It has recently been shown that Bayesian networks with hidden variables represent a wider range of probabilistic distributions than Bayesian networks without hidden variables. After introducing the general concept of a hidden variable and how it can be understood in Bayesian networks, we present a distinction between optimizing and essential hidden variables. We propose that it is only essential hidden variables that add representational power to Bayesian networks. We then explain past research with hidden variables in light of this new distinction and implement an exploratory algorithm to find essential hidden variables and to examine the conditions on the distribution that hint at their existence.