We applied the thin–fat Kanizsa shape discrimination task invented by D. L. Ringach and R. Shapley (1996) to study perceptual completion by measuring whether the discrimination was more accurate for illusory than for occluded shapes. Differently from Ringach and Shapley, we tested naive observers with stereoscopic displays. Discrimination was consistently more accurate for illusory than for occluded shapes under a variety of stimulus conditions. However, the absolute performance was worse than Ringach and Shapley's experienced observers, who discriminated illusory and occluded shapes equally well. When our naive observers were trained, their performance approached that in Ringach and Shapley, and their performance difference diminished between the illusory and occluded. The more precise discrimination of the illusory shapes by untrained observers is consistent with the subjective impression that illusory contours appear clearer and positionally better defined. This makes sense from the perspective of Bayesian decision theory: the location of an illusory contour that is closer to an observer might be more important than an occluded contour, and hence obligatorily represented more precisely. We conclude the paper by discussing implications of our results on the current debate on mechanisms of perceptual completion (M. K. Albert, 2007; B. L. Anderson, 2007; P. J. Kellman, P. Garrigan, T. F. Shipley, & B. P. Keane, 2007).