Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach. Part II: Theoretical Analysis

The present paper describes the empirical Bayesian approach applied in the estimation of several small rates. Modeling by empirical Bayesian approach the probabilities of several rare events, it is assumed that the frequencies of events follow to Poisson’s law with different parameters, which are correlated Gaussian random values. The unknown parameters are estimated by the maximum likelihood method computing the integrals appeared here by Hermite–Gauss quadratures. The equations derived that are satisfied by maximum likelihood estimates of model parameters.

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TREND-CYCLE DECOMPOSITIONS OF REAL GDP REVISITED: CLASSICAL AND BAYESIAN PERSPECTIVES ON AN UNSOLVED PUZZLE

Macroeconomic Dynamics ◽

10.1017/s1365100520000218 ◽

2020 ◽

pp. 1-25

Author(s):

Chang-Jin Kim ◽

Jaeho Kim

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Bayesian Approach ◽

Stationary Process ◽

Likelihood Estimation ◽

Empirical Results ◽

Real Gdp ◽

Pile Up ◽

Out Of Sample ◽

The Bayesian Approach

While Perron and Wada (2009) maximum likelihood estimation approach suggests that postwar US real GDP follows a trend stationary process (TSP), our Bayesian approach based on the same model and the same sample suggests that it follows a difference stationary process (DSP). We first show that the results based on the approach should be interpreted with caution, as they are relatively more subject to the ‘pile-up problem’ than those based on the Bayesian approach. We then directly estimate and compare the two competing TSP and DSP models of real GDP within the Bayesian framework. Our empirical results suggest that a DSP model is preferred to a TSP model both in terms of in-sample fits and out-of-sample forecasts.

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A comparison of salmon escapement estimates using a hierarchical Bayesian approach versus separate maximum likelihood estimation of each year's return

Canadian Journal of Fisheries and Aquatic Sciences ◽

10.1139/f01-100 ◽

2001 ◽

Vol 58 (8) ◽

pp. 1663-1671 ◽

Cited By ~ 11

Author(s):

Milo D Adkison ◽

Zhenming Su

Keyword(s):

Maximum Likelihood ◽

Maximum Likelihood Estimation ◽

Bayesian Approach ◽

Count Data ◽

Likelihood Estimation ◽

Data Sets ◽

Hierarchical Bayesian ◽

Hierarchical Approach ◽

Bayesian Hierarchical ◽

The Bayesian Approach

In this simulation study, we compared the performance of a hierarchical Bayesian approach for estimating salmon escapement from count data with that of separate maximum likelihood estimation of each year's escapement. We simulated several contrasting counting schedules resulting in data sets that differed in information content. In particular, we were interested in the ability of the Bayesian approach to estimate escapement and timing in years where few or no counts are made after the peak of escapement. We found that the Bayesian hierarchical approach was much better able to estimate escapement and escapement timing in these situations. Separate estimates for such years could be wildly inaccurate. However, even a single postpeak count could dramatically improve the estimability of escapement parameters.

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