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This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions.Gaussian process regression is a powerful, non-parametric Bayesian approach towards regression problems that can be utilized in exploration and exploitation scenarios. This tutorial aims to provide an accessible introduction to these techniques. We will introduce Gaussian processes which generate distributions over functions used for Bayesian non-parametric regression, and demonstrate their use in applications and didactic examples including simple regression problems, a demonstration of kernel-encoded prior assumptions and compositions, a pure exploration scenario within an optimal design framework, and a bandit-like exploration-exploitation scenario where the goal is to recommend movies. Beyond that, we describe a situation modelling risk-averse exploration in which an additional constraint (not to sample below a certain threshold) needs to be accounted for.Lastly, we summarize recent psychological experiments utilizing Gaussian processes. Software and literature pointers are also provided.

From foraging for food to learning complex games, many aspects of human behaviour can be framed as a search problem with a vast space of possible actions. Under finite search horizons, optimal solutions are generally unobtainable. Yet how do humans navigate vast problem spaces, which require intelligent exploration of unobserved actions? Using a variety of bandit tasks with up to 121 arms, we study how humans search for rewards under limited search horizons, where the spatial correlation of rewards (in both generated and natural environments) provides traction for generalization. Across a variety of different probabilistic and heuristic models, we find evidence that Gaussian Process function learning-combined with an optimistic Upper Confidence Bound sampling strategy-provides a robust account of how people use generalization to guide search. Our modelling results and parameter estimates are recoverable, and can be used to simulate human-like performance, providing insights about human behaviour in complex environments.All rights reserved. No reuse allowed without permission.was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.The copyright holder for this preprint (which . http://dx.doi.org/10.1101/171371 doi: bioRxiv preprint first posted online Aug. 1, 2017; previous work exploring inductive biases in pure function learning contexts 21,22 and human behaviour in univariate function optimization 23 , we present a comprehensive approach using a robust computational modelling framework to understand how humans generalize in an active search task.Across three studies using uni-and bivariate multi-armed bandits with up to 121 arms, we compare a diverse set of computational models in their ability to predict individual human behaviour. In all experiments, the majority of subjects are best captured by a model combining function learning using Gaussian Process (GP) regression, with an optimistic Upper Confidence Bound (UCB) sampling strategy that directly balances expectations of reward with the reduction of uncertainty. Importantly, we recover meaningful and robust estimates about the nature of human generalization, showing the limits of traditional models of associative learning 24 in tasks where the environmental structure supports learning and inference.The main contributions of this paper are threefold:1. We introduce the spatially correlated multi-armed bandit as a paradigm for studying how people use generalization to guide search in larger problems space than traditionally used for studying human behaviour.2. We find that a Gaussian Process model of function learning robustly captures how humans generalize and learn about the structure of the environment, where an observed tendency towards undergeneralization is shown to sometimes be beneficial.3. We show that participants solve the exploration-exploitation dilemma by optimistically inflating expectations of reward by the underlying uncertainty, with recoverable evidence for the separate phenome...

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We introduce the Berlin Numeracy Test, a new psychometrically sound instrument that quickly assesses statistical numeracy and risk literacy. We present 21 studies (n=5336) showing robust psychometric discriminability across 15 countries (e.g., Germany, Pakistan, Japan, USA) and diverse samples (e.g., medical professionals, general populations, Mechanical Turk web panels). Analyses demonstrate desirable patterns of convergent validity (e.g., numeracy, general cognitive abilities), discriminant validity (e.g., personality, motivation), and criterion validity (e.g., numerical and non-numerical questions about risk). The Berlin Numeracy Test was found to be the strongest predictor of comprehension of everyday risks (e.g., evaluating claims about products and treatments; interpreting forecasts), doubling the predictive power of other numeracy instruments and accounting for unique variance beyond other cognitive tests (e.g., cognitive reflection, working memory, intelligence). The Berlin Numeracy Test typically takes about three minutes to complete and is available in multiple languages and formats, including a computer adaptive test that automatically scores and reports data to researchers (http://www.riskliteracy.org). The online forum also provides interactive content for public outreach and education, and offers a recommendation system for test format selection. Discussion centers on construct validity of numeracy for risk literacy, underlying cognitive mechanisms, and applications in adaptive decision support.