This work casts light upon a pair of restrictions inherent to the basic weighted updating model, which is a generalization of Bayesian updating that allows for biased learning. Relaxing the restrictions allows for the study of individuals who discriminate between observations or who treat information in a dynamically inconsistent manner. These generalizations augment the set of cognitive biases that can be studied using new versions of the weighted updating model to include the availability heuristic, order effects, self-attribution bias, and base-rate neglect in light of irrelevant information.