The likelihood ratio (LR) is largely used to evaluate the relative weight of forensic data regarding two hypotheses, and for its assessment, Bayesian methods are widespread in the forensic field. However, the Bayesian ‘recipe’ for the LR presented in most of the literature consists of plugging‐in Bayesian estimates of the involved nuisance parameters into a frequentist‐defined LR: frequentist and Bayesian methods are thus mixed, giving rise to solutions obtained by hybrid reasoning. This paper provides the derivation of a proper Bayesian approach to assess LRs for the ‘rare type match problem’, the situation in which the expert wants to evaluate a match between the DNA profile of a suspect and that of a trace from the crime scene, and this profile has never been observed before in the database of reference. LR assessment using the two most popular Bayesian models (beta‐binomial and Dirichlet‐multinomial) is discussed and compared with corresponding plug‐in versions.