Point Estimation - Bayesian Point-estimation

Bayesian Point-estimation

Bayesian inference is based on the posterior distribution. Many Bayesian point-estimators are the posterior distribution's statistics of central tendency, e.g., its mean, median, or mode:

• Posterior mean, which minimizes the (posterior) risk (expected loss) for a squared-error loss function; in Bayesian estimation, the risk is defined in terms of the posterior distribution.
• Posterior median, which minimizes the posterior risk for the absolute-value loss function.
• maximum a posteriori (MAP), which finds a maximum of the posterior distribution; for a uniform prior probability, the MAP estimator coincides with the maximum-likelihood estimator;

The MAP estimator has good asymptotic properties, even for many difficult problems, on which the maximum-likelihood estimator has difficulties. For regular problems, where the maximum-likelihood estimator is consistent, the maximum-likelihood estimator ultimately agrees with the MAP estimator. Bayesian estimators are admissible, by Wald's theorem.

Special cases of Bayesian estimators are important:

• Kalman filter
• Wiener filter

Several methods of computational statistics have close connections with Bayesian analysis:

• particle filter
• Markov chain Monte Carlo (MCMC)