Weighted average pooling algorithm

A decision variable Di was obtained by taking the weighted average of the confidence ratings/ neural decision variables for every trial i for a group of participants (refer to¹⁹). Mathematically, Di is represented as

where Ri, the column vector, consists of the ratings or neural decision variables of the participants in a group, and the column vector w contains the linear weights.

For each random group of N participants, the behavioural ratings/ neural decision variables corresponding to 90% of the trials were used for training and rest for testing the performance of the algorithm. This was performed 10 times. The weights w were estimated from the training data by maximizing the Fisher’s criterion function as follows:

where μ1 and μ2 are the mean rating/neural decision variable for present trials and absent trials, respectively, for N participants, and

with C1 and C2 denoting the class of present trials and absent trials, respectively. But before inverting Sw, to tackle the issue of singularity (if any), it is regularized as

where γ is the ridge parameter. In our case, we chose γ=2. IN×N is the identity matrix, and N is the total number of participants in the group. The decision criterion C, which maximized the proportion correct, was also estimated from the training data. The group decision Gi, defined below, was obtained by comparing Di to criterion C. Define

The outcome ‘1’ denotes the presence of the target and ‘2’ denotes the absence.

We have limited the group decision rules to simple majority and weighted average pooling since these are the two most commonly used pooling algorithms^19,20. Linear combination has been shown to be superior to majority voting on visual search tasks²⁰ based on behavioural and eye tracking data. We have used both the pooling algorithms to fuse data from multiple brains using both behavioural and neural data.