In this paper, the use of closed-form expressions is compared to the BIC approximation, with respect to speaker clustering. We first show that the particular BIC setting which is commonly used in this task, namely the approximation of the marginal - with respect to the model parameters - and conditional - with respect to the latent variables - likelihood, belongs to the exponential family, and hence admits a closed-form expression by attaching conjugate priors. We then formalize the role of the tuning parameter as a hyperparameter of the prior and finally we explain the several proposed setting - global, local and segmental - based on the strength of the prior. Experiments are carried out for the speaker clustering task and improvement over the BIC approximation is reported.