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Dynamic Tempered Transitions for Exploring Multimodal Posterior Distributions

Published online by Cambridge University Press:  04 January 2017

Jeff Gill
Affiliation:
Department of Political Science, University of California, Davis, One Shields Avenue, Davis, CA 95616. e-mail: [email protected]
George Casella
Affiliation:
Department of Statistics, University of Florida, Griffin-Floyd Hall, P.O. Box 118545, Gainesville, FL 32611. e-mail: [email protected]

Abstract

Multimodal, high-dimension posterior distributions are well known to cause mixing problems for standard Markov chain Monte Carlo (MCMC) procedures; unfortunately such functional forms readily occur in empirical political science. This is a particularly important problem in applied Bayesian work because inferences are made from finite intervals of the Markov chain path. To address this issue, we develop and apply a new MCMC algorithm based on tempered transitions of simulated annealing, adding a dynamic element that allows the chain to self-tune its annealing schedule in response to current posterior features. This important feature prevents the Markov chain from getting trapped in minor modal areas for long periods of time. The algorithm is applied to a probabilistic spatial model of voting in which the objective function of interest is the candidate's expected return. We first show that such models can lead to complex target forms and then demonstrate that the dynamic algorithm easily handles even large problems of this kind.

Type
Research Article
Copyright
Copyright © Society for Political Methodology 2004 

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References

Adams, James, and Samuel, Merrill III. 2001. “A Theory of Spatial Competition with Biased Voters: Party Policies Viewed Temporally and Comparatively.” British Journal of Political Science 31: 121158.Google Scholar
Aldrich, John H., Sullivan, John L., and Borgida, Eugene. 1989. “Foreign Affairs and Issue Voting: Do Presidential Candidates ‘Waltz Before A Blind Audience?’ American Political Science Review 83: 123141.Google Scholar
Altman, Micah, Gill, Jeff, and McDonald, Michael. 2004. Numerical Issues in Statistical Computing for the Social Scientist. New York: John Wiley & Sons.Google Scholar
Alvarez, R. Michael, Nagler, Jonathan, and Bowler, Shaun. 2000. “Issues, Economics, and the Dynamics of Multiparty Elections: The 1997 British General Election.” American Political Science Review 94: 131149.Google Scholar
Bartels, Larry. 1988. “Issue Voting under Uncertainty: An Empirical Test.” American Journal of Political Science 30: 709728.Google Scholar
Burden, Barry C. 1997. “Deterministic and Probabilistic Voting Models.” American Journal of Political Science 41: 11501169.Google Scholar
Calvert, Randall L. 1986. Models of Imperfect Information in Politics. London: Harwood.Google Scholar
Campbell, James E. 1983. “Ambiguity in the Issue Positions of Presidential Elections: A Causal Analysis.” American Journal of Political Science 27: 284293.Google Scholar
Carmines, Edward G., and Stimson, James A. 1980. “The Two Faces of Issue Voting.” American Political Science Review 74: 7891.Google Scholar
Celeux, G., Hurn, M., and Robert, C. P. 2000. “Computational and Inferential Difficulties with Mixture Posterior Distributions.” Journal of the American Statistical Association 95: 957970.Google Scholar
Cho, Sungdai, and Endersby, James W. 2003. “Issues, the Spatial Theory of Voting, and British General Elections: A Comparison of Proximity and Directional Models.” Public Choice 114: 275293.Google Scholar
Coughlin, Peter J. 1982. “Pareto Optimality of Policy Proposals with Probabilistic Voting.” Public Choice 39: 427433.Google Scholar
Coughlin, Peter J. 1992. Probabilistic Voting Theory. Cambridge: Cambridge University Press.Google Scholar
Coughlin, Peter J., and Nitzan, Samuel. 1981. “Election Outcomes with Probabilistic Voting and Nash Social Welfare Maxima.” Journal of Public Economics 15: 113122.Google Scholar
de Palma, A., Hong, G.-S., and Thisse, J.-F. 1990. “Equilibria in Multi-party Competition under Uncertainty.” Social Choice Welfare 7: 247259.Google Scholar
Enelow, James M., Endersby, James W., and Munger, Michael C. 1993. “A Revised Spatial Model of Elections: Theory and Evidence.” In Information, Participation, and Choice, ed. Grofman, Bernard. Ann Arbor: University of Michigan Press, pp. 125140.Google Scholar
Enelow, James M., and Hinich, Melvin J. 1982. “Nonspatial Candidate Characteristics and Electoral Competition.” Journal of Politics 44: 115130.Google Scholar
Enelow, James M., and Hinich, Melvin J. 1984. The Spatial Theory of Voting. Cambridge: Cambridge University Press.Google Scholar
Enelow, James M., Mendell, Nancy R., and Ramesh, Subha. 1988. “A Comparison of Two Distance Metrics through Regression Diagnostics of a Model of Relative Candidate Evaluation.” Journal of Politics 50: 10571071.Google Scholar
Erikson, Robert S., and Romero, David W. 1990. “Candidate Equilibrium and the Behavioral Model of the Vote.” American Political Science Review 84: 11031126.Google Scholar
Gelfand, A. E., and Smith, A. F. M. 1990. Sampling-Based Approaches to Calculating Marginal Densities. Journal of the American Statistical Association 85: 389409.Google Scholar
Geyer, C. 1991. “Markov Chain Monte Carlo Maximum Likelihood.” Computing Science and Statistics, Proceedings of the 23rd Symposium on the Interface, 156163.Google Scholar
Geyer, C., and Thompson, E. 1995. “Annealing Markov Chain Monte Carlo with Applications to Ancestral Inference.” Journal of the American Statistical Association 90: 909920.Google Scholar
Gill, Jeff, and King, Gary. Forthcoming. “Alternatives to Model Respecification in Nonlinear Estimation.” Sociological Methods and Research.Google Scholar
Goren, Paul. 1997. “Political Expertise and Issue Voting in Presidential Elections.” Political Research Quarterly 50: 387412.Google Scholar
Herzberg, Roberta Q., and Wilson, Rick K. 1988. “Results on Sophisticated Voting in an Experimental Setting.” Journal of Politics 50: 7186.Google Scholar
Hinich, Melvin. 1977. “Equilibrium in Spatial Voting: The Median Voter is an Artifact.” Journal of Economic Theory 16: 208219.Google Scholar
Hinich, Melvin, Ledyard, John, and Ordeshook, Peter. 1972. “Nonvoting and the Existence of Equilibrium under Majority Rule.” Journal of Economic Theory 14: 144153.Google Scholar
Hinich, Melvin, Ledyard, John, and Ordeshook, Peter. 1973. “A Theory of Electoral Equilibrium: A Spatial Analysis Based on the Theory of Games.” Journal of Politics 35: 154193.Google Scholar
Hinich, Melvin, and Munger, Michael C. 1994. Ideology and the Theory of Political Choice. Ann Arbor: University of Michigan Press.Google Scholar
Iverson, Torben. 1994. “Political Leadership and Representation in West European Democracies: A Test of Three Models of Voting. American Journal of Political Science 38: 4574.Google Scholar
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P. 1983. “Optimization by simulated annealing.” Science 220: 671680.Google Scholar
Kollman, Ken, Miller, John H., and Page, Scott E. 1992a. “Adaptive Parties in Spatial Elections.” American Political Science Review 86: 929937.Google Scholar
Kollman, Ken, Miller, John H., and Page, Scott E. 1992b. “Political Institutions and Sorting in a Tiebout Model.” American Economic Review 87: 977992.Google Scholar
Ledyard, J. O. 1984. “The Pure Theory of Large Two-Candidate Elections.” Public Choice 44: 741.Google Scholar
Lewis, Jeffrey, and King, Gary. 2000. “No Evidence on Directional vs. Proximity Voting.” Political Analysis 8: 2133.Google Scholar
Lin, Tse-Min, Enelow, James M., and Dorussen, Han. 1999. “Equilibrium in Multicandidate Probabilistic Voting.” Public Choice 98: 5982.Google Scholar
Liu, J. S., and Sabatti, C. 1999. “Simulatated Sintering: Markov Chain Monte Carlo with Spaces Varying Dimension.” In Bayesian Statistics, eds. Bernardo, J. M., Smith, A. F. M., Dawid, A. P., and Berger, J. O. Oxford: Oxford University Press, pp. 389414.Google Scholar
Luce, R. D., and Raiffa, H. 1957. Games and Decisions. New York: John Wiley & Sons.Google Scholar
Macdonald, Stuart Elaine, Rabinowitz, George, and Listhaug, Olga. 1998. “On Attempting to Rehabilitate the Proximity Model: Sometimes the Patient Just Can't be Helped.” Journal of Politics 60: 653690.Google Scholar
Marinari, E., and Parisi, G. 1992. “Simulated Tempering: A New Monte Carlo Scheme.” Europhysics Letters 19: 451458.Google Scholar
Martinelli, César. 2001. “Elections with Privately Informed Parties and Voters.” Public Choice 108: 147167.Google Scholar
McKelvey, Richard D., and Ordeshook, Peter C. 1985. “Sequential Elections with Limited Information.” American Journal of Political Science 29: 480512.Google Scholar
Merrill, Samuel III, and Adams, James. 2002. “Centrifugal Incentives in Multi-candidate Elections.” Journal of Theoretical Politics 14: 275300.Google Scholar
Meyn, S. P., and Tweedie, R. L. 1994. “State-Dependent Criteria for Convergence of Markov Chains.” Annals of Applied Probability 4: 149168.Google Scholar
Morton, Rebecca B. 1993. “Incomplete Information and Ideological Explanations of Platform Divergence.” American Political Science Review 382407.Google Scholar
Mueller, Dennis C. 1989. Public Choice II. Cambridge: Cambridge University Press.Google Scholar
Neal, R. 1996. “Sampling from Multimodal Distributions Using Tempered Transitions.” Statistics and Computing 4: 353366.Google Scholar
Nelder, J. A., and Wedderburn, R. W. M. 1972. “Generalized Linear Models.” Journal of the Royal Statistical Society, Series A 135: 370385.Google Scholar
Ordeshook, Peter C. 1986 Game Theory and Political Theory. Cambridge: Cambridge University Press.Google Scholar
Patty, John W. 2002. “Equivalence of Objectives in Two Candidate Elections.” Public Choice 112: 151166.Google Scholar
Rasmussen, Carl Edward. 2003. “Gaussian Processes to Speed up Hybrid Monte Carlo for Expensive Integrals.” In Bayesian Statistics 7, eds. Bernardo, J. M., Bayarri, M. J., Berger, J. O., and Dawid, A. P. Oxford: Oxford University Press.Google Scholar
Robert, C. P., and Casella, G. 1999. Monte Carlo Statistical Methods. New York: Springer-Verlag.Google Scholar
Shepsle, Kenneth A. 1972. “The Strategy of Ambiguity: Uncertainty and Electoral Competition.” American Political Science Review 66: 555568.Google Scholar
Tanner, M. A., and Wong, W. H. 1987. “The Calculation of Posterior Distributions by Data Augmentation.” Journal of the American Statistical Society 82: 528550.Google Scholar
Westholm, Anders. 1997. “Distance Versus Direction: The Illusory Defeat of the Proximity Theory of Electoral Choice.” American Political Science Review 91: 865885.Google Scholar
Yang, C. C. 1995. “Endogenous Tariff Formation under Representative Democracy: A Probabilistic Voting Model.” American Economic Review 85: 956963.Google Scholar