Observer models.

<p><b>A</b>: Observer models consist of three model factors: Causal inference strategy, Shape of sensory noise, and Type of prior over stimuli (see text). <b>B</b>: Graphical representation of the observer model. In the left panel (<i>C</i> = 1), the visual (<i>s</i>_vis) and vestibular (<i>s</i>_vest) heading direction have a single, common cause. In the right panel (<i>C</i> = 2), <i>s</i>_vis and <i>s</i>_vest have separate sources, although not necessarily statistically independent. The observer has access to noisy sensory measurements <i>x</i>_vis, <i>x</i>_vest, and knows the visual reliability level of the trial <i>c</i>_vis. The observer is either asked to infer the causal structure (unity judgment, explicit causal inference), or whether the vestibular stimulus is rightward of straight ahead (inertial discrimination, implicit causal inference). Model factors affect different stages of the observer model: the strategy used to combine the two causal scenarios; the type of prior over stimuli <i>p</i>_prior(<i>s</i>_vis, <i>s</i>_vest|<i>C</i>); and the shape of sensory noise distributions <i>p</i>(<i>x</i>_vis|<i>s</i>_vis, <i>c</i>_vis) and <i>p</i>(<i>x</i>_vest|<i>s</i>_vest) (which affects equally both how noisy measurements are generated and the observer’s beliefs about such noise). <b>C</b>: Example decision boundaries for the Bay-X-E model (for the three reliability levels), and for the Fix model, for a representative observer. The observer reports ‘unity’ when the noisy measurements <i>x</i>_vis, <i>x</i>_vest fall within the boundaries. Note that the Bayesian decision boundaries expand with larger noise. Nonlinearities are due to the interaction between eccentricity-dependence of the noise and the prior (wiggles are due to the discrete empirical prior).