This article examines the potential for misleading inferences from conventional analyses and sensitivity analyses of observational data, and describes some proposed solutions based on specifying prior distributions for uncontrolled sources of bias. The issues are illustrated in a sensitivity analysis of confounding in a study of residential wire code and childhood leukemia and in a pooled analysis of 12 studies of magnetic-field measurements and childhood leukemia. Both analyses have been interpreted as evidence in favor of a causal effect of magnetic fields on leukemia risk. This interpretation is contrasted with results from analyses based on prior distributions for the unidentified bias parameters used in the original sensitivity-analysis model. These analyses indicate that accounting for uncontrolled confounding and response bias under a reasonable prior can substantially alter inferences about the existence of a magnetic-field effect. More generally, analyses with informative priors for unidentified bias parameters can help avoid misinterpretation of conventional results and ordinary sensitivity analyses.