We make this formulation slightly more general by allowing for a set of auxiliary variables
U with corresponding weights
u. Later, we will impose a prior on
w but not on
u. This formulation allows for the inclusion of factors that affect the observer's responses other than the noise stimulus. For example, in Tadin et al. (
2006), the question of interest is the time-varying influence of visual motion in the stimulus surround on judgments of motion direction in the center. Here, the time-varying surround stimulus would be put into the
X matrix while the signal sign in the center would take one column of the
U matrix. Another possibility, in detection tasks, is to include the signal sign into the
U matrix, as in Knoblauch and Maloney (
2008b, p.5, Equation 16), rather than summing noise and signal in the
X matrix. We found that the
U strategy generally led to faster optimization while yielding equivalent estimates of internal templates. A final possibility is to include trial-delayed responses in the
U matrix to model observer's tendency to respond similarly or dissimilarly on subsequent trials, as in the commonly used method of modelling refractory periods and burstiness in neurons (Pillow et al.,
2008). We eliminate the constant
c by adding a constant column to the matrix
U. Under this new parameterization, we have the negative log-likelihood function for the linear observer model: