While the serotypes of Streptococcus pneumoniae are known to compete during colonization in human hosts, our knowledge of how competition occurs is still incomplete. New insights of pneumococcal between-type competition could be generated from carriage data obtained by molecular-based detection methods, which record more complete sets of serotypes involved in co-carriage than when detection is done by culture. Here, we develop a Bayesian estimation method for inferring between-type interactions from longitudinal data recording the presence/absence of the types at discrete observation times. It allows inference from data containing co-carriage of two or more serotypes, which is often the case when pneumococcal presence is determined by molecular-based methods. The computational burden posed by the increased number of types detected in co-carriage is addressed by approximating the likelihood under a multi-state model with the likelihood of only those trajectories with minimum number of acquisition and clearance events between observation times. The proposed method's performance was validated on simulated data. The estimates of the interaction parameters of acquisition and clearance were unbiased in settings with short sampling intervals between observation times. With less frequent sampling, the estimates of the interaction parameters became more biased, but their ratio, which summarizes the total interaction, remained unbiased. Confounding due to unobserved heterogeneity in exposure could be corrected by including individual-level random effects. In an application to empirical data about pneumococcal carriage in infants, we found new evidence for between-serotype competition in clearance, although the effect size was small.