Our immediate observations must be supplemented with contextual information to resolve ambiguities. However, the context is often ambiguous too, and thus it should be inferred itself to guide behavior. Here, we introduce a novel hierarchical task (airplane task) in which participants should infer a higher-level, contextual variable to inform probabilistic inference about a hidden dependent variable at a lower level. By controlling the reliability of past sensory evidence through varying the sample size of the observations, we find that humans estimate the reliability of the context and combine it with current sensory uncertainty to inform their confidence reports. Behavior closely follows inference by probabilistic message passing between latent variables across hierarchical state representations. Commonly reported inferential fallacies, such as sample size insensitivity, are not present, and neither did participants appear to rely on simple heuristics. Our results reveal uncertainty-sensitive integration of information at different hierarchical levels and temporal scales.

While previous studies have shown that human behavior adjusts in response to uncertainty, it is still not well understood how uncertainty is estimated and represented. As probability distributions are high dimensional objects, only constrained families of distributions with a low number of parameters can be specified from finite data. However, it is unknown what the structural assumptions are that the brain uses to estimate them. We introduce a novel paradigm that requires human participants of either sex to explicitly estimate the dispersion of a distribution over future observations. Judgments are based on a very small sample from a centered, normally distributed random variable that was suggested by the framing of the task. This probability density estimation task could optimally be solved by inferring the dispersion parameter of a normal distribution. We find that although behavior closely tracks uncertainty on a trial-by-trial basis and resists an explanation with simple heuristics, it is hardly consistent with parametric inference of a normal distribution. Despite the transparency of the simple generating process, participants estimate a distribution biased towards the observed instances while still strongly generalizing beyond the sample. The inferred internal distributions can be well approximated by a nonparametric mixture of spatially extended basis distributions. Thus, our results suggest that fluctuations have an excessive effect on human uncertainty judgments because of representations that can adapt overly flexibly to the sample. This might be of greater utility in more general conditions in structurally uncertain environments.

Spatially resolved luminescence and Raman spectroscopy investigations are applied to a series of (112¯2)-GaN samples grown by hydride vapor phase epitaxy (HVPE) grown over an initial layer deposited by metal organic vapor phase epitaxy on patterned sapphire substrates. Whereas these two differently grown GaN layers are crystallographically homogeneous, they differ largely in their doping level due to high unintentional oxygen uptake in the HVPE layer. This high doping shows up in luminescence spectra, which can be explained by a free-electron recombination band for which an analytical model considering the Burstein-Moss shift, conduction band tailing, and the bandgap renormalization is included. Secondary ion mass spectrometry, Raman spectroscopy, and Hall measurements concordantly determine the electron density to be above 1019 cm−3. In addition, the strain state is assessed by Raman spectroscopy and compared to a finite element analysis.

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