  In this paper, we introduce the increasing belief criterion in association
rule mining. The criterion uses a recursive application of Bayes' theorem to
compute a rule's belief. Extracted rules are required to have their belief
increase with their last observation. We extend the taxonomy of association
rule mining algorithms with a new branch for Bayesian rule mining~(BRM), which
uses increasing belief as the rule selection criterion. In contrast, the
well-established frequent association rule mining~(FRM) branch relies on the
minimum-support concept to extract rules. We derive properties of the
increasing belief criterion, such as the increasing belief boundary,
no-prior-worries, and conjunctive premises. Subsequently, we implement a BRM
algorithm using the increasing belief criterion, and illustrate its
functionality in three experiments: (1)~a proof-of-concept to illustrate BRM
properties, (2)~an analysis relating socioeconomic information and chemical
exposure data, and (3)~mining behaviour routines in patients undergoing
neurological rehabilitation. We illustrate how BRM is capable of extracting
rare rules and does not suffer from support dilution. Furthermore, we show that
BRM focuses on the individual event generating processes, while FRM focuses on
their commonalities. We consider BRM's increasing belief as an alternative
criterion to thresholds on rule support, as often applied in FRM, to determine
rule usefulness.
