A set of sensors is used to identify which of the users, from a pre-specified set of users, is currently using a device. Each sensor provides a name of a user and a real number representing its level of confidence in the assessment. However, the sensors measure different signals for different traits that are largely unrelated. To be able to implement a policy based on these measurements, one needs to aggregate the information provided by all sensors. We use an axiomatic approach to provide several reasonable trust functions. We show that by providing a few desirable properties we can derive several solutions that are characterized by these properties. Our analysis makes use of an important result by Kolmogorov (1930).
dp739.pdf
In the standard Bayesian framework the data are assumed to be generated by a distribution parametrized by θ in a parameter space Θ, over which a prior distribution π is defined. A Bayesian statistician quantifies the belief that the true parameter is θ_0 in Θ by its posterior probability given the observed data. We investigate the behavior of the posterior belief in θ_0 when the data are generated under some parameter θ_1, which may or may not be be the same as θ_0. Starting from stochastic orders, specifically, likelihood ratio dominance, that obtain for resulting distributions of posteriors, we consider monotonicity properties of the posterior probabilities as a function of the sample size when data arrive sequentially. While the θ_0-posterior is monotonically increasing (i.e., it is a submartingale) when the data are generated under that same θ_0, it need not be monotonically decreasing in general, not even in terms of its overall expectation, when the data are generated under a different θ_1; in fact, it may keep going up and down many times. In the framework of simple iid coin tosses, we show that under certain conditions the overall expected posterior of θ_0 eventually becomes monotonically decreasing when the data are generated under θ_1≠θ_0. Moreover, we prove that when the prior is uniform this expected posterior is a log-concave function of the sample size, by developing an inequality that is related to Turán's inequality for Legendre polynomials.
A stumper is a riddle whose solution is typically so elusive that it does not come to mind, at least initially - leaving the responder stumped. Stumpers work by eliciting a (typically visual) representation of the narrative, in which the solution is not to be found. In order to solve the stumper, the blocking representation must be changed, which does not happen to most respondents. I have collected all the riddles I know at this time that qualify, in my opinion, as stumpers. I have composed a few, and tested many. Whenever rates of correct solutions were available, they are included, giving a rough proxy for difficulty.
Reports in the 1970s of cognitive illusions in judgments of uncertainty had been anticipated by Laplace 150 years earlier. We discuss Miller and Gelman's remark that Laplace's anticipation of the main ideas of the heuristics and biases approach "gives us a new perspective on these ideas as more universal and less contingent on particular developments [that came much] later."
Stafford (2018) found that female chess players outperform expectations when playing against men, in a study of data from over 5.5 million official games around the world. I examined whether that result could stem from not controlling for the ages of both players, as female players tend to be much younger than male players. Using the same data as Stafford, I was able to replicate his main result only when the opponent’s age was ignored. When the ages of both players were included in the analysis, the gender-composition effect was reversed. Further analyses using other data demonstrated the robustness of this pattern, re-establishing that female chess players underperform when playing against men. Prior to Stafford’s paper, the leading premise was that women encounter psychological obstacles that prevent them from performing at their normal capacity against men. My commentary continues that line of evidence and is consistent with the stereotype-threat explanation.
The Cognitive Reflection Test (CRT) allegedly measures the tendency to override the prepotent incorrect answers to some special problems, and to engage in further reflection. A growing literature suggests that the CRT is a powerful predictor of performance in a wide range of tasks. This research has mostly glossed over the fact that the CRT is composed of math problems. The purpose of this paper is to investigate whether numerical CRT items do indeed call upon more than is required by standard math problems, and whether the latter predict performance in other tasks as well as the CRT. In Study 1 we selected from a bank of standard math problems items that, like CRT items, have a fast lure, as well as others which do not. A 1-factor model was the best supported measurement model for the underlying abilities required by all three item types. Moreover, the quality of all these items – CRT and math problems alike – as predictors of performance on a set of choice and reasoning tasks did not depend on whether or not they had a fast lure, but rather only on their quality as math items. In other words, CRT items seem not to be a “special” category of math problems, although they are quite excellent ones. Study 2 replicated these results with a different population and a different set of math problems.
