Numerical Judgments Research Articles

Decision-making from temporally accumulated conflicting evidence: The more the merrier.

How do humans evaluate temporally accumulated discrete pieces of evidence and arrive at a decision despite the presence of conflicting evidence? In the present study, we showed human participants a sequential presentation of objects drawn from two novel object categories and asked them to decide whether a given presentation contained more objects from one or the other category. We found that both a more disparate ratio and greater numerosity of objects improved both reaction time (RT) and accuracy. The effect of numerosity was separate from ratio, where with a fixed object ratio, sequences with more total objects had lower RT and lower error rates than those with fewer total objects. We replicated these results across three experiments. Additionally, even with the total presentation duration equated and with the motor response assignment varied from trial to trial, an effect of numerosity was still found in RT. The same RT benefit was also present when objects were shown simultaneously, rather than sequentially. Together, these results showed that, for comparative numerosity judgment involving sequential displays, there was a benefit of numerosity, such that showing more objects independent of the object ratio and the total presentation time led to faster decision performance.

Four Approaches to Supposition

Suppositions can be introduced in either the indicative or subjunctive mood. There are also two types of judgments that may be initiated by suppositions of either sort: qualitative (binary) judgments and quantitative (numerical) ones. The former are judgments about whether a given proposition is acceptable and the latter are about how acceptable it is. We systematically explicate the relationships between canonical representatives of each of the four available types of theories. For the qualitative accounts, our representative theories of indicative and subjunctive supposition are based on the belief change operations given by AGM revision and KM update respectively; for the quantitative ones, we consider those given by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as ‘provisional belief revision’ wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we appeal to a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update.

EEG-representational geometries and psychometric distortions in approximate numerical judgment.

When judging the average value of sample stimuli (e.g., numbers) people tend to either over- or underweight extreme sample values, depending on task context. In a context of overweighting, recent work has shown that extreme sample values were overly represented also in neural signals, in terms of an anti-compressed geometry of number samples in multivariate electroencephalography (EEG) patterns. Here, we asked whether neural representational geometries may also reflect a relative underweighting of extreme values (i.e., compression) which has been observed behaviorally in a great variety of tasks. We used a simple experimental manipulation (instructions to average a single-stream or to compare dual-streams of samples) to induce compression or anti-compression in behavior when participants judged rapid number sequences. Model-based representational similarity analysis (RSA) replicated the previous finding of neural anti-compression in the dual-stream task, but failed to provide evidence for neural compression in the single-stream task, despite the evidence for compression in behavior. Instead, the results indicated enhanced neural processing of extreme values in either task, regardless of whether extremes were over- or underweighted in subsequent behavioral choice. We further observed more general differences in the neural representation of the sample information between the two tasks. Together, our results indicate a mismatch between sample-level EEG geometries and behavior, which raises new questions about the origin of common psychometric distortions, such as diminishing sensitivity for larger values.

Neurophysiological signatures of approximate number system acuity in preschoolers

BackgroundA hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers. MethodERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task. ResultsOnly children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks. ConclusionResults demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

The influence of visual illusion perception on numerosity estimation could be evolutionarily conserved: exploring the numerical Delboeuf illusion in humans (Homo sapiens) and fish (Poecilia reticulata).

Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.

Visual P2p component responds to perceived numerosity

Numerosity perception is a key ability for human and non-human species, probably mediated by dedicated brain mechanisms. Electrophysiological studies revealed the existence of both early and mid-latency components of the Electrophysiological (EEG) signal sensitive to numerosity changes. However, it is still unknown whether these components respond to physical or perceived variation in numerical attributes. We here tackled this point by recording electrophysiological signal while participants performed a numerosity adaptation task, a robust psychophysical method yielding changes in perceived numerosity judgments despite physical numerosity invariance. Behavioral measures confirmed that the test stimulus was consistently underestimated when presented after a high numerous adaptor while perceived as veridical when presented after a neutral adaptor. Congruently, EEG results revealed a potential at around 200 ms (P2p) which was reduced when the test stimulus was presented after the high numerous adaptor. This result was much prominent over the left posterior cluster of electrodes and correlated significantly with the amount of adaptation. No earlier modulations were retrievable when changes in numerosity were illusory while both early and mid-latency modulations occurred for physical changes. Taken together, our results reveal that mid-latency P2p mainly reflects perceived changes in numerical attributes, while earlier components are likely to be bounded to the physical characteristics of the stimuli. These results suggest that short-term plastic mechanisms induced by numerosity adaptation may involve a relatively late processing stage of the visual hierarchy likely engaging cortical areas beyond the primary visual cortex. Furthermore, these results also indicate mid-latency electrophysiological correlates as a signature of the internal representation of numerical information.

Vibrotactile Spatiotemporal Pattern Recognition in Two-Dimensional Space Around Hand.

This study investigated vibrotactile spatiotemporal pattern recognition in the two-dimensional space around a hand. The participants placed their hands on the medium, and identified the recognized pattern presented in the medium. There were 64 rotational patterns, presented with sequential impulse vibrations. We investigated how well humans recognized the patterns presented around their hand, and identified the pattern factors (e.g., rotational direction) that affected recognition accuracy. The probability of obtaining correct answers was 48.9 %. It was observed that the start and end points, rotational direction, and the number of vibrations affected recognition accuracy. It was also found that patterns starting or ending on the ulnar side ( 0°) of the hand were difficult to recognize, whereas those starting or ending on the distal ( 90°) or proximal side ( 270°) of the hand were easily recognizable. Furthermore, we found a type of the oblique effect. Patterns starting or ending in the oblique direction were more difficult to recognize than those in the cardinal direction. We also found that the clockwise rotational pattern was slightly easier to recognize than the counterclockwise rotational pattern. Finally, the underestimation of the judgment of tactile numerosity explains how the number of vibrations in the patterns affected the recognition accuracy. This result can be used as a baseline when the recognition of spatiotemporal patterns outside the body under other conditions is examined in future studies.

Mental control of uncertainty.

Can you reduce uncertainty by thinking? Intuition suggests that this happens through the elusive process of attention: if we expend mental effort, we can increase the reliability of our sensory data. Models based on "rational inattention" formalize this idea in terms of a trade-off between the costs and benefits of attention. This paper surveys the origin of these models in economics, their connection to rate-distortion theory, and some of their recent applications to psychology and neuroscience. We also report new data from a numerosity judgment task in which we manipulate performance incentives. Consistent with rational inattention, people are able to improve performance on this task when incentivized, in part by increasing the reliability of their sensory data.

Attention to number requires magnitude-specific inhibition

Recent studies have shown that the ability to process number in the face of conflicting dimensions of magnitude is a crucial aspect of numerosity judgments, relying in part on the inhibition of the non-numerical dimensions. Here we report, for the first time, that these inhibitory control processes are specific to the conflicting dimension of magnitude. Using a non-symbolic numerical comparison task adapted to a conflict adaptation paradigm on a group of 82 adults, we show that congruency effects between numerical and non-numerical information were reduced only when the conflicting dimension was the same in the preceding incongruent trial. Attention to number thus involves inhibitory control processes acting at a specific level of information. These results contribute to better characterize the domain general abilities involved in numerical cognition, and provide evidence for a specific interaction between numerosity perception and inhibitory control.

Number comparison under the Ebbinghaus illusion.

A series of studies show interest in how visual attributes affect the estimate of object numbers in a scene. In comparison tasks, it is suggested that larger patches are perceived as more numerous. However, the inequality of density, which changes inversely with the area when numerosity remains constant, may mediate the influence of area on numerosity perception. This study aims to explore the role of area and density in the judgment of numerosity. The Ebbinghaus illusion paradigm was adopted to induce differences in the perceived, rather than the physical, area of the two patches to be compared. Participants were asked to compare the area, density, and the number of the two patches in three tasks. To this end, no PSE (point of subjective equality) bias was found in number comparison with randomly distributed dots, although a significant difference was revealed in the perceived area of the two patches. No PSE bias was found in the density comparison, either. For a comparison, density and number tasks were also conducted with regularly distributed dots. No PSE bias was found in density comparison. By contrast, significant PSE bias showed up in number comparison, and larger patches appeared to be more numerous than smaller patches. The density mechanism was proposed as the basis for number comparison with regular patterns. The individual Weber fractions for regular patterns were not correlated with those for random patterns in the number task, but they were correlated with those for both patterns in the density task. To summarize, numerosity is directly sensed, and numerosity perception is not affected by area inequality induced by the Ebbinghaus illusion. In contrast, density and area are combined to infer numerosity when the approximate numerosity mechanism is disrupted by dot distribution.

Underestimation in temporal numerosity judgments computationally explained by population coding model

The ability to judge numerosity is essential to an animal’s survival. Nevertheless, the number of signals presented in a sequence is often underestimated. We attempted to elucidate the mechanism for the underestimation by means of computational modeling based on population coding. In the model, the population of neurons which were selective to the logarithmic number of signals responded to sequential signals and the population activity was integrated by a temporal window. The total number of signals was decoded by a weighted average of the integrated activity. The model predicted well the general trends in the human data while the prediction was not fully sufficient for the novel aging effect wherein underestimation was significantly greater for the elderly than for the young in specific stimulus conditions. Barring the aging effect, we can conclude that humans judge the number of signals in sequence by temporally integrating the neural representations of numerosity.

Social relevance modulates multisensory integration.

When interacting with the environment, humans exhibit robust biases toward information that pertains to themselves: Self-relevant information is processed faster and yields more accurate responses than information linked to others. Recent studies have shown that simple social associations can lead to the instant deployment of this benefit in the processing of abstract stimuli. However, how self-prioritization evolves across the processing hierarchy has been a subject of intense debate. Furthermore, there is little empirical evidence about the functional efficiency of social relevance in natural environments in which information is present across multiple senses. Across three experiments (each n = 40), the present study shows that self-prioritization effects (a) can arise in simple audio-visual numerosity judgements, (b) can be efficiently deployed across the senses by funneling perception toward self-relevant information in the more reliable sensory modality, and (c) modulate the integration of auditory and visual information into a multisensory representation. Taken together, the present findings suggest that social relevance can influence multisensory processing at both perceptual and postperceptual stages via early attentional modulations of sensory integration and later, task-dependent attentional control. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Cognitive control is modulated by hierarchical complexity of task switching: An event-related potential study

The study aimed to explore the effect of hierarchical complexity on task switching. The participants (n = 36) were asked to perform a magnitude or parity judgement on digits (1−9) in the hierarchical simple or complex block. In the simple block, participants made a numerical judgement on the presented digit (1−9) in each trial, whereas in the complex block, they had to first identify whether the digit in the current trial belonged to a predefined category (e.g., whether it was an even number), then perform a numerical judgement or not respond. The behavioural results revealed a significant interaction between hierarchical complexity and transition type (repeat vs. switch), with greater switch cost in the complex than in the simple block. Event-related potentials (ERPs) locked in the cue stage did not reveal this interaction, whereas the ERPs locked in the target stage revealed this interaction during the N2 and P3 time windows, with a larger switch negativity (switch minus repeat) in the complex than in the simple block. These findings demonstrate that an increase in hierarchical complexity triggers increased reactive control in the inhibition of the old task-set and reconfiguration of the new task-set during task switching.

Preserved time but altered numerosity processing in epileptic patients with postoperative lesion in the inferior frontal gyrus

Previous researches have shown that the inferior frontal gyrus (IFG) is involved in time and numerosity processing. This study aimed at examining (i) interval timing and (ii) interaction between duration and numerosity processing in four drug-resistant epileptic patients with postoperative lesions in the IFG in comparison with thirteen healthy controls. The duration reproduction and discrimination tasks performed in the sub- and supra-second ranges did not reveal any significant differences between patients and controls. The duration discrimination task of stimuli varying in numerosity (DurN) and the numerosity discrimination task of stimuli varying in duration (NumD) revealed that only numerosity judgment was altered in IFG patients. A time-order effect was notably observed in the NumD task but in opposite directions for the two groups: The second patch was perceived as more numerous than the first patch in controls and conversely as less numerous in patients. Finally in the DurN task, we observed a congruency effect which was dependent on numerical distance in patients but not in controls. These converging results suggest that the IFG would be more specifically involved in numerosity than in duration processing, possibly playing a role in numerical decision.

Medical and sanitary provision of convicts in places of detention in the mechanism of exercising the right to health care

The article reveals the place of medical and sanitary provision of convicts in places of detention in the mechanism of exercising the right to health care. It is noted that in accordance with current legislation and international standards in the mechanism of exercising the right to health care, the state is responsible for the life and health of persons under its control, regardless of their legal status.
 An analysis of national legislation and international documents in the field of protection of the right of a convict to adequate medical care in places of detention showed that medical care for convicts in places of detention should be provided and organized by the administration of penal institutions according to such quality standards and high professional level used in health care facilities outside places of detention.
 It was stressed that the Ministry of Justice of Ukraine, as the successor of the State Penitentiary Service of Ukraine, announced a number of initiatives in the field of execution of criminal penalties, including improving the functioning of the state institution «Health Center of the State Criminal and Executive Service of Ukraine».
 However, the attempts of the Ministry of Justice to make positive changes in this system should be properly perceived by both convicts and staff of penal institutions. Unfortunately, they are not accepted, as there is no clear algorithm of actions of the medical service of the penal institution for medical care of convicts in places of detention.
 That is why, at present, among the many problems in the functioning of the penitentiary system of Ukraine, the issue of proper medical care of convicts in the mechanism of exercising the right to health care remains on the sidelines.
 Thus, despite the rather extensive regulation of the rights of convicts to health care, the Ministry of Justice of Ukraine is still unable to ensure the implementation of its commitments in the field of human rights. This is confirmed by numerous judgments of the European Court of Human Rights, which confirm the systematic violation of Article 3 of the Convention for the Protection of Human Rights and Fundamental Freedoms in the context of effective and timely medical care in prisons.
 The author’s definition of medical and sanitary provision of convicts in places of detention is formulated – the mechanism provided by the current legislation and guaranteed by a democratic society for the exercising of the convict’s right to health care through the application of medical and sanitary and health prevention measures, as well as a combination of free and paid forms of medical care in accordance with international standards and procedures in this area.

The impact of stimulus format on task inhibition during task switching.

Switching between competing tasks is supported by active inhibition of the preceding task. The level of task stimulus processing at which interference between competing tasks must occur for inhibition to be recruited is still unclear. Here, we investigated whether inhibition is recruited by task conflict occurring at an early or late (semantic) stage of task stimulus processing by dissociating the task stimulus format from its meaning. In two experiments, participants performed three different numerical judgment tasks on numerical stimuli that could be presented as digits or number words (e.g., "6" or "six") in a cued task-switching procedure. The effects of the change of stimulus format for the inhibition of the previous task were investigated and assessed by the N-2 task repetition cost, an index of the extent to which task representations are inhibited. The N-2 task repetition cost observed in the same stimulus format condition disappeared when target stimuli on task N-1 were presented in a different format from stimuli of task N-2 and N. This occurred both when the format changed from digits to number words stimuli (Experiment 1) as well as when it changed from number words to digits stimuli (Experiment 2). Results indicated that task set inhibition is recruited very early during the stimulus processing stage. They also provided evidence that task inhibition is not tied to task preparation processes but operates as a reactive, rather than proactive mechanism of conflict resolution.

Visual Duration but Not Numerosity Is Distorted While Running

There is increasing evidence that action and perception interact in the processing of magnitudes such as duration and numerosity. Sustained physical exercise (such as running or cycling) increases the apparent duration of visual stimuli presented during the activity. However, the effect of exercise on numerosity perception has not yet been investigated. Here, we asked participants to make either a temporal or a numerical judgment by comparing the duration or numerosity of standard stimuli displayed at rest with those presented while running. The results support previous reports in showing that physical activity significantly expands perceived duration; however, it had no effect on perceived numerosity. Furthermore, the distortions of the perceived durations vanished soon after the running session, making it unlikely that physiological factors such as heart rate underlie the temporal distortion. Taken together, these results suggest a domain-selective influence of the motor system on the perception of time, rather than a general effect on magnitude.

The dynamics of grouping-induced biases in apparent numerosity revealed by a continuous tracking technique.

Connecting pairs of items causes robust underestimation of the numerosity of an ensemble, presumably by invoking grouping mechanisms. Here we asked whether this underestimation in numerosity judgments could be revealed and further explored by continuous tracking, a newly developed technique that allows for fast and efficient data acquisition and monitors the dynamics of the responses. Participants continuously reproduced the perceived numerosity of a cloud of dots by moving a cursor along a number line, while the number of dots and the proportion connected by lines varied over time following two independent random walks. The technique was robust and efficient, and correlated well with results obtained with a standard psychophysics task. Connecting objects with lines caused an underestimation of approximately 15% during tracking, agreeing with previous studies. The response to the lines was slower than the response to the physical numerosity, with a delay of approximately 150 ms, suggesting that this extra time is necessary for processing the grouping effect.

Selective interference of hand posture with grasping capability estimation.

Previous studies have shown that judgments about how one would perform an action are affected by the current body posture. Hence, judging one's capability to grasp an object between index and thumb is influenced by their aperture at the time of the judgment. This finding can be explained by a modification of the internal representation of one's hand through the effect of sensorimotor input. Alternatively, the influence of grip aperture might be mediated by a response congruency effect, so that a "less" vs. "more" open grip would bias the judgment toward a "less" vs. "more" capable response. To specify the role of sensorimotor input in prospective action judgments, we asked participants to estimate their capability to grasp circles between index and thumb while performing a secondary task that requires them to squeeze a ball with these two fingers (precision grip) or with a different hand configuration (palm grip). Experiment 1 showed that participants underestimated their grasping capability when the squeezing task involved the same grip as the judged action (precision grip) and their estimates were bound to the relative size of objects as revealed by size-contrast illusions (Ebbinghaus). Experiment 2 showed that the grip reduction caused by the squeezing task also interfered with the discrimination of large numbers in magnitude judgments, but this incongruency effect was only observed for the palm grip. The dissociated effects of the two grips in graspability and numerical judgments indicate that sensorimotor input may affect the perceived ability to grasp objects, independently of response congruency, by modifying the representation of the hand in action.

Cross-Format Integration of Auditory Number Words and Visual-Arabic Digits: An ERP Study.

Converting visual-Arabic digits to auditory number words and vice versa is seemingly effortless for adults. However, it is still unclear whether this process takes place automatically and whether accessing the underlying magnitude representation is necessary during this process. In two event-related potential (ERP) experiments, adults were presented with identical (e.g., “one” and 1) or non-identical (e.g., “one” and 9) number pairs, either unimodally (two visual-Arabic digits) or cross-format (an auditory number word and a visual-Arabic digit). In Experiment 1 (N=17), active task demands required numerical judgments, whereas this was not the case in Experiment 2 (N=19). We found pronounced early ERP markers of numerical identity unimodally in both experiments. In the cross-format conditions, however, we only observed late neural correlates of identity and only if the task required semantic number processing (Experiment 1). These findings suggest that unimodal pairs of digits are automatically integrated, whereas cross-format integration of numerical information occurs more slowly and involves semantic access.