The second model is a Sum of Weighted Likelihoods model based on a Bayesian observer (
Green and Swets, 1974), and has been presented previously by Eckstein, et al. (
2002), and Shimozaki, et al. (
2001).
Figure 2 gives a schematic of the model, and
gives the mathematical equations for predicting performance for the model. As in the linear model, a response is generated at each location (
x_{c},
x_{uc}), and perturbed by internal noise (
N). Then, the model determines the likelihoods of the responses
x_{c} and
x_{uc}, given a signal at the cued location (upper branch, valid trial, signal at cued location =
s_{c}, noise at uncued location =
n_{uc}), given a signal at the uncued location (middle branch, invalid trial, noise at cued location =
n_{c}, signal at uncued location =
s_{uc}), and given signal absence (lower branch, noise at cued location =
n_{c}, noise at uncued location =
n_{uc}). The likelihoods are computed with assumed probability density functions for
x_{c} and
x_{uc} given signal presence and absence; these probability density functions are assumed to be Gaussian. The likelihoods of the responses given a signal at the cued and uncued locations are then weighted separately (
w_{c},
w_{uc}) and summed to give an overall weighted likelihood given signal presence (weighted
L_{s}).
L_{s} is divided by the likelihood of the responses given signal absence (
L_{n}), resulting in a weighted likelihood ratio (
L_{s}/
L_{n}). The weighted likelihood ratio is then compared to a criterion to make a decision on signal presence.