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Regularization and Kernel Methods
Published in Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman, Data Science and Machine Learning, 2019
Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman
One approach to setting the hyperparameter θ is to determine its posterior p(θ|y) and obtain a point estimate, for instance via its maximum a posteriori estimate. However, this can be a computationally demanding exercise. What is frequently done in practice is to consider instead the marginal likelihood p(y | θ) and maximize this with respect to θ. This procedure is called empirical Bayes. empiricalBayes
Fitting Continuous Models
Published in Norman Matloff, Probability and Statistics for Data Science, 2019
By contrast, there is no controversy if the prior makes use of real data, termed empirical Bayes. Actually, many Bayesian analyses one sees in practice are of this kind, and again, thre is no controversy here. So, our use of the term here Bayesian refers only to subjective priors.
Empirical Bayes Transfer Learning for Uncertainty Characterization in Predicting Parkinson’s Disease Severity
Published in IISE Transactions on Healthcare Systems Engineering, 2018
The integration in Eq. (6) cannot be calculated analytically, and therefore we propose an empirical Bayes approach (type-II maximum likelihood) for hyper-parameters estimation. Empirical Bayes is an approximation procedure for a fully Bayesian model to estimate parameters at the highest level of the hierarchy around their most likely values, instead of being integrated out. In our setting, the first step is to seek optimal values for hyper-parameters and then plug them into to obtain an approximation for , i.e., where are obtained by a Maximum a Posterior (MAP) from , i.e.,
Bayesian Predictive Inference for Zero-Inflated Poisson (ZIP) Distribution with Applications
Published in American Journal of Mathematical and Management Sciences, 2018
Suntaree Unhapipat, Montip Tiensuwan, Nabendu Pal
Here, we use the natural individual parameter priors as follows. The joint prior for (π, λ), denoted by is, where C1 = 1/{B(a, b)Γ(α)βα}. The joint distribution is, Since the hyperparameters a, b, α, and β are unknown, the above marginal is also unknown. As a result, the posterior based on (10) is also unknown. Henceforth, the estimated posterior distribution is as follows. where , , , and are the estimates of the hyper parameters, which maximize the marginal , i.e., the denominator of (11). The empirical Bayes method chooses the hyperparameters, as dictated by the marginal distribution.