Recommender systems based on collaborative filtering have received a great deal of interest over the last two decades. In particular, recently proposed methods based on dimensionality reduction techniques and using a symmetrical representation of users and items have shown promising results. Following this line of research, we propose a probabilistic collaborative filtering model that explicitly represents all items and users simultaneously in the model. Experimental results show that the proposed system obtains significantly better results than other collaborative filtering systems (evaluated on the MovieLens data set). Furthermore, the explicit representation of all users and items allows the model to, e.g., make group-based recommendations balancing the preferences of the individual users.