In theory, the probabilistic linkage method provides two distinct advantages over non-probabilistic methods, including minimal rates of linkage error and accurate measures of these rates for data users. However, implementations can fall short of these expectations either because the conditional independence assumption is made, or because a model with interactions is used but lacks the identification property. In official statistics, this is currently the main challenge to the automated production and use of linked data. To address this challenge, a new methodology is described for proper linkage problems, where matched records may be identified with a probability that is bounded away from zero, regardless of the population size. It models the number of neighbours of a given record, i.e. the number of resembling records. To be specific, the proposed model is a finite mixture where each component is the sum of a Bernoulli variable with an independent Poisson variable. It has the identification property and yields solutions for many longstanding problems, including the evaluation of blocking criteria and the estimation of linkage errors for probabilistic or non-probabilistic linkages, all without clerical reviews or conditional independence assumptions. Thus it also enables unsupervised machine learning solutions for record linkage problems.