Bayesian Parameter Estimation Algorithm
A Bayesian Parameter Estimation Algorithm is a parameter estimation algorithm that is a Bayesian algorithm.
- Context:
- It can be implements by a Bayesian Parameter Estimation System (to solve a Bayesian parameter estimation task).
- Example(s):
- MCMC.
- …
- Counter-Example(s):
- See: Probability Distribution Parameter Estimation Algorithm.
References
2014
- http://en.wikipedia.org/wiki/Bayesian_inference#Estimates_of_parameters_and_predictions
- It is often desired to use a posterior distribution to estimate a parameter or variable. Several methods of Bayesian estimation select measurements of central tendency from the posterior distribution. For one-dimensional problems, a unique median exists for practical continuous problems. The posterior median is attractive as a robust estimator.[1] If there exists a finite mean for the posterior distribution, then the posterior mean is a method of estimation.[citation needed] :[math]\displaystyle{ \tilde \theta = \operatorname{E}[\theta] = \int_\theta \theta \, p(\theta \mid \mathbf{X},\alpha) \, d\theta }[/math] Taking a value with the greatest probability defines maximum a posteriori (MAP) estimates:[citation needed] :[math]\displaystyle{ \{ \theta_{\text{MAP}}\} \subset \arg \max_\theta p(\theta \mid \mathbf{X},\alpha) . }[/math]
There are examples where no maximum is attained, in which case the set of MAP estimates is empty.
There are other methods of estimation that minimize the posterior risk (expected-posterior loss) with respect to a loss function, and these are of interest to statistical decision theory using the sampling distribution ("frequentist statistics").[citation needed]
The posterior predictive distribution of a new observation [math]\displaystyle{ \tilde{x} }[/math] (that is independent of previous observations) is determined by[citation needed] :[math]\displaystyle{ p(\tilde{x}|\mathbf{X},\alpha) = \int_\theta p(\tilde{x},\theta \mid \mathbf{X},\alpha) \, d\theta = \int_\theta p(\tilde{x} \mid \theta) p(\theta \mid \mathbf{X},\alpha) \, d\theta . }[/math]
- It is often desired to use a posterior distribution to estimate a parameter or variable. Several methods of Bayesian estimation select measurements of central tendency from the posterior distribution. For one-dimensional problems, a unique median exists for practical continuous problems. The posterior median is attractive as a robust estimator.[1] If there exists a finite mean for the posterior distribution, then the posterior mean is a method of estimation.[citation needed] :[math]\displaystyle{ \tilde \theta = \operatorname{E}[\theta] = \int_\theta \theta \, p(\theta \mid \mathbf{X},\alpha) \, d\theta }[/math] Taking a value with the greatest probability defines maximum a posteriori (MAP) estimates:[citation needed] :[math]\displaystyle{ \{ \theta_{\text{MAP}}\} \subset \arg \max_\theta p(\theta \mid \mathbf{X},\alpha) . }[/math]
- ↑ Sen, Pranab K.; Keating, J. P.; Mason, R. L. (1993). Pitman's measure of closeness: A comparison of statistical estimators. Philadelphia: SIAM.
2012
- (Levy, 2012) ⇒ Roger Levy. (2012). “Probabilistic Models in the Study of Language - Chapter 4: Parameter Estimation."
- QUOTE: … In this chapter we delve more deeply into the theory of probability density estimation, focusing on inference within parametric families of probability distributions (see discussion in Section 2.11.2). We start with some important properties of estimators, then turn to basic frequentist parameter estimation (maximum-likelihood estimation and corrections for bias), and finally basic Bayesian parameter estimation.