Studies sometimes estimate the prevalence of a disease from the results of one or more diagnostic tests that are applied to individuals of unknown disease status. This approach invariably means that, in the absence of a gold standard and without external constraints, more parameters must be estimated than the data permit. Two assumptions are regularly made in the literature, namely that the test characteristics (sensitivity and specificity) are constant over populations and the tests are conditionally independent given the true disease status. These assumptions have been criticized recently as being unrealistic. Nevertheless, to estimate the prevalence, some restrictions on the parameter estimates need to be imposed. We consider 2 types of restrictions: deterministic and probabilistic restrictions, the latter arising in a Bayesian framework when expert knowledge is available. Furthermore, we consider 2 possible parameterizations allowing incorporation of these restrictions. The first is an extension of the approach of Gardner et al and Dendukuri and Joseph to more than 2 diagnostic tests and assuming conditional dependence. We argue that this system of equations is difficult to combine with expert opinions. The second approach, based on conditional probabilities, looks more promising, and we develop this approach in a Bayesian context. To evaluate the combination of data with the (deterministic and probabilistic) constraints, we apply the recently developed Deviance Information Criterion and effective number of parameters estimated (pD) together with an appropriate Bayesian P value. We illustrate our approach using data collected in a study on the prevalence of porcine cysticercosis with verification from external data.