Chances and frequencies in probabilistic reasoning: rejoinder to Hoffrage,Gigerenzer, Krauss,and Martignon

Do individuals unfamiliar with probability and statistics need a specific type of data in order to draw correct inferences about uncertain events? Girotto and Gonzalez (Cognition 78 (2001) 247) showed that naive individuals solve frequency as well as probability problems, when they reason extensionally, in particular when probabilities are represented by numbers of chances. Hoffrage, Gigerenzer, Krauss, and Martignon (Cognition 84 (2002) 343) argued that numbers of chances are natural frequencies disguised as probabilities, though lacking the properties of true probabilities. They concluded that we failed to demonstrate that naive individuals can deal with true probabilities as opposed to natural frequencies. In this paper, we demonstrate that numbers of chances do represent probabilities, and that naive individuals do not confuse numbers of chances with frequencies. We conclude that there is no evidence for the claim that natural frequencies have a special cognitive status, and the evolutionary argument that the human mind is unable to deal with probabilities.     

Errors in subtractions: judged and observed frequencies

In a first study 10 adults, aged 24-44 years, solved all 105 subtraction problems in the form M - N = , where 0 < or = M < or = 13, 0 < or = N < or = 13 and N < or = M. Each participant solved every problem 10 times and in total there were 10 500 answers. Answers, response latencies and errors were registered. Retrospective verbal reports were also given, indicating how a solution was reached: (1) via a (conscious) reconstructive cognitive process or (2) via an (unconscious) reproductive (retrieval) process. The participants made 291 errors (2.8%) when solving the subtractions in study 1. The rate of self-correction was very high, 92%. In a second study 27 undergraduate students estimated overall error rates, including self-corrected errors for the 105 subtraction problems used in the first study. Judged and actual error rates were compared. The participants systematically underestimated error rates for error prone problems and overestimated error rates for error free problems. The participants were fairly accurate when they predicted problems that were most error prone, with a hit rate of 0.67 for the (18) problems predicted as the most error prone ones. In contrast, predictions of which problems were error free were very poor with a hit rate of only 0.20 of the problems predicted as error free really having no errors in study 1. The correlation between judged error rates and frequencies for actually made errors was 0.69 for answers belonging to reconstructive solutions. In contrast, there was no significant correlation between judged and actual error rates at all for retrieved solutions, possibly reflecting the inaccessibility to consciousness of quick retrieval processes.     

Distinguishing indeterminate belief from “risk-averse” preferences

I focus my discussion on the well-known Ellsberg paradox. I find good normative reasons for incorporating non-precise belief, as represented by sets of probabilities, in an Ellsberg decision model. This amounts to forgoing the completeness axiom of expected utility theory. Provided that probability sets are interpreted as genuinely indeterminate belief (as opposed to “imprecise” belief), such a model can moreover make the “Ellsberg choices” rationally permissible. Without some further element to the story, however, the model does not explain how an agent may come to have unique preferences for each of the Ellsberg options. Levi (1986, Hard choices: Decision making under unresolved conflict. Cambridge, New York: Cambridge University Press) holds that the extra element amounts to innocuous secondary “risk” or security considerations that are used to break ties when more than one option is rationally permissible. While I think a lexical choice rule of this kind is very plausible, I argue that it involves a greater break with xpected utility theory than mere violation of the ordering axiom.     

Probability Biases in Genetic Problem Solving: A Comparison of Undergraduates, Genetic Counseling Graduate Students, and Genetic Counselors

Heuristics are mental shortcuts that aid people in everyday problem-solving and decision-making. Although numerous studies have demonstrated their use in contexts ranging from consumers’ shopping decisions to experts’ estimations of experimental validity, virtually no published research has addressed heuristics use in problems involving genetic conditions and associated risk probabilities. The present research consists of two studies. In the first study, 220 undergraduates attempted to solve four genetic problems—two common heuristic problems modified to focus on genetic likelihood, and two created to study heuristics and probability rule application. Results revealed that the vast majority of undergraduates used heuristics and also demonstrated a complete misuse of probability rules. In the second study, 156 practicing genetic counselors and 89 genetic counseling students solved slightly modified versions of the genetic problems used in Study 1. Results indicated that a large percentage of both genetic counselors and students used heuristics, but the counselors demonstrated superior problem-solving performance compared to both the genetic counseling students and the undergraduates from Study 1. Research, training, and practice recommendations are presented.     

等级效标分数的概率神经网络预测方法研究

针对基于统计学的等级效标分数预测存在的问题,提出了运用概率神经网络进行等级效标分数预测的方法。在20种条件下进行了计算机模拟实验,包括测验分数为单变量和多变量,以及各种水平的白噪声干扰条件,结果表明在测验分数为多变量的情况下,或者在有白噪声干扰的条件下,运用概率神经网络方法可以比统计学方法更好地对等级效标分数进行预测。     

Dissimilarity cumulation theory and subjective metrics

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a≠b?Dab>0; (2) Daa=0; (3) If and , then ; and (4) for any sequence {a_nX_nb_n}_n∈N, where X_n is a chain of stimuli, Da_nX_nb_n→0?Da_nb_n→0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb+DbYa across all possible chains X and Y inserted between a and b.     

The origins of probabilistic inference in human infants

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.     

The Equiprobability Bias from a Mathematical and Psychological Perspective

The equiprobability bias (EB) is a tendency to believe that every process in which randomness is involved corresponds to a fair distribution, with equal probabilities for any possible outcome. The EB is known to affect both children and adults, and to increase with probability education. Because it results in probability errors resistant to pedagogical interventions, it has been described as a deep misconception about randomness: the erroneous belief that randomness implies uniformity. In the present paper, we show that the EB is actually not the result of a conceptual error about the definition of randomness. On the contrary, the mathematical theory of randomness does imply uniformity. However, the EB is still a bias, because people tend to assume uniformity even in the case of events that are not random. The pervasiveness of the EB reveals a paradox: The combination of random processes is not necessarily random. The link between the EB and this paradox is discussed, and suggestions are made regarding educational design to overcome difficulties encountered by students as a consequence of the EB.