First,
May and Solomon (2013) factorize the shape parameter of the psychometric functions into the product of an internal noise component and a transducer component. According to this product, data may be fitted in different (equivalent) ways. Our expression,
Equation 6 (or more clearly in the less detailed
Equations 10 and
A2), which considers the product of the covariance of the noise and the Jacobian of the transducer, is an alternative to the product in
May and Solomon (2013) to describe such ambiguity. For instance, in
Equations 10 and
A2, it is obvious that a variation in the covariance of the inner noise can be compensated by a corresponding variation in the Jacobian; therefore, one has an ambiguity. However,
Equation 6 explicitly includes the contributions of the early and external noises (not considered by
May & Solomon, 2013), and this (1) clarifies that different models
S′ affect the estimates of the late noise, but not of the early noise, and (2) suggests eventual ways to break the ambiguity. On the one hand, consider first a model
S′ with a linearly scaled sensitivity so that ∇
S′ = α∇
S. In that case,
Equation 6 says that the distance
D stays the same if the late noise is scaled accordingly, that is, if
\({\boldsymbol n^{\prime }_l} = \alpha \, {\boldsymbol n_l}\) and
\(\Sigma ^{\prime }_l = \alpha ^2 \Sigma _l\). In the general case with a more complicated
S′, α will not be constant but input-dependent. This local/adaptive scaling will affect the late noise in an equivalent adaptive way. However, the (same) early and external noises will be automatically scaled when transferred to the inner representation by
S′. As a result, only the late noise is affected to keep the relative scale of all the noise sources in the inner domain. On the other hand, note that the distances stay the same (equivalent reproduction of the data) only if the Jacobian covaries with the late noise while the early noise is fixed, which is a rather specific situation. However, we have not pursued this possibility given the low accuracy of the results obtained using this
threshold-only method (flat plateaus in
Figures 5 and
A2), even for a fixed nonlinearity. Similarly to the discussion between
Kontsevich, Chen, and Tyler (2002) and
Georgeson and Meese (2006), accuracy and significance of the results are critical, and hence, it is better not to address this issue with methods that intrinsically have low accuracy, as shown here for the threshold-only method.