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Empirical Bayes methods can be seen as an approximation to a fully Bayesian treatment of a hierarchical Bayes model.. In, for example, a two-stage hierarchical Bayes model, observed data = {,, …,} are assumed to be generated from an unobserved set of parameters = {,, …,} according to a probability distribution ().
The term relates to the notion that the improved estimate is made closer to the value supplied by the 'other information' than the raw estimate. In this sense, shrinkage is used to regularize ill-posed inference problems. Shrinkage is implicit in Bayesian inference and penalized likelihood inference, and explicit in James–Stein-type
In general, the risk () cannot be computed because the distribution (,) is unknown to the learning algorithm. However, given a sample of iid training data points, we can compute an estimate, called the empirical risk, by computing the average of the loss function over the training set; more formally, computing the expectation with respect to the empirical measure:
A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator. Empirical Bayes methods enable the use of auxiliary empirical data, from observations of related parameters, in the development of a Bayes estimator. This is done under the assumption that the estimated parameters are obtained from a common prior.
VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability , which is an alternative approach for characterizing generalization.
Mathematical models of social learning aim to model opinion dynamics in social networks.Consider a social network in which people (agents) hold a belief or opinion about the state of something in the world, such as the quality of a particular product, the effectiveness of a public policy, or the reliability of a news agency.
Bayesian learning mechanisms are probabilistic causal models [1] used in computer science to research the fundamental underpinnings of machine learning, and in cognitive neuroscience, to model conceptual development. [2] [3]
Best linear unbiased predictions are similar to empirical Bayes estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates.