Optimal response selection and decisional separability in Gaussian general recognition theory

• Optimal Gaussian general recognition theory models can mimic decisional separability.
• Unequal stimulus base rates and payoff schemes do not affect mimicry of decisional separability.
• The distinction between psychological processes and mathematical constructs is important.

We provide the necessary and sufficient conditions for a Gaussian general recognition theory (GRT) model with an optimal response selection rule to be empirically indistinguishable from a model with linear decision bounds and decisional separability. General recognition theory assumes noisy, multidimensional perception and deterministic, multidimensional response selection; decisional separability holds if, and only if, the decision bounds that define response regions are parallel to the coordinate axes of the cognitive space of interest (e.g., perceptual space). The analysis of decisional separability is complicated by the fact that multiple response rules are possible in GRT. Recent work showed that failure of decisional separability is not identifiable in Gaussian GRT models with linear or piecewise linear decision bounds (Silbert & Thomas 2013). In the present work, we analyze the role of apparent decisional separability (and failures thereof) in an optimal GRT model. In addition to describing the necessary and sufficient conditions for optimal responding to mimic decisional separability, we show that invertible linear transformations of optimal models that explicitly mimic decisional separability produce models that implicitly mimic decisional separability. We end with a brief discussion of the effects of unequal prior stimulus probabilities and biased payoff schemes on the presence or absence of decisional separability.