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理解不确定性：儿童概率表征的认知发展机制*
Understanding the Uncertainties: The Cognitive Developmental Mechanisms of Children’s Probabilistic Representation
概率表征反映个体对不确定性的感知与判断的能力。以往研究表明，人类在生命早期就表现出了概率表征能力，部分非人类动物也具有概率表征能力，且该能力被应用于各个领域，如语言学习、情绪理解等。由此可见，概率表征能力具有跨物种一致性与领域一般性，其对个体的生存与繁衍具有重要适应意义。基于对以往研究证据的系统梳理，本研究阐明概率表征的认知发展理论解释，其中包含数量加工和逻辑推断等不同视角的分析；提出概率表征认知发展机制的整合性假说，该假说认为数量加工与逻辑推断分别反映概率表征的不同认知加工成分，共同构成概率表征能力的认知基础；最后指出未来的可能研究方向。
Probabilistic representation refers to the ability to perceive, judge, and infer about uncertainties. Human infants, children, and several nonhuman animals, such as great apes, monkeys, and birds, are sensitive to probabilistic information, as well as capable of making probabilistic judgments and inferences, even learning based on probabilities, suggesting inter-species consistency of probabilistic representation. Also, human and nonhuman animals are capable of integrating different-domain information into probabilistic representation, such as spatio-temporal information, physical constraints, and mental states, suggesting domain-generality of probabilistic representation. Domain-generality and inter-species consistency indicate that probabilistic representation is of great importance for organisms’ survival and reproduction. However, how human represent probabilities remains controversial for a long time. Understanding what cognitive mechanisms underlie probabilistic representation and how children acquire the ability to represent probabilities would be of great importance for understanding how human learn and reason about the world, as probabilities must be the core information of the world in which human and animals live.
Existing empirical research mostly used two paradigms to examine probabilistic representation, violation of expectation (VoE) and two-alternative forced-choice (2AFC) task. Violation of expectation paradigm was mostly applied to infant and nonhuman animal studies, as infants and nonhuman animals were not able to give meaningful verbal responses. Two-alternative forced-choice paradigm was mostly applied to child and nonhuman animal studies. Compared with two-alternative forced-choice paradigm, violation of expectation paradigm could only indicate infants’ sensitivity to probabilities, as it always showed all the information to capture infants’ visual attention patterns toward expected and unexpected consequences, instead of making infants choose uncertain choices based on their predictions.
Researchers have examined human and nonhuman animals’ probabilistic representation by the two paradigms mentioned above, and have discussed how human and nonhuman animals represent probabilities from different theoretical perspectives. Two theoretical explanations have been proposed to construct the cognitive developmental mechanisms of probabilistic representation, numerical processing, and logic inferences. On the one hand, theoretical perspective based on numerical processing, or intuitive statistics, suggests that we represent probabilities based on numerical information, such as the proportions. Some research indeed showed that children and nonhuman animals exhibited similar characteristics to numerical processing in probabilistic representations, and numerical representation acuity was positively correlated with performances of probabilistic representation. Children showed increasingly better abilities to represent probabilities based on numerical information with age. On the other hand, according to the theoretical perspective based on logic inferences, we represent probabilities by enumeration of all the possible and exclusive consequences, which was referred to as modal concept or modal logic. However, some researchers argued that we were not born with the ability to represent probabilities with modal logic. Instead, they proposed that infants and toddlers represented probabilities by simulating a random consequence from all the possible consequences till age three, whereas older children and adults represent probabilities by enumerating all the consequences, or modal logic.
Combined with the empirical evidence and existing theoretical explanations, we expound the theoretical explanations for the cognitive developmental mechanisms of probabilistic representation, including different perspectives from numerical processing and logical inferences. This article proposed an integrative hypothesis of probabilistic representation, and suggested the possible directions for future research. Additionally, the two aforementioned proposals may not be exclusive to each other. Each explanation focused on one aspect of probability representation. Therefore, we proposed the integrated theory to construct the cognitive developmental mechanism of probability representation. Logic inferences, or modal logic, allows us to understand that all the possible consequences might happen and they are exclusive. Meanwhile, numerical processing helps us estimate the numerical information of probabilities. In general, both modal logic and numerical processing underly human’s probabilistic representation. Future research could focus on explore the integrated cognitive developmental mechanism of probability representation and apply different approaches in this field.
概率表征 / 认知发展 / 数量加工 / 逻辑推断 / 整合性假说
probabilistic representation / cognitive development / numerical processing / logic inferences / integrated hypothesis
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People's emotional experiences depend not only on what actually happened, but also on what they thought would happen. However, these expectations about future outcomes are not always communicated explicitly. Thus, the ability to infer others' expectations in context and understand how these expectations influence others' emotions is an important aspect of our social intelligence. Prior work suggests that an abstract understanding of how expectations modulate emotional responses may not emerge until 7 to 8 years of age. Using a novel paradigm that capitalizes on intuitive physics to generate contextually plausible expectations, we present evidence for expectation-based emotion inference in preschool-aged children. Given two bowlers who experienced identical final outcomes (hitting 3 of 6 pins), we varied the trajectory of their balls such that one would initially expect to hit all pins (high-expectation), while the other would expect to hit none (low-expectation). In Experiment 1, both 4- and 5-year-olds appropriately adjusted characters' happiness ratings upward (low-expectation) or downward (high-expectation) relative to their initial emotions; however, only 5-year-olds made adjustments robust enough to manifest as higher final ratings for the low-expectation than the high-expectation character. In Experiments 2-3, we replicate these results and show that 5-year-olds reliably differentiate the characters' emotions even when their expectations must be inferred from context. An internal meta-analysis revealed a robust and consistent effect across the three experiments. Together, these findings provide the earliest evidence for expectation-based emotion reasoning and suggest that the ability to spontaneously generate and consider others' expectations continues to develop during preschool years. (PsycINFO Database Record (c) 2019 APA, all rights reserved).
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The current experiments investigate how infants use goal-directed action to reason about intentionally sampled outcomes in a probabilistic inference paradigm. Older infants and young children are flexible in their expectations of sampling: They expect random samples to reflect population statistics and non-random samples to reflect an agent's preferences or goals (Kushnir, Xu, & Wellman, 2010; Xu & Denison, 2009). However, more recent work shows that probabilistic inference comes online at approximately 6 months (Denison, Reed, & Xu, 2013; Kayhan, Gredebäck, & Lindskog, 2017; Ma & Xu, 2011; Wellman, Kushnir, Xu, & Brink, 2016), and thus, these sampling assumptions can be investigated at the age probabilistic reasoning first emerges. Results indicate that 6-month-old infants expect a human agent to sample in accord with their goal and do not expect the same of an unintentional agent-a mechanical claw. By 9.5 months, infants expect the mechanical claw to sample in accord with random sampling. These results suggest that infants use goals to make inferences about intentional sampling, under appropriate conditions at 6 months, and they have expectations of the kinds of samples a mechanical device should obtain by 9.5 months.© 2019 International Congress of Infant Studies (ICIS).
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When perceptually available information is scant, we can leverage logical connections among hypotheses to draw reliable conclusions that guide our reasoning and learning. We investigate whether this function of logical reasoning is present in infancy and aid understanding and learning about the social environment. In our task, infants watch reaching actions directed toward a hidden object whose identity is ambiguous between two alternatives and has to be inferred by elimination. Here we show that infants apply a disjunctive inference to identify the hidden object and use this logical conclusion to assess the consistency of the actions with a preference previously demonstrated by the agent and, importantly, also to acquire new knowledge regarding the preferences of the observed actor. These findings suggest that, early in life, preverbal logical reasoning functions as a reliable source of evidence that can support learning by offering a logical route for knowledge acquisition.
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Contrasting possibilities has a fundamental adaptive value for prediction and learning. Developmental research, however, has yielded controversial findings. Some data suggest that preschoolers might have trouble in planning actions that take into account mutually exclusive possibilities, while other studies revealed an early understanding of alternative future outcomes based on infants' looking behaviour. To better understand the origin of such abilities, here we use pupil dilation as a potential indicator of infants' representation of possibilities. Ten- and 14-month-olds were engaged in an object-identification task by watching video animations where three different objects with identical top parts moved behind two screens. Importantly, a target object emerged from one of the screens but remained in partial occlusion, revealing only its top part, which was compatible with a varying number of possible identities. Just as adults' pupil diameter grows monotonically with the amount of information held in memory, we expected that infants' pupil size would increase with the number of alternatives sustained in memory as candidate identities for the partially occluded object. We found that pupil diameter increased with the object's potential identities in 14- but not in 10-month-olds. We discuss the implications of these results for the foundation of humans' capacities to represent alternatives.
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Research has shown that some forms of inferential reasoning are likely widespread throughout the animal kingdom (e.g., exclusion, in which a subject infers the placement of a reward by eliminating potential alternative sites), but other types of inferential tasks have not been extensively tested. We examined whether a nonhuman might succeed in an experiment based on probabilistic reasoning, specifically, the ability to make inferences about a sample based on information about a population. A Grey parrot (Psittacus erithacus), previously trained to use English labels referentially to identify objects, observed a human researcher deposit 2 different types of items in a 3:1 ratio (e.g., 3 corks and 1 piece of paper) into an opaque bucket. One item was then randomly withdrawn while hidden from the parrot's view. When asked to identify the still-hidden object, the parrot's vocal responses tracked this 3:1 ratio over a large number of trials. Some levels of probabilistic reasoning therefore are not limited to humans, nonhuman primates, or even mammals. (PsycINFO Database Record(c) 2018 APA, all rights reserved).
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Humans can use an intuitive sense of statistics to make predictions about uncertain future events, a cognitive skill that underpins logical and mathematical reasoning. Recent research shows that some of these abilities for statistical inferences can emerge in preverbal infants and non-human primates such as apes and capuchins. An important question is therefore whether animals share the full complement of intuitive reasoning abilities demonstrated by humans, as well as what evolutionary contexts promote the emergence of such skills. Here, we examined whether free-ranging rhesus macaques () can use probability information to infer the most likely outcome of a random lottery, in the first test of whether primates can make such inferences in the absence of direct prior experience. We developed a novel expectancy-violation looking time task, adapted from prior studies of infants, in order to assess the monkeys' expectations. In Study 1, we confirmed that monkeys (n = 20) looked similarly at different sampled items if they had no prior knowledge about the population they were drawn from. In Study 2, monkeys (n = 80) saw a dynamic 'lottery' machine containing a mix of two types of fruit outcomes, and then saw either the more common fruit () or the relatively rare fruit () fall from the machine. We found that monkeys looked longer when they witnessed the unexpected outcome. In Study 3, we confirmed that this effect depended on the causal relationship between the sample and the population, not visual mismatch: monkeys (n = 80) looked equally at both outcomes if the experimenter pulled the sampled item from her pocket. These results reveal that rhesus monkeys spontaneously use information about probability to reason about likely outcomes, and show how comparative studies of nonhumans can disentangle the evolutionary history of logical reasoning capacities.
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How do people make rich inferences from such sparse data? Recent research has explored this inferential ability by investigating probabilistic reasoning in infancy. For example, 8- and 11-month-old infants can make inferences from samples to populations and vice versa (Denison & Xu, 2010a; Xu & Denison, 2009; Xu & Garcia, 2008a). The current experiment investigates the developmental origins of this probabilistic inference mechanism with 4.5- and 6-month-old infants. Infants were shown 2 large boxes, 1 containing a ratio of 4 pink to 1 yellow balls, the other containing the opposite ratio. The experimenter sampled from, for example, the mostly pink box and removed a sample of either 4 pink and 1 yellow balls or 4 yellow and 1 pink balls on alternating trials. Six-month-olds but not 4.5-month-olds looked longer at the 4 yellow and 1 pink sample (the improbable outcome) than at the 4 pink and 1 yellow sample (the probable outcome).(c) 2013 APA, all rights reserved.
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A rich tradition in developmental psychology explores physical reasoning in infancy. However, no research to date has investigated whether infants can reason about physical objects that behave probabilistically, rather than deterministically. Physical events are often quite variable, in that similar-looking objects can be placed in similar contexts with different outcomes. Can infants rapidly acquire probabilistic physical knowledge, such as some leaves fall and some glasses break by simply observing the statistical regularity with which objects behave and apply that knowledge in subsequent reasoning? We taught 11-month-old infants physical constraints on objects and asked them to reason about the probability of different outcomes when objects were drawn from a large distribution. Infants could have reasoned either by using the perceptual similarity between the samples and larger distributions or by applying physical rules to adjust base rates and estimate the probabilities. Infants learned the physical constraints quickly and used them to estimate probabilities, rather than relying on similarity, a version of the representativeness heuristic. These results indicate that infants can rapidly and flexibly acquire physical knowledge about objects following very brief exposure and apply it in subsequent reasoning.PsycINFO Database Record (c) 2014 APA, all rights reserved.
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Reasoning under uncertainty is the bread and butter of everyday life. Many areas of psychology, from cognitive, developmental, social, to clinical, are interested in how individuals make inferences and decisions with incomplete information. The ability to reason under uncertainty necessarily involves probability computations, be they exact calculations or estimations. What are the developmental origins of probabilistic reasoning? Recent work has begun to examine whether infants and toddlers can compute probabilities; however, previous experiments have confounded quantity and probability-in most cases young human learners could have relied on simple comparisons of absolute quantities, as opposed to proportions, to succeed in these tasks. We present four experiments providing evidence that infants younger than 12 months show sensitivity to probabilities based on proportions. Furthermore, infants use this sensitivity to make predictions and fulfill their own desires, providing the first demonstration that even preverbal learners use probabilistic information to navigate the world. These results provide strong evidence for a rich quantitative and statistical reasoning system in infants.Copyright © 2013 Elsevier B.V. All rights reserved.
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Humans frequently make inferences about uncertain future events with limited data. A growing body of work suggests that infants and other primates make surprisingly sophisticated inferences under uncertainty. First, we ask what underlying cognitive mechanisms allow young learners to make such sophisticated inferences under uncertainty. We outline three possibilities, the, and views, and assess the empirical evidence for each. We argue that the weight of the empirical work favors the probabilistic view, in which early reasoning under uncertainty is grounded in inferences about the relationship between samples and populations as opposed to being grounded in simple heuristics. Second, we discuss the apparent contradiction between this early-emerging sensitivity to probabilities with the decades of literature suggesting that adults show limited use of base-rate and sampling principles in their inductive inferences. Third, we ask how these early inductive abilities can be harnessed for improving later mathematics education and inductive inference. We make several suggestions for future empirical work that should go a long way in addressing the many remaining open questions in this growing research area.
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Improbable events are surprising. However, it is unknown whether children consider probability when attributing surprise to other people. We conducted four experiments that investigate this issue. In the first three experiments, children saw stories in which two characters received a red gumball from two gumball machines with different distributions, and children then judged which character was more surprised. Experiment 1 (N = 120) shows development in children's use of probability to infer surprise. Children aged 7 correctly inferred that the character with a lower chance of getting a red gumball would be more surprised, but 4- to 6-year-olds did not. Experiment 2 (N = 120) shows that children's performance does not improve when the probability of getting a red gumball is zero and should be maximally surprising. Experiment 3 (N = 120) demonstrates that 6-year-olds' performance improves when they are prompted to consider probabilities, but not when they are prompted to consider the characters' beliefs. Experiment 4 (N = 60) replicates this finding, but using a new design in which children attributed emotions to just a single character. Together these findings suggest that by age 6, a conceptual shift occurs, in which children begin to integrate their understanding of probability with their understanding of surprise. (PsycINFO Database Record (c) 2018 APA, all rights reserved).
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Happiness with an outcome often depends on whether better or worse outcomes were initially more likely. In five experiments, we found that young children ( = 620, Experiments 1-4) and adults ( = 254, Experiment 5) used probability to infer emotions and assess outcome quality. In Experiments 1 and 2, 5- and 6-year-olds (but not 4-year-olds) inferred that an agent would be less happy with an outcome if a better outcome were initially more likely. In Experiment 3, 4- to 6-year-olds used probability to assess quality. These findings suggest a developmental lag between 4-year-olds' assessments of quality and happiness. We replicated this lag in Experiment 4. In Experiment 5, adults used probability to assess both quality and happiness. We suggest that children and adults may use probability to establish a standard against which actual outcomes are compared. Doing so might allow them to make probability-based inferences of happiness without drawing on counterfactual reasoning.
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In pursuing goals, people seek favorable odds. We investigated whether young children use this fact to infer goals from people's actions across two experiments on Canadian 3- to 7-year-old children ( = 316; 167 girls, 149 boys). Participants' demographic information was not formally collected, but the region is predominantly middle-class and White. In Experiment 1, 3-year-old children saw a story where one agent went to a gumball machine with mostly red gumballs and another agent went to a machine with mostly purple ones. When asked which agent wanted a red gumball, children mostly selected the agent who chose the mostly red machine. Moreover, children responded at chance in a control condition where they judged which agent knew they would get a red gumball. In Experiment 2a, 3- to 7-year-old children saw a story where an agent either chose between two gumball machines or two open bowls of gumballs. In both conditions, the agent chose a location with mostly red gumballs over one with mostly blue gumballs but ended up with a blue gumball. Children were more likely to infer the agent had wanted a red gumball when the agent had made a probabilistic choice (machines) than a determinative choice (bowls), though inferences that the red gumball was preferred also increased with age. In Experiment 2b, a preregistered follow-up, American adults responded similarly to the older children. Together our findings suggest that children infer goals by drawing on the understanding that people seek favorable odds, though the clearest findings come from children aged 6 years and older. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
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Great apes have been shown to be intuitive statisticians: they can use proportional information within a population to make intuitive probability judgments about randomly drawn samples [1, J.E., J.C., J.H., E.H., and H.R., unpublished data]. Humans, from early infancy onward, functionally integrate intuitive statistics with other cognitive domains to judge the randomness of an event [2-6]. To date, nothing is known about such cross-domain integration in any nonhuman animal, leaving uncertainty about the origins of human statistical abilities. We investigated whether chimpanzees take into account information about psychological states of experimenters (their biases and visual access) when drawing statistical inferences. We tested 21 sanctuary-living chimpanzees in a previously established paradigm that required subjects to infer which of two mixed populations of preferred and non-preferred food items was more likely to lead to a desired outcome for the subject. In a series of three experiments, we found that chimpanzees chose based on proportional information alone when they had no information about experimenters' preferences and (to a lesser extent) when experimenters had biases for certain food types but drew blindly. By contrast, when biased experimenters had visual access, subjects ignored statistical information and instead chose based on experimenters' biases. Lastly, chimpanzees intuitively used a violation of statistical likelihoods as indication for biased sampling. Our results suggest that chimpanzees have a random sampling assumption that can be overridden under the appropriate circumstances and that they are able to use mental state information to judge whether this is necessary. This provides further evidence for a shared statistical inference mechanism in apes and humans.Copyright © 2018 Elsevier Ltd. All rights reserved.
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What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.
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Psychological scientists use statistical information to determine the workings of human behavior. We argue that young children do so as well. Over the course of a few years, children progress from viewing human actions as intentional and goal directed to reasoning about the psychological causes underlying such actions. Here, we show that preschoolers and 20-month-old infants can use statistical information-namely, a violation of random sampling-to infer that an agent is expressing a preference for one type of toy instead of another type of toy. Children saw a person remove five toys of one type from a container of toys. Preschoolers and infants inferred that the person had a preference for that type of toy when there was a mismatch between the sampled toys and the population of toys in the box. Mere outcome consistency, time spent with the toys, and positive attention toward the toys did not lead children to infer a preference. These findings provide an important demonstration of how statistical learning could underpin the rapid acquisition of early psychological knowledge.
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Sometimes we accept propositions, sometimes we reject them, and sometimes we take propositions to be worth considering but not yet established, as merely possible. The result is a complex representation with logical structure. Is the ability to mark propositions as merely possible part of our innate representational toolbox or does it await development, perhaps relying on language acquisition? Several lines of inquiry show that preverbal infants manage possibilities in complex ways, while others find that preschoolers manage possibilities poorly. Here, we discuss how this apparent conflict can be resolved by distinguishing modal representations of possibility, which mark possibility symbolically, from minimal representations of possibility, which do not encode any modal status and need not have a logical structure.Copyright © 2019 Elsevier Ltd. All rights reserved.
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Human adults have a strong bias to invoke intentional agents in their intuitive explanations of ordered wholes or regular compositions in the world. Less is known about the ontogenetic origin of this bias. In 4 experiments, we found that 9- to 10-month-old infants expected a human hand, but not a mechanical tool with similar affordances, to be the primary cause of nonrandom sampling events that resulted in regular color patterns in visual displays. Infants did not have such expectations when the sampling appeared random with no regular compositions in the outcome. These findings provide the first evidence that by about 9 months of age, infants infer the presence of an intentional agent from the perception of regularity.
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A crucial task in social interaction involves understanding subjective mental states. Here we report two experiments with toddlers exploring whether they can use statistical evidence to infer the subjective nature of preferences. We found that 2-year-olds were likely to interpret another person's nonrandom sampling behavior as a cue for a preference different from their own. When there was no alternative in the population or if the sampling was random, 2-year-olds did not ascribe a preference and persisted in their initial beliefs that the person would share their own preference. We found similar but weaker patterns of responses in 16-month-olds. These results suggest that the ability to infer the subjectivity of preferences based on sampling information begins to emerge between 16 months and 2 years. Our findings provide some of the first evidence that from early in development, young children can use statistical evidence to make rational inferences about the social world.Copyright © 2011 Elsevier B.V. All rights reserved.
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Logical inference is often seen as an exclusively human and language-dependent ability, but several nonhuman animal species search in a manner that is consistent with a deductive inference, the disjunctive syllogism: when a reward is hidden in one of two cups, and one cup is shown to be empty, they will search for the reward in the other cup. In Experiment 1, we extended these results to toddlers, finding that 23-month-olds consistently approached the non-empty location. However, these results could reflect non-deductive approaches of simply avoiding the empty location, or of searching in any location that might contain the reward, rather than reasoning through the disjunctive syllogism to infer that the other location must contain the reward. Experiment 2 addressed these alternatives, finding evidence that 3- to 5-year-olds used the disjunctive syllogism, while 2.5-year-olds did not. This suggests that younger children may not easily deploy this logical inference, and that a non-deductive approach may be behind the successful performance of nonhuman animals and human infants.Copyright © 2016 Elsevier B.V. All rights reserved.
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Two experiments were designed to investigate the developmental trajectory of children's probability approximation abilities. In Experiment 1, results revealed 6- and 7-year-old children's (N = 48) probability judgments improve with age and become more accurate as the distance between two ratios increases. Experiment 2 replicated these findings with 7- to 12-year-old children (N = 130) while also accounting for the effect of the size and the perceived numerosity of target objects. Older children's performance suggested the correct use of proportions for estimating probability; but in some cases, children relied on heuristic shortcuts. These results suggest that children's nonsymbolic probability judgments show a clear distance effect and that the acuity of probability estimations increases with age.© 2019 Society for Research in Child Development.
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Inductive learning and reasoning, as we use it both in everyday life and in science, is characterized by flexible inferences based on statistical information: inferences from populations to samples and vice versa. Many forms of such statistical reasoning have been found to develop late in human ontogeny, depending on formal education and language, and to be fragile even in adults. New revolutionary research, however, suggests that even preverbal human infants make use of intuitive statistics. Here, we conducted the first investigation of such intuitive statistical reasoning with non-human primates. In a series of 7 experiments, Bonobos, Chimpanzees, Gorillas and Orangutans drew flexible statistical inferences from populations to samples. These inferences, furthermore, were truly based on statistical information regarding the relative frequency distributions in a population, and not on absolute frequencies. Intuitive statistics in its most basic form is thus an evolutionarily more ancient rather than a uniquely human capacity. Copyright © 2014 Elsevier B.V. All rights reserved.
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Recent experiments reveal that 6- to 12-month-old infants can learn probabilities and reason with them. In this work, we present a novel computational system called Neural Probability Learner and Sampler (NPLS) that learns and reasons with probabilities, providing a computationally sufficient mechanism to explain infant probabilistic learning and inference. In 24 computer simulations, NPLS shows how probability distributions can emerge naturally from neural-network learning of event sequences, providing a novel explanation of infant probabilistic learning and reasoning. Three mathematical proofs show how and why NPLS simulates the infant results so accurately. The results are situated in relation to seven other active research lines. This work provides an effective way to integrate Bayesian and neural-network approaches to cognition. (PsycInfo Database Record (c) 2021 APA, all rights reserved).
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Looking time experiments based on the violation-of-expectation (VOE) method have consistently demonstrated that infants look longer when their expectations are violated. However, it remains an open question whether similar effects will be observed in infants' approach behaviours. Specifically, do infants selectively approach and explore sources that violate their expectations? In this study, we address this question by examining how infants' looking times are related to their approach and exploration behaviours. Using a traditional VOE method and a crawling paradigm, we demonstrate a strong correspondence between looking time and approach behaviours, which indicates that 13-month-old infants preferentially explore sources of unexpected events. Such spontaneous exploration may provide learning opportunities and allow infants to play an active role in driving their own development. Statement of contribution What is already known on this subject? Infants look longer when their expectations are violated. There is some evidence that infants also preferentially explore objects that violate their 'core' physical expectations. What the present study adds? There is a clear correspondence between infants' looking behaviour and their approach behaviour. Expectancy violations involving non-core knowledge can similarly influence infants' exploration.© 2017 The British Psychological Society.
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Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.© 2021 Society for Research in Child Development.
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The ability to reason about probabilities has ecological relevance for many species. Recent research has shown that both preverbal infants and non-human great apes can make predictions about single-item samples randomly drawn from populations by reasoning about proportions. To further explore the evolutionary origins of this ability, we conducted the first investigation of probabilistic inference in a monkey species (capuchins; Sapajus spp.). Across four experiments, capuchins (N = 19) were presented with two populations of food items that differed in their relative distribution of preferred and non-preferred items, such that one population was more likely to yield a preferred item. In each trial, capuchins had to select between hidden single-item samples randomly drawn from each population. In Experiment 1 each population was homogeneous so reasoning about proportions was not required; Experiments 2-3 replicated previous probabilistic reasoning research with infants and apes; and Experiment 4 was a novel condition untested in other species, providing an important extension to previous work. Results revealed that at least some capuchins were able to make probabilistic inferences via reasoning about proportions as opposed to simpler quantity heuristics. Performance was relatively poor in Experiment 4, so the possibility remains that capuchins may use quantity-based heuristics in some situations, though further work is required to confirm this. Interestingly, performance was not at ceiling in Experiment 1, which did not involve reasoning about proportions, but did involve sampling. This suggests that the sampling task posed demands in addition to reasoning about proportions, possibly related to inhibitory control, working memory, and/or knowledge of object permanence.
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Recent research shows that preverbal infants can reason about single-case probabilities without relying on observed frequencies, adapting their predictions to relevant dynamic parameters of the situation (Téglás, Vul, Girotto, Gonzalez, Tenenbaum & Bonatti, ; Téglás, Girotto, Gonzalez & Bonatti, ). Here we show that intuitions of probabilities may derive from the ability to represent a limited number of possibilities. After watching a scene containing moving objects of two ensembles, 12-month-olds looked longer at an unlikely than at a likely single-case outcome when the objects were within the parallel individuation range. However, they did not do so when the scene contained the same ratio between ensembles but a larger number of objects. At the same time, they could form rational expectations about single-case outcomes in scenes containing the same large number of objects when they could exploit subtle physical parameters induced by the objects' movements and their spatial configuration. Our findings demonstrate that at early stages of development the mental representations involved in probability estimations of future individual situations are powerful and sophisticated, but at the same time they depend on infants' overall cognitive architecture, being constrained by the numerical representations spontaneously induced by the situations. © 2014 John Wiley & Sons Ltd.
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Many organisms can predict future events from the statistics of past experience, but humans also excel at making predictions by pure reasoning: integrating multiple sources of information, guided by abstract knowledge, to form rational expectations about novel situations, never directly experienced. Here, we show that this reasoning is surprisingly rich, powerful, and coherent even in preverbal infants. When 12-month-old infants view complex displays of multiple moving objects, they form time-varying expectations about future events that are a systematic and rational function of several stimulus variables. Infants' looking times are consistent with a Bayesian ideal observer embodying abstract principles of object motion. The model explains infants' statistical expectations and classic qualitative findings about object cognition in younger babies, not originally viewed as probabilistic inferences.
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| [51] |
Research on initial conceptual knowledge and research on early statistical learning mechanisms have been, for the most part, two separate enterprises. We report a study with 11-month-old infants investigating whether they are sensitive to sampling conditions and whether they can integrate intentional information in a statistical inference task. Previous studies found that infants were able to make inferences from samples to populations, and vice versa [Xu, F., & Garcia, V. (2008). Intuitive statistics by 8-month-old infants. Proceedings of the National Academy of Sciences of the United States of America, 105, 5012-5015]. We found that when employing this statistical inference mechanism, infants are sensitive to whether a sample was randomly drawn from a population or not, and they take into account intentional information (e.g., explicitly expressed preference, visual access) when computing the relationship between samples and populations. Our results suggest that domain-specific knowledge is integrated with statistical inference mechanisms early in development.
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| [52] |
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| [53] |
We report a new study testing our proposal that word learning may be best explained as an approximate form of Bayesian inference (Xu & Tenenbaum, in press). Children are capable of learning word meanings across a wide range of communicative contexts. In different contexts, learners may encounter different sampling processes generating the examples of word-object pairings they observe. An ideal Bayesian word learner could take into account these differences in the sampling process and adjust his/her inferences about word meaning accordingly. We tested how children and adults learned words for novel object kinds in two sampling contexts, in which the objects to be labeled were sampled either by a knowledgeable teacher or by the learners themselves. Both adults and children generalized more conservatively in the former context; that is, they restricted the label to just those objects most similar to the labeled examples when the exemplars were chosen by a knowledgeable teacher, but not when chosen by the learners themselves. We discuss how this result follows naturally from a Bayesian analysis, but not from other statistical approaches such as associative word-learning models.
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