Magic, Bayes and wows: a Bayesian account of magic tricks Magic, Bayes and wows: a Bayesian account of magic tricks Pablo Rodrigo Grassi1, 2, 3, * and Andreas Bartels1, 2, 3 1 Department of Psychology, University of Tübingen, Germany 2 Centre for Integrative Neurosciences, Tübingen, Germany 3 Max-Planck Institute for Biological Cybernetics, Tübingen, Germany * Corresponding author: [email protected] Abstract Magic tricks have enjoyed an increasing interest by scientists. However, most research in magic focused on isolated aspects of it and a conceptual understanding of magic, encompassing its distinct components and varieties, is missing. Here, we present an account of magic within the theory of Bayesian predictive coding. We present the “wow” effect of magic as an increase in surprise evoked by the prediction error between expected and observed sensory data. We take into account prior knowledge of the observer, attention, and (mis-)direction of perception and beliefs by the magician to bias the observer’s predictions and present two examples for the modelling of the evoked surprise. The role of misdirection is described as everything that aims to maximize the surprise a trick evokes by the generation of novel beliefs, the exploitation of background knowledge and attentional control of the incoming information. Understanding magic within Bayesian predictive coding allows unifying all aspects of magic tricks within one framework, making it tractable, comparable and unifiable with other models in psychology and neuroscience. Keywords: Magic, Bayesian surprise, predictive coding, misdirection, attention 1 Magic, Bayes and wows: a Bayesian account of magic tricks 1. Introduction Magic is about violating expectations: a freely selected card is guessed from a random deck, a coin magically appears in a hand, a chair levitates on stage. We are surprised, because our expectations are suddenly questioned by our senses. This makes magic fascinating for the viewer and also for the researcher. The effect magic evokes has been described as an extraordinary phenomenological experience, namely, as the “sense of wonder induced by seeing an apparently impossible event” (Rensink and Kuhn, 2015a). This experience of wonder and “impossibility” is induced by the profound mismatch between memory-based expectations and a sensory occurrence (Smith et al., 2016). Notably, it is exactly this – the prediction error between expected and actual sensory information – that has increasingly become one of central elements of recent developments in cognitive neuroscience: Bayesian predictive coding theories of brain function (Mumford, 1992; Rao and Ballard, 1999; Friston and Kiebel, 2009; Clark, 2013). Bayesian theories of brain function understand the brain as an inference organ that aims to minimize the prediction error between expected and actual sensory information. Magic, in turn, seems to exploit these prediction errors to maximize surprise. Intuitively, it appears promising to frame magic and misdirection within this influential theory of brain function and describe its effect as an increase of prediction error or surprise. However, although there has been an increasing effort in propagating the scientific investigation of magic using the tools of cognitive science (see Kuhn et al., 2008; Macknik et al., 2008; Rensink and Kuhn, 2015a; Thomas et al., 2015; Quiroga, 2016; Garcia-pelegrin et al., 2020 for an overview), the connection of magic research to Bayesian predictive coding and cognitive neuroscience in general is still thin. In particular, most studies of magic focused their analyses on behavioural responses and eye movements (e.g. Kuhn and Tatler, 2004; Kuhn and Land, 2006; Cui et al., 2011; Barnhart and Goldinger, 2014; Mohr et al., 2015; Ekroll et al., 2016). The link to neural signals has been largely neglected. Only three studies used neural signals to study our response to magic: two closely-related functional magnetic resonance imaging (fMRI) studies localized brain areas involved in the perception of magic (Parris et al., 2009; Danek et al., 2015), while one electroencephalogram (EEG) study focused in the evoked potential related to a “magical” event (Caffaratti et al., 2016). The purpose of this paper is to close this gap. We aim to integrate the study of magic with cognitive neuroscience using an account of magic within the context of Bayesian predictive coding. The use of this established framework can help linking the study of magic to cognitive neuroscience and to develop hypothesis-driven physiological studies. The relation of magic to Bayesian predictive coding is not new (see e.g. Kuhn 2019). Previous studies have related magic to Bayesian inference and predictive coding in different formulations (e.g. Barnhart, 2010; Danek et al., 2015; Van de Cruys et al., 2015; Kuhn and Rensink, 2016). However, these references were usually given as a possible interpretation of certain magic effects and as an interesting note of something worth delving deep into. We will proceed as follows: in a first section, we shortly introduce Bayesian predictive coding and describe the Bayesian brain’s aim of minimization of prediction error and surprise. Then, we will show how magic effects can be understood as an increase of (Bayesian) surprise resulting from the mismatch between expectations and sensory information. We exemplify its computation using Bayesian models applied to two simple magic tricks. Thereafter, we describe misdirection as all elements of the preparation phase that aim to maximize the surprise, and we present three types of misdirection based on their effects on Bayesian inference. Finally, we argue that the understanding of experience of magic as surprise within a Bayesian predictive coding framework enables a variety of research questions that can be addressed using magic tricks as stimuli exploiting known neural and behavioural responses. 2. Bayesian predictive coding A key function of our sensory and cognitive systems is to provide us with trustworthy representations about physical properties of the world in order to allow for a meaningful interaction with it. Yet, the sensory information is incomplete, noisy, and ambiguous, making it 2 Magic, Bayes and wows: a Bayesian account of magic tricks an incredibly complex task to derive the underlying physical sources from it. As an example, the whole process of transforming the visual input from a scene received in both our retinas back again into a representation of the scene consisting of segregated elements is an sophisticated achievement of our visual system – but also one thought to require more than a third of the human brain (Felleman and Van Essen, 1991). For any given retinal signal there is an infinite number of possible sources that could have caused it (Kersten and Yuille, 2003). To cope with this fundamental ambiguity of perception our brain employs prior knowledge about the world based on experienced statistical regularities to predict the most likely causes of the incoming information. Recent cognitive theories of brain function understand the brain as a probabilistic inference system that attempts to explain the input it receives by generating hypotheses of the possible causes of it by means of Bayesian inference (Kersten et al., 2004; Friston, 2005; Friston and Kiebel, 2009; Clark, 2013). Bayesian inference involves the combination of observed information (likelihood) with probabilistic predictions based on previous knowledge (prior beliefs) to calculate a posterior probability, which corresponds to the most likely interpretation. This inference process has been hypothesized to be implemented in the brain through predictive coding (Mumford, 1992; Rao and Ballard, 1999). Bayesian predictive coding theories understand the brain as a hierarchical system whose aim is to maximize the evidence for its probabilistic model of the world by minimizing the difference between its predictions (i.e. prior beliefs) with the sensory data. The difference between prior beliefs and incoming sensory information is referred to as prediction error, which, in short, reports the “surprise” induced by the mismatch. If the predictions fail to explain the incoming information, then the prediction error gives novel information to update the probabilistic beliefs about the world and is therefore the fundamental basis of learning (Schultz et al., 1997; Friston, 2005; Friston and Kiebel, 2009). In sum, following this account, a key task of the brain is to acquire knowledge and build a world model, such that prediction errors between its probabilistic world model and actual observations are minimized (“not to be surprised”). However, this very process, updating and improving the model, and adapting it throughout the dynamics of human life, necessitates seeking out the unexpected and unknown, and thus, being surprised (Schwartenbeck et al., 2013; Clark, 2018). Accordingly, curiosity-driven learning and novelty- seeking explorative human behaviour necessarily accompanies the Bayesian predictive coding imperative of “minimize long-term prediction error”. This essential aspect of human behaviour is expressed in the practices of, just to name a few, literature, humour, games, puzzles, visual arts and, relevant for us, magic. In the following, we will present three main features of Bayesian predictive coding, which are important for the understanding of magic: its hierarchical structure, contextual dependency and attentional modulation. 2.1. Inference: across the cortical hierarchy In