drift rates in the decision process. The two tasks produce qualitatively different patterns of RTs: One model fits results from one task, but the results are mixed for the other task. The effects of age on model parameters show a modest decrease in evidence driving the decision process, an increase in the duration of processes outside the decision process (nondecision time), and an increase in the amount of evidence needed to make a decision (boundary separation). (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Non-informative vision improves spatial tactile discrimination on the shoulder but does not influence detection sensitivity

Vision of the body has been reported to improve tactile acuity even when vision is not informative about the actual tactile stimulation. However, it is currently unclear whether this effect is limited to body parts such as hand, forearm or foot that can be normally viewed, or it also generalizes to body locations, such as the shoulder, that are rarely before our own eyes. In this study, subjects consecutively performed a detection threshold task and a numerosity judgment task of tactile stimuli on the shoulder. Meanwhile, they watched either a real-time video showing their shoulder or simply a fixation cross as control condition. We show that non-informative vision improves tactile numerosity judgment which might involve tactile acuity, but not tactile sensitivity. Furthermore, the improvement in tactile accuracy modulated by vision seems to be due to an enhanced ability in discriminating the number of adjacent active electrodes. These results are consistent with the view that bimodal visuotactile neurons sharp tactile receptive fields in an early somatosensory map, probably via top-down modulation of lateral inhibition.

Do estimates of numerosity really adhere to Weber\u2019s law? A reexamination of two case studies

Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber’s law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate “number sense.” We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber’s law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber’s law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.