Reconciling similarity across models of continuous selections

Recently-developed models of decision making have provided accounts of the cognitive processes underlying choice on tasks where responses can fall along a continuum, such as identifying the color or orientation of a stimulus. Even though nearly all of these models seek to extend diffusion decision processes to a continuum of response options, they vary in terms of complexity, tractability, and their ability to predict patterns of data such as multimodal distributions of responses. We suggest that these differences are almost entirely due to differences in how these models account for the similarity among response options. In this theoretical note, we reconcile these differences by characterizing the existing models under a common framework, where the assumptions about psychological representations of similarity, and their implications for behavioral data (e.g., multimodal responses), are made explicit. Furthermore, we implement a simulation-based approach to computing model likelihoods that allows for greater freedom in constructing and implementing continuous response models. The resulting geometric similarity representation (GSR) can supplement approaches like the circular / spherical diffusion models by allowing them to generate multimodal distributions of responses, or simplify models like the spatially continuous diffusion model by condensing their representations of similarity and allowing them to generate simulations more efficiently. To illustrate its utility, we apply this approach to multimodal distributions responses, two-dimensional responses (such as locations on a computer screen), and continuua of response options with nontrivial, nonlinear similarity relations between response options.