Skip to main content Accessibility help

Fast Estimation of Ideal Points with Massive Data



Estimation of ideological positions among voters, legislators, and other actors is central to many subfields of political science. Recent applications include large data sets of various types including roll calls, surveys, and textual and social media data. To overcome the resulting computational challenges, we propose fast estimation methods for ideal points with massive data. We derive the expectation-maximization (EM) algorithms to estimate the standard ideal point model with binary, ordinal, and continuous outcome variables. We then extend this methodology to dynamic and hierarchical ideal point models by developing variational EM algorithms for approximate inference. We demonstrate the computational efficiency and scalability of our methodology through a variety of real and simulated data. In cases where a standard Markov chain Monte Carlo algorithm would require several days to compute ideal points, the proposed algorithm can produce essentially identical estimates within minutes. Open-source software is available for implementing the proposed methods.


Corresponding author

Kosuke Imai is Professor, Department of Politics and Center for Statistics and Machine Learning, Princeton University, Princeton, NJ 08544. Phone: 609-258-6601 (, URL:
James Lo is Assistant Professor, Department of Political Science, University of Southern California, Los Angeles, CA 90089 (
Jonathan Olmsted is Solutions Manager, NPD Group, Port Washington, NY 11050 (


Hide All
Bafumi, Joseph, Gelman, Andrew, Park, David K., and Kaplan, Noah. 2005. “Practical Issues in Implementing and Understanding Bayesian Ideal Point Estimation.” Political Analysis 13: 171–87.
Bafumi, Joseph, and Herron, Michael. 2010. “Leapfrog Representation and Extremism: A Study of American Voters and Their Members in Congress.” American Political Science Review 104: 519–42.
Bailey, Michael. 2007. “Comparable Preferences across Time and Institutions for the Court, Congress, and Presidency.” American Journal of Political Science 51: 433–48.
Bailey, Michael A. 2013. “Is Today’s Court the Most Conservative in Sixty Years? Challenges and Opportunities in Measuring Judicial Preferences.” Journal of Politics 75: 821–34.
Bailey, Michael, and Chang, Kelly H.. 2001. “Comparing Presidents, Senators, and Justices: Interinstitutional Preference Estimation.” The Journal of Law, Economics, and Organization 17: 477506.
Bailey, Michael A., Kamoie, Brian, and Maltzman, Forrest. 2005. “Signals from the Tenth Justice: The Political Role of the Solicitor General in the Supreme Court Decision Making.” American Journal of Political Science 49: 7285.
Bailey, Michael A., Strezhnev, Anton, and Voeten, Erik. 2015. “Estimating Dynamic State Preferences from United Nations Voting Data.” Journal of Conflict Resolution.
Barberá, Pablo. 2015. “Birds of the Same Feather Tweet Together: Bayesian Ideal Point Estimation Using Twitter Data.” Political Analysis 23: 7691.
Battista, James Coleman, Peress, Michael, and Richman, Jesse. 2013. “Common-Space IdealPoints, Committee Assignments, and Financial Interests in the State Legislatures.” State Politics & Policy Quarterly 13: 7087.
Bock, R. Darrell, and Aitkin, Murray. 1981. “Marginal Maximum Likelihood Estimation of Item Parameters: Application of an EM Algorithm.” Psychometrika 46: 443–59.
Bond, Robert, and Messing, Solomon. 2015. “Quantifying Social Medias Political Space: Estimating Ideology from Publicly Revealed Preferences on Facebook.” American Political Science Review 109: 6278.
Bonica, Adam. 2013. “Ideology and Interests in the Political Marketplace.” American Journal of Political Science 57: 294311.
Bonica, Adam. 2014. “Mapping the Ideological Marketplace.” American Journal of Political Science 58: 367–87.
Carroll, Royce, Lewis, Jeffrey B., Lo, James, and Poole, Keith T.. 2009. “Measuring Bias and Uncertainty in DW-NOMINATE Ideal Point Estimates via the Parametric Bootstrap.” Political Analysis 17: 261–75.
Carroll, Royce, Lewis, Jeffrey B., Lo, James, Poole, Keith T., and Rosenthal, Howard. 2009. “Comparing NOMINATE and IDEAL: Points of difference and Monte Carlo tests.” Legislative Studies Quarterly 34: 555–91.
Carroll, Royce, Lewis, Jeffrey B., Lo, James, Poole, Keith T., and Rosenthal, Howard. 2013. “The Structure of Utility in Spatial Models of Voting.” American Journal of Political Science 57: 1008–28.
Clark, Tom S., and Lauderdale, Benjamin. 2010. “Locating Supreme Court Opinions in Doctrine Space.” American Journal of Political Science 54: 871–90.
Clinton, Joshua D., Bertelli, Anthony, Grose, Christian R., Lewis, David E., and Nixon, David C.. 2012. “Separated Powers in the United States: The Ideology of Agencies, Presidents, and Congress.” American Journal of Political Science 56: 341–54.
Clinton, Joshua, Jackman, Simon, and Rivers, Douglas. 2004. “The Statistical Analysis of Roll Call Data.” American Political Science Review 98: 355–70.
Clinton, Joshua D., and Lewis, David E.. 2008. “Expert Opinion, Agency Characteristics, and Agency Preferences.” Political Analysis 16: 320.
Clinton, Joshua D., and Meirowitz, Adam. 2003. “Integrating Voting Theory andRoll Call Analysis: A Framework.” Political Analysis 11: 381–96.
Dempster, Arthur P., Laird, Nan M., and Rubin, Donald B.. 1977. “Maximum Likelihood from Incomplete Data Via the EM Algorithm (with Discussion).” Journal of the Royal Statistical Society, Series B, Methodological 39: 137.
Gelman, Andrew. 2006. “Prior Distributions for Variance Parameters in Hierarchical Models.” Bayesian Analysis 1: 515–33.
Gerber, Elisabeth R., and Lewis, Jeffrey B.. 2004. “Beyond the Median: Voter Preferences, District Heterogeneity, and Political Representation.” Journal of Political Economy 112: 1364–83.
Gerrish, Sean M., and Blei, David M.. 2012. “How They Vote: Issue-Adjusted Models of Legislative Behavior.” Advances in Neural Information Processing Systems 25: 2762–70.
Grimmer, Justin. 2011. “An Introduction to Bayesian Inference via Variational Approximations.” Political Analysis 19: 3247.
Hirano, Shigeo, Imai, Kosuke, Shiraito, Yuki, and Taniguchi, Masaaki. 2011. “Policy Positions in Mixed Member Electoral Systems:Evidence from Japan.” Working Paper available at
Hix, Simon, Noury, Abdul, and Roland, Gérard. 2006. “Dimensions of Politics in the European Parliament.” American Journal of Political Science 50: 494511.
Ho, Daniel E., and Quinn, Kevin M.. 2010. “Did a Switch in Time Save Nine?Journal of Legal Analysis 2: 145.
Imai, Kosuke, Lo, James, and Olmsted, Jonathan. 2015. “emIRT: EM Algorithms for Estimating Item Response Theory Models.” available at the Comprehensive R Archive Network (CRAN).
Imai, Kosuke, Lo, James, and Olmsted, Jonathan. 2016. “Replication data for: Fast Estimation of Ideal Points with Massive Data.” doi:10.7910/DVN/HAU0EX. The Dataverse Network.
Jackman, Simon. 2001. “Multidimensional Analysis of Roll Call Data via Bayesian Simulation: Identification, Estimation, Inference, and Model Checking.” Political Analysis 9: 227–41.
Jackman, Simon. 2012. pscl: Classes and Methods for R Developed in the Political Science Computational Laboratory, Stanford University. Department of Political Science, Stanford University, Stanford, California: Stanford University. R package version 1.04.4.
Kim, In Song, Londregan, John, and Ratkovic, Marc. 2014. Voting, Speechmaking, and the Dimensions of Conflict in the US Senate. Technical Report. Department of Politics, Princeton University.
Lauderdale, Benjamin E., and Herzog, Alexander. 2014. Measuring Political Positions from Legislative. Technical Report. London School of Economics and Political Science.
Lewandowski, Jirka, Merz, Nicolas, Regel, Sven, and Lehmann, Pola. 2015. manifestoR: Access and Process Data and Documents of the Manifesto Project. R package version 1.1-1.
Lewis, Jeffrey B., and Poole, Keith T.. 2004. “Measuring Bias and Uncertainty in Ideal Point Estimates via the Parametric Boostrap.” Political Analysis 12 (2): 105–27.
Londregan, John B. 1999. “Estimating Legislators’ Preferred Points.” Political Analysis 8: 3556.
Londregan, John B. 2007. Legislative Institutions and Ideology in Chile. Cambridge, England: Cambridge University Press.
Lowe, Will, Benoit, Kenneth, Mikhaylov, Slava, and Laver, Michael. 2011. “Scaling Policy Preferences from Coded Political Texts.” Legislative Studies Quarterly 36: 123–55.
Martin, Andrew D., and Quinn, Kevin M.. 2002. “Dynamic Ideal Point Estimation via Markov chain Monte Carlo for the U.S. Supreme Court, 1953–1999.” Political Analysis 10: 134–53.
Martin, Andrew D., Quinn, Kevin M., and Park, Jong Hee. 2013. MCMCpack: Markov chain Monte Carlo MCMC Package.
McCarty, Nolan, Poole, Keith T., and Rosenthal, Howard. 2006. Polarized America: The Dance of Ideology and Unequal Riches. Cambridge, MA: MIT Press.
Morgenstern, Scott. 2004. Patterns of Legislative Politics: Roll-Call Voting in Latin America and the United States. Cambridge, England: Cambridge University Press.
Poole, Keith T. 2000. “Nonparametric Unfolding of Binary Choice Data.” Political Analysis 8: 211–37.
Poole, Keith, Lewis, Jeffrey, Lo, James, and Carroll, Royce. 2011. “Scaling Roll Call Votes with wnominate in R.” Journal of Statistical Software 42: 121.
Poole, Keith, Lewis, Jeffrey, Lo, James, and Carroll, Royce. 2012. oc: OC Roll Call Analysis Software. R package version 0.93.
Poole, Keith T., and Rosenthal, Howard. 1997. Congress: A Political Economic History of Roll Call Voting. New York: Oxford University Press.
Poole, Keither T., and Rosenthal, Howard. 1991. “Patterns of Congressional Voting.” American Journal of Political Science 35: 228–78.
Proksch, Sven-Oliver, and Slapin, Jonathan B.. 2010. “Position Taking in European Parliament Speeches.” British Journal of Political Science 40: 587611.
Quinn, Kevin M. 2004. “Bayesian Factor Analysis for Mixed Ordinal and Continuous Responses.” Political Analysis 12: 338–53.
Rosas, Guillermo, and Shomer, Yael. 2008. “Models of Nonresponse in Legislative Politics.” Legislative Studies Quarterly 33: 573601.
Shor, Boris, Berry, Christopher, and McCarty, Nolan. 2011. “A Bridge to Somewhere: Mapping State and Congressional Ideology on a Cross-institutional Common Space.” Legislative Studies Quarterly 35: 417–48.
Shor, Boris, and McCarty, Nolan. 2011. “The Ideological Mapping of American Legislatures.” American Political Science Review 105: 530–51.
Slapin, Jonathan B., and Proksch, Sven-Oliver. 2008. “A Scaling Model for Estimating Time-Series Party Positions from Texts.” American Journal of Political Science 52: 705–22.
Spirling, Arthur, and McLean, Iain. 2007. “UK OC OK? Interpreting Optimal Classification Scores for the U.K. House of Commons.” Political Analysis 15: 8596.
Tausanovitch, Chris, and Warshaw, Christopher. 2013. “Measuring Constituent Policy Preferences in Congress, State Legislatures, and Cities.” Journal of Politics 75: 330–42.
Voeten, Erik. 2000. “Clashes in the Assembly.” International Organization 54: 185215.
Wainwright, Martin J., and Jordan, Michael I.. 2008. “Graphical Models, Exponential Families, and Variational Inference.” Foundations and Trends in Machine Learning 1: 1310.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

American Political Science Review
  • ISSN: 0003-0554
  • EISSN: 1537-5943
  • URL: /core/journals/american-political-science-review
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
Type Description Title
Supplementary materials

Imai supplementary material
Imai supplementary material 1

 PDF (308 KB)
308 KB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed