The paper develops a Bayesian procedure for estimating and inferring spatial models of roll call voting, which is flexible and applicable to any legislative setting. The model can be extended to incorporate various sources of information, such as the nature of underlying dimensions, party discipline, and the evolution of the legislative agenda. The Bayesian approach provides a coherent framework for estimation and inference, allowing for the assessment of uncertainty and hypothesis testing. The authors demonstrate the method's effectiveness through several examples, including the estimation of ideal points, the identification of pivotal legislators, and the analysis of party switchers and the "party influence" hypothesis. The Bayesian simulation method is shown to be robust and flexible, making it suitable for handling complex models and providing accurate posterior distributions.The paper develops a Bayesian procedure for estimating and inferring spatial models of roll call voting, which is flexible and applicable to any legislative setting. The model can be extended to incorporate various sources of information, such as the nature of underlying dimensions, party discipline, and the evolution of the legislative agenda. The Bayesian approach provides a coherent framework for estimation and inference, allowing for the assessment of uncertainty and hypothesis testing. The authors demonstrate the method's effectiveness through several examples, including the estimation of ideal points, the identification of pivotal legislators, and the analysis of party switchers and the "party influence" hypothesis. The Bayesian simulation method is shown to be robust and flexible, making it suitable for handling complex models and providing accurate posterior distributions.