Modern studies of legislative behavior focus upon the relationship among the policy preferences of legislators, institutional arrangements, and legislative outcomes. In spatial models of legislatures, policies are represented geometrically, as points in a low-dimensional Euclidean space. Each legislator has a most preferred policy or ideal point in this space and his or her utility for a policy declines with the distance of the policy from his or her ideal point; see Davis, Hinich, and Ordeshook (1970) for an early survey.The primary use of roll call data-the recorded votes of deliberative bodies 1 -is the estimation of ideal points. The appeal and importance of ideal point estimation arises in two ways. First, ideal point estimates let us describe legislators and legislatures. The distribution of ideal points estimates reveals how cleavages between legislators reflect partisan affiliation or region or become more polarized over time (e.g., McCarty, Poole, and Rosenthal 2001). Roll call data serve similar purposes for interest groups, such as Americans for Democratic Action, the National Taxpayers Union, and the Sierra Club, to produce "ratings" of legislators along different policy dimensions. Second, estimates from roll call analysis can be used to test theories of legislative behavior. For instance, roll call analysis has been used Joshua Clinton is Assistant Professor, Department of Politics, Princeton University, Princeton, NJ 08540 (clinton@princeton.edu).Simon Jackman is Associate Professor (Voeten 2000). In short, roll call analysis make conjectures about legislative behavior amenable to quantitative analysis, helping make the study of legislative politics an empirically grounded, cumulative body of scientific knowledge.Current methods of estimating ideal points in political science suffer from both statistical and theoretical deficiencies. First, any method of ideal point estimation embodies an explicit or implicit model of legislative behavior. Generally, it is inappropriate to use ideal points estimated under one set of assumptions (such as sincere voting over a unidimensional policy space) to test a different behavioral model (such as log-rolling). Second, the computations required for estimating even the simplest roll call model are very difficult and extending these models to incorporate more realistic behavioral assumptions is nearly impossible with extant methods. Finally, the statistical basis of current methods for ideal point estimation is, to be polite, questionable. Roll call analysis involves very large numbers of parameters, since each legislator has an ideal point and each bill has a policy location that must be estimated. Popular methods of roll call analysis compute standard errors that are admittedly invalid (Poole and Rosenthal 1997, 246) and one cannot appeal to standard statistical theory to ensure the consistency and other properties of estimators (we revisit this point below).In this paper we develop and illustrate Bayesian methods for ideal point estimation and the analysis of...