# Bayesian Analysis Essay

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Bayesian analysis generally refers to statistical analysis that relies on Bayes’ theorem. Bayes’ theorem provides a solution to the general problem of induction, explaining how to rationally update prior beliefs in light of evidence so as to yield posterior beliefs. In statistical inference, if θ is a parameter of interest, then Bayes’ theorem implies that if p(θ) is the researcher’s a priori probability density over θ (or a priori mass function, in the case of a discrete parameter), and p(y|θ) is the likelihood function for some data y, then the posterior probability density p(θ|y) is proportional to the likelihood multiplied by the prior density. Bayesian statistical inference consists of computing, summarizing, and communicating features of the posterior density. Once considered controversial because of the subjectivism implicit in specifying a prior density, Bayesian analysis has become popular in recent years via the widespread availability of relatively powerful computers and algorithms (e.g., Markov chain Monte Carlo algorithms) that generate arbitrarily precise characterizations of posterior densities, making Bayesian analysis feasible for statistical models with large numbers of parameters or nonstandard probability distributions. Examples in political science include the analysis of roll call data, hierarchical models, and multinomial choice models.

Bibliography:

1. Gelman, Andrew, John B. Carlin, Hal S. Stern, and Donald B. Rubin. Bayesian Data Analysis, 2nd ed. Boca Raton, Fla.: Chapman Hall/CRC, 2004.
2. Gill, Jeff. Bayesian Methods: A Social and Behavioral Sciences Approach. Boca Raton, FL: Chapman Hall/CRC, 2002.
3. Jackman, Simon. “Bayesian Analysis for Political Research.” Annual Review of Political Science 7 (2004): 483–505.