In the era of big data, the increasing availability of huge data sets can paradoxically be harmful when our causal inference method is designed to search for a causal model that is faithful to our data. Under the commonly made Causal Faithfulness Assumption, we look for patterns of dependencies and independencies in the data and match them with causal models that imply the same patterns. However, given enough data, we start picking up on the fact that everything is ultimately connected. These interactions are not normally picked up in small samples. The only faithful causal model in the limit of a large number of samples (the large-sample limit) therefore becomes the one where everything is connected. Alas, we cannot extract any useful causal information from a completely connected structure without making additional (strong) assumptions. We propose an alternative approach (RoCELL) that replaces the Causal Faithfulness Assumption with a prior that reflects the existence of many "weak" (irrelevant) and "strong" interactions. RoCELL outputs a posterior distribution over the target causal effect estimator that leads to good estimates even in the large-sample limit.