Probabilistic cognition in two indigenous Mayan groups

Rational number representation by the approximate number system

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known "connectedness illusion" to provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers.

Debiasing thinking among non-WEIRD reasoners

Human reasoning has been shown to be biased in a variety of situations. While most studies have focused on samples of WEIRD participants (from Western Educated Industrialized Rich and Democratic societies), the sparse non-WEIRD data on the topic suggest an even stronger propensity for biased reasoning. This could be explained by a competence issue (people lack the ability to integrate logical knowledge into their reasoning) or a performance issue (people possess the logical knowledge but do not know it is relevant). We addressed this question using a debiasing paradigm with the base-rate task on a sample of non-industrialized people, the Himba of Namibia. After a short training, most participants were debiased, lending credence to the performance account. Debiasing was however to some extent boosted by schooling and living environment suggesting that competence also plays a role (in that more acquired knowledge allows for a higher training benefit). Results imply that debias interventions can be successfully employed to boost sound reasoning around the world.

Crows flexibly apply statistical inferences based on previous experience

Statistical inference, the ability to use limited information to draw conclusions about the likelihood of an event, is critical for decision-making during uncertainty. The ability to make statistical inferences was thought to be a uniquely human skill requiring verbal instruction and mathematical reasoning.1 However, basic inferences have been demonstrated in both preliterate and pre-numerate individuals,2,3,4,5,6,7 as well as non-human primates.8 More recently, the ability to make statistical inferences has been extended to members outside of the primate lineage in birds.9,10 True statistical inference requires subjects use relative rather than absolute frequency of previously experienced events. Here, we show that crows can relate memorized reward probabilities to infer reward-maximizing decisions. Two crows were trained to associate multiple reward probabilities ranging from 10% to 90% to arbitrary stimuli. When later faced with the choice between various stimulus combinations, crows retrieved the reward probabilities associated with individual stimuli from memory and used them to gain maximum reward. The crows showed behavioral distance and size effects when judging reward values, indicating that the crows represented probabilities as abstract magnitudes. When controlling for absolute reward frequency, crows still made reward-maximizing choices, which is the signature of true statistical inference. Our study provides compelling evidence of decision-making by relative reward frequency in a statistical inference task.

Numeracy, gist, literal thinking and the value of nothing in decision making

The onus on the average person is greater than ever before to make sense of large amounts of readily accessible quantitative information, but the ability and confidence to do so are frequently lacking. Many people lack practical mathematical skills that are essential for evaluating risks, probabilities and numerical outcomes such as survival rates for medical treatments, income from retirement savings plans or monetary damages in civil trials. In this Review, we integrate research on objective and subjective numeracy, focusing on cognitive and metacognitive factors that distort human perceptions and foment systematic biases in judgement and decision making. Paradoxically, an important implication of this research is that a literal focus on objective numbers and mechanical number crunching is misguided. Numbers can be a matter of life and death but a person who uses rote strategies (verbatim representations) cannot take advantage of the information contained in the numbers because 'rote' strategies are, by definition, processing without meaning. Verbatim representations (verbatim is only surface form, not meaning) treat numbers as data as opposed to information. We highlight a contrasting approach of gist extraction: organizing numbers meaningfully, interpreting them qualitatively and making meaningful inferences about them. Efforts to improve numerical cognition and its practical applications can benefit from emphasizing the qualitative meaning of numbers in context - the gist - building on the strengths of humans as intuitive mathematicians. Thus, we conclude by reviewing evidence that gist training facilitates transfer to new contexts and, because it is more durable, longer-lasting improvements in decision making.

Viruses, Vaccines, and COVID-19: Explaining and Improving Risky Decision-making

Risky decision-making lies at the center of the COVID-19 pandemic and will determine future viral outbreaks. Therefore, a critical evaluation of major explanations of such decision-making is of acute practical importance. We review the underlying mechanisms and predictions offered by expectancy-value and dual-process theories. We then highlight how fuzzy-trace theory builds on these approaches and provides further insight into how knowledge, emotions, values, and metacognitive inhibition influence risky decision-making through its unique mental representational architecture (i.e., parallel verbatim and gist representations of information). We discuss how social values relate to decision-making according to fuzzy-trace theory, including how categorical gist representations cue core values. Although gist often supports health-promoting behaviors such as vaccination, social distancing, and mask-wearing, why this is not always the case as with status-quo gist is explained, and suggestions are offered for how to overcome the "battle for the gist" as it plays out in social media.

The Number Sense Represents (Rational) Numbers

On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique-the arguments from congruency, confounds, and imprecision-and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as "numerosities" or "quanticals," as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

A description-experience gap in statistical intuitions: Of smart babies, risk-savvy chimps, intuitive statisticians, and stupid grown-ups

Comparison of different lines of research on statistical intuitions and probabilistic reasoning reveals several puzzling contradictions. Whereas babies seem to be intuitive statisticians, surprisingly capable of statistical learning and inference, adults' statistical inferences have been found to be inconsistent with the rules of probability theory and statistics. Whereas researchers in the 1960s concluded that people's probability updating is "conservatively" proportional to normative predictions, probability updating research in the 1970s suggested that people are incapable of following Bayes's rule. And whereas animals appear to be strikingly risk savvy, humans often seem "irrational" when dealing with probabilistic information. Drawing on research on the description-experience gap in risky choice, we integrate and systematize these findings from disparate fields of inquiry that have, to date, operated largely in parallel. Our synthesis shows that a key factor in understanding inconsistencies in statistical intuitions research is whether probabilistic inferences are based on symbolic, abstract descriptions or on the direct experience of statistical information. We delineate this view from other conceptual accounts, consider potential mechanisms by which attributes of first-hand experience can facilitate appropriate statistical inference, and identify conditions under which they improve or impair probabilistic reasoning. To capture the full scope of human statistical intuition, we conclude, research on probabilistic reasoning across the lifespan, across species, and across research traditions must bear in mind that experience and symbolic description of the world may engage systematically distinct cognitive processes.

The numerator bias exists in millions of real-world comparisons

Fractions are crucial, from math and science education to daily activities, but they are hard. A puzzling aspect of fractions is that people over-rely on the numerator when comparing a pair of fractions. Previous work has considered this numerator bias mostly as a reasoning mishap. Still, in a vast amount of pairwise comparisons, across many real-world domains, not just education textbooks, we report a high prior probability that the larger fraction has the larger numerator, and, for a relevant case, we provide formal arguments why. The existence of such a regularity suggests that the numerator bias may reflect a rational adaptation that detects and exploits likely events. In a pair of visual-proportion tasks (discrete and continuous fractions), we confirm that the numerator bias in participants adapts to experimented regularities. Even though weak education and math abilities play a role, adaptation to informative priors outside the classroom poses a challenge to educators, learners, and decision-makers.

Do infants and preschoolers quantify probabilities based on proportions?

Most statistical problems encountered throughout life require the ability to quantify probabilities based on proportions. Recent findings on the early ontogeny of this ability have been mixed: For example, when presented with jars containing preferred and less preferred items, 12-month-olds, but not 3- and 4-years-olds, seem to rely on the proportions of objects in the jars to predict the content of samples randomly drawn out of them. Given these contrasting findings, it remains unclear what the probabilistic reasoning abilities of young children are and how they develop. In our study, we addressed this question and tested, with identical methods across age groups and similar methods to previous studies, whether 12-month-olds and 3- and 4-years-olds rely on proportions of objects to estimate probabilities of random sampling events. Results revealed that neither infants nor preschoolers do. While preschoolers' performance is in line with previous findings, infants' performance is difficult to interpret given their failure in a control condition in which the outcomes happened with certainty rather than a graded probability. More systematic studies are needed to explain why infants succeeded in a previous study but failed in our study.

Of Viruses, Vaccines, and Variability: Qualitative Meaning Matters

Deaths from COVID-19 depend on millions of people understanding risk and translating this understanding into risk-reduction behaviors. Although numerical information about risk is helpful, numbers are surprisingly ambiguous, and there are predictable mismatches in risk perception between laypeople and experts. Hence, risk communication should convey the qualitative, contextualized meaning of risk.

Long-tailed macaques extract statistical information from repeated types of events to make rational decisions under uncertainty

Human children and apes seem to be intuitive statisticians when making predictions from populations of objects to randomly drawn samples, whereas monkeys seem not to be. Statistical reasoning can also be investigated in tasks in which the probabilities of different possibilities must be inferred from relative frequencies of events, but little is known about the performance of nonhuman primates in such tasks. In the current study, we investigated whether long-tailed macaques extract statistical information from repeated types of events to make predictions under uncertainty. In each experiment, monkeys first experienced the probability of rewards associated with different factors separately. In a subsequent test trial, monkeys could then choose between the different factors presented simultaneously. In Experiment 1, we tested whether long-tailed macaques relied on probabilities and not on a comparison of absolute quantities to make predictions. In Experiment 2 and 3 we varied the nature of the predictive factors and the complexity of the covariation structure between rewards and factors. Results indicate that long-tailed macaques extract statistical information from repeated types of events to make predictions and rational decisions under uncertainty, in more or less complex scenarios. These findings suggest that the presentation format affects the monkeys’ statistical reasoning abilities.

Emotions and beliefs about morality can change one another

A dual-process theory postulates that belief and emotions about moral assertions can affect one another. The present study corroborated this prediction. Experiments 1, 2 and 3 showed that the pleasantness of a moral assertion - from loathing it to loving it - correlated with how strongly individuals believed it, i.e., its subjective probability. But, despite repeated testing, this relation did not occur for factual assertions. To create the correlation, it sufficed to change factual assertions, such as, "Advanced countries are democracies," into moral assertions, "Advanced countries should be democracies". Two further experiments corroborated the two-way causal relations for moral assertions. Experiment 4 showed that recall of pleasant memories about moral assertions increased their believability, and that the recall of unpleasant memories had the opposite effect. Experiment 5 showed that the creation of reasons to believe moral assertions increased the pleasantness of the emotions they evoked, and that the creation of reasons to disbelieve moral assertions had the opposite effect. Hence, emotions can change beliefs about moral assertions; and reasons can change emotions about moral assertions. We discuss the implications of these results for alternative theories of morality.

Infant Statisticians: The Origins of Reasoning Under Uncertainty

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 logic, probabilistic, and heuristics 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.

Rhesus macaques use probabilities to predict future events

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 (Macaca mulatta) 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 (expected trial) or the relatively rare fruit (unexpected trial) 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.

Long-tailed macaques (Macaca fascicularis) can use simple heuristics but fail at drawing statistical inferences from populations to samples

Human infants, apes and capuchin monkeys engage in intuitive statistics: they generate predictions from populations of objects to samples based on proportional information. This suggests that statistical reasoning might depend on some core knowledge that humans share with other primate species. To aid the reconstruction of the evolution of this capacity, we investigated whether intuitive statistical reasoning is also present in a species of Old World monkey. In a series of four experiments, 11 long-tailed macaques were offered different pairs of populations containing varying proportions of preferred versus neutral food items. One population always contained a higher proportion of preferred items than the other. An experimenter simultaneously drew one item out of each population, hid them in her fists and presented them to the monkeys to choose. Although some individuals performed well across most experiments, our results imply that long-tailed macaques as a group did not make statistical inferences from populations of food items to samples but rather relied on heuristics. These findings suggest that there may have been convergent evolution of this ability in New World monkeys and apes (including humans).

Human behavioral complexity peaks at age 25

Random Item Generation tasks (RIG) are commonly used to assess high cognitive abilities such as inhibition or sustained attention. They also draw upon our approximate sense of complexity. A detrimental effect of aging on pseudo-random productions has been demonstrated for some tasks, but little is as yet known about the developmental curve of cognitive complexity over the lifespan. We investigate the complexity trajectory across the lifespan of human responses to five common RIG tasks, using a large sample (n = 3429). Our main finding is that the developmental curve of the estimated algorithmic complexity of responses is similar to what may be expected of a measure of higher cognitive abilities, with a performance peak around 25 and a decline starting around 60, suggesting that RIG tasks yield good estimates of such cognitive abilities. Our study illustrates that very short strings of, i.e., 10 items, are sufficient to have their complexity reliably estimated and to allow the documentation of an age-dependent decline in the approximate sense of complexity.

It has been unclear how this ability evolves over a person’s lifetime and it had not been possible to be assessed with previous classical tools for statistical randomness. To better understand how age impacts behavior, we have assessed more than 3,400 people aged 4 to 91 years old. Each participant performed a series of online tasks that assessed their ability to behave randomly. The five tasks included listing the hypothetical results of a series of 12 coin flips so that they would “look random to somebody else,” guessing which card would appear when selected from a randomly shuffled deck, and listing the hypothetical results of 10 rolls of a die. We analyzed the participants’ choices according to their algorithmic randomness, which is based on the idea that patterns that are more random are harder to encode in a short computer program. After controlling for characteristics such as gender, language, and education. We have found that age was the only factor that affected the ability to behave randomly. This ability peaked at age 25, on average, and declined from then on. We also demonstrate that a relatively short list of choices, say 10 hypothetical coin flips, can be used to reliably gauge randomness of human behavior. A similar approach could be then used to study potential connections between the ability to behave randomly, cognitive decline, neurodegenerative diseases and abilities such as human creativity.

Intuitive probabilistic inference in capuchin monkeys

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.

Collective Intelligence: Aggregation of Information from Neighbors in a Guessing Game

Complex systems show the capacity to aggregate information and to display coordinated activity. In the case of social systems the interaction of different individuals leads to the emergence of norms, trends in political positions, opinions, cultural traits, and even scientific progress. Examples of collective behavior can be observed in activities like the Wikipedia and Linux, where individuals aggregate their knowledge for the benefit of the community, and citizen science, where the potential of collectives to solve complex problems is exploited. Here, we conducted an online experiment to investigate the performance of a collective when solving a guessing problem in which each actor is endowed with partial information and placed as the nodes of an interaction network. We measure the performance of the collective in terms of the temporal evolution of the accuracy, finding no statistical difference in the performance for two classes of networks, regular lattices and random networks. We also determine that a Bayesian description captures the behavior pattern the individuals follow in aggregating information from neighbors to make decisions. In comparison with other simple decision models, the strategy followed by the players reveals a suboptimal performance of the collective. Our contribution provides the basis for the micro-macro connection between individual based descriptions and collective phenomena.

Comparison of discrete ratios by rhesus macaques (Macaca mulatta)

Perceiving and comparing ratios are crucial skills for humans. Little is known about whether other animals can compare ratios. We trained two rhesus macaques (Macaca mulatta) to choose arrays that contained the greater ratio of positive to negative stimuli, regardless of the absolute number of stimuli in each of the two choice arrays. Subjects learned this task, and their performance generalized to novel ratios. Moreover, performance was modulated by the ratio between ratios; subjects responded more quickly and accurately when the ratio between ratios was higher. Control conditions ruled out the possibility that subjects were relying on surface area, although the ratio between ratios of surface area did seem to influence their choices. Our results demonstrate that rhesus monkeys can compare discrete ratios, demonstrating not only proportional reasoning ability but also the ability to reason about relations between relations.