This paper presents a Bayesian approach for estimating and inferring spatial models of roll call voting. The method is flexible and applicable to any legislative setting, regardless of size, the extremism of voting histories, or the number of roll calls. It allows for the integration of various sources of information, such as the nature of policy dimensions, party discipline, and legislative agendas. The Bayesian approach provides a coherent framework for estimation and inference with roll call data, enabling uncertainty assessments and hypothesis testing. The paper demonstrates how the method can be extended to accommodate complex models of legislative behavior. Roll call data are primarily used to estimate ideal points, which represent legislators' policy preferences. Ideal point estimates help describe legislators and legislatures, revealing cleavages and polarization. They also allow testing theories of legislative behavior, such as in studies of the U.S. Congress, state legislatures, courts, and international relations. Current methods for estimating ideal points have statistical and theoretical limitations. They often assume sincere voting, which may not align with models involving log-rolling or party discipline. Estimating roll call models is computationally intensive, and extending them to incorporate realistic behavioral assumptions is challenging. Additionally, the statistical basis of current methods is questionable due to the large number of parameters involved. The paper develops and illustrates Bayesian methods for ideal point estimation and roll call analysis. Bayesian inference provides a coherent method for assessing uncertainty and hypothesis testing in the presence of many parameters. Recent advances in computing make Bayesian modeling feasible for social scientists. The approach allows extending the standard voting model to accommodate complex behavioral assumptions. The paper presents a statistical model for roll call analysis, assuming quadratic utility functions with normal errors. The model is compared to other specifications, such as Gaussian utilities with extreme value errors. The likelihood function is derived based on the model's assumptions, and the model is identified through a priori restrictions on the ideal point matrix. The paper discusses identification issues, noting that the model is not identified without additional restrictions. Identification is essential for estimation methods like maximum likelihood. Bayesian methods can be applied to unidentified models, though data are only informative about identified parameters. The paper compares the Bayesian approach with other methods of ideal point estimation, such as W-NOMINATE and factor analysis. It illustrates the differences and similarities between these approaches using data from the 106th U.S. House of Representatives. The Bayesian approach is shown to produce accurate ideal point estimates and to handle lop-sided votes effectively. The paper also discusses estimation and inference for auxiliary quantities of interest, such as pivotal legislators. The Bayesian approach allows for posterior distributions over any function of the model parameters, enabling inference about these quantities. The paper provides examples of how the Bayesian approach can be used to estimate the identity and location of pivotal legislators in the 106th U.S. Senate. The paper also examines party switchers and the "party influence" hypothesis. The Bayesian approach allows for the extension of the model to accommodate changes in ideal points, facilitating the investigationThis paper presents a Bayesian approach for estimating and inferring spatial models of roll call voting. The method is flexible and applicable to any legislative setting, regardless of size, the extremism of voting histories, or the number of roll calls. It allows for the integration of various sources of information, such as the nature of policy dimensions, party discipline, and legislative agendas. The Bayesian approach provides a coherent framework for estimation and inference with roll call data, enabling uncertainty assessments and hypothesis testing. The paper demonstrates how the method can be extended to accommodate complex models of legislative behavior. Roll call data are primarily used to estimate ideal points, which represent legislators' policy preferences. Ideal point estimates help describe legislators and legislatures, revealing cleavages and polarization. They also allow testing theories of legislative behavior, such as in studies of the U.S. Congress, state legislatures, courts, and international relations. Current methods for estimating ideal points have statistical and theoretical limitations. They often assume sincere voting, which may not align with models involving log-rolling or party discipline. Estimating roll call models is computationally intensive, and extending them to incorporate realistic behavioral assumptions is challenging. Additionally, the statistical basis of current methods is questionable due to the large number of parameters involved. The paper develops and illustrates Bayesian methods for ideal point estimation and roll call analysis. Bayesian inference provides a coherent method for assessing uncertainty and hypothesis testing in the presence of many parameters. Recent advances in computing make Bayesian modeling feasible for social scientists. The approach allows extending the standard voting model to accommodate complex behavioral assumptions. The paper presents a statistical model for roll call analysis, assuming quadratic utility functions with normal errors. The model is compared to other specifications, such as Gaussian utilities with extreme value errors. The likelihood function is derived based on the model's assumptions, and the model is identified through a priori restrictions on the ideal point matrix. The paper discusses identification issues, noting that the model is not identified without additional restrictions. Identification is essential for estimation methods like maximum likelihood. Bayesian methods can be applied to unidentified models, though data are only informative about identified parameters. The paper compares the Bayesian approach with other methods of ideal point estimation, such as W-NOMINATE and factor analysis. It illustrates the differences and similarities between these approaches using data from the 106th U.S. House of Representatives. The Bayesian approach is shown to produce accurate ideal point estimates and to handle lop-sided votes effectively. The paper also discusses estimation and inference for auxiliary quantities of interest, such as pivotal legislators. The Bayesian approach allows for posterior distributions over any function of the model parameters, enabling inference about these quantities. The paper provides examples of how the Bayesian approach can be used to estimate the identity and location of pivotal legislators in the 106th U.S. Senate. The paper also examines party switchers and the "party influence" hypothesis. The Bayesian approach allows for the extension of the model to accommodate changes in ideal points, facilitating the investigation