An agent wants to derive her belief over outcomes based on past observations collected in her

database (memory). There is well establish evidence in the psychology and marketing literature

that agents consistently fail (or choose not) to process all available information. An agent might

be constraint to pay attention (recall) and consider only parts of her potentially available information

due to unawareness, cognitive or psychological limitations or intentionally for effort-efficiency.

Based on this insight, we axiomatize a two-stage belief formation process in which in a first step

agents filter ((un)intentionally) the available information. In a second step individuals employ the

remaining observations to express a belief. We impose cognitively and normatively desirable properties

on the filtering process. The axioms on the belief formation stage describe the relationship

between databases and their induced beliefs. The axiomatized belief induced by a filtered databases

is representable by a similarity weighted average of the estimations induced by each past attentiongrabbing

observation. An appealing application is a satisficing filter that induces a filtered belief

that relies only on past experiences that are sufficiently relevant for a current problem. For the

specific situation that agents (are able to) always pay attention to all available information, our

axiomatization coincides with the axiomatization of a belief formation in Billot et al. (Econometrica

(2005)).