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Variational Bayesian Super-Resolution Reconstruction
Published in Peyman Milanfar, Super-Resolution Imaging, 2017
S. Derin Babacan, Rafael Molina, Aggelos K. Katsaggelos
It is clear that using a degenerate distribution for x in Algorithm 2 removes the uncertainty terms in the image and hyperparameters estimates. However, the incorporation of these uncertainties through the covariance of x improves the restoration performance, especially in cases when the observation noise is high. This is mainly due to the fact that poor estimations of one variable (due to noise or outliers) can influence the estimation of other unknowns, and as a result the overall performance can significantly be affected. By estimating the full posterior distribution of the unknowns instead of point estimates corresponding to the maximum probability (such as MAP estimates), the uncertainty of the estimates is incorporated into the estimation procedure to ameliorate the propagation of estimation errors among unknowns.
Modeling Images with Undirected Graphical Models
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
The term MAP inference is an acronym for Maximum A Posteriori inference and has its roots in probabilistic estimation theory. For the purposes of the discussion here, the MAP inference strategy begins with the definition of a probability distribution p(x|y)1, where x is the vector of random variables being estimated from the observations y. In MAP inference, the actual estimate x* is found by finding the vector x* that maximizes p(x|y). This is sometimes referred to as a point estimate because the estimate is limited to a single point, as opposed to actually calculating the posterior distribution p(x|y).
Optimal Signal Processing
Published in Peter M. Clarkson, Optimal and Adaptive Signal Processing, 2017
Contrasting ML and MAP estimation, we see that the MAP estimate is derived using a combination of prior and posterior knowledge. In MAP estimation we assume something is known about θ and we formulate this knowledge into the prior density f(θ). ML estimation is potentially more useful than MAP estimation in practical problems because no prior knowledge of the parameter is required. It must be remembered, however, that both procedures require knowledge of the joint posterior density of the observations.
Improved calibration of building models using approximate Bayesian calibration and neural networks
Published in Journal of Building Performance Simulation, 2023
For illustrative purposes, three separate examples were selected to represent what results may be expected following a standard calibration procedure. Examples 1 through 3 were selected by iterating the design variables until the NMBE and cvRMSE of all three energy meters were within acceptable limits per ASHRAE Guideline 14. Also included for comparison are the baseline pre-calibration case and maximum a posteriori (MAP) estimate. The MAP estimate is the value most likely value for each parameter following Bayesian inference (inference results are described in further detail in Section 3.3.2). The error metrics are shown in Table 4 for each case, metric, and energy meter. Note that the MAP does not perform significantly better than the other three ‘calibrated’ examples in terms of calibration metrics.
A correlated likelihood function for Bayesian model calibration to harness gradient information
Published in Mechanics of Advanced Materials and Structures, 2022
Krishna Kenja, Arun Srinivasa, Kumar Vemaganti
It should be noted that Eq. (10) does not account for the correlations between the data points when not conditioned on the parameters. This equation will henceforth be referred to as the uncorrelated log-likelihood. The value of that maximizes the likelihood function is called the maximum likelihood estimate (MLE). The value of that maximizes the posterior is called the maximum a posteriori (MAP) estimate. In the case of a uniform prior, the MLE and MAP estimate coincide. To fully realize the power of the Bayesian approach, of course, one must consider not just the MLE or MAP estimate but the entire posterior distribution of the model parameters [13].
Inverse discrete choice modelling: theoretical and practical considerations for imputing respondent attributes from the patterns of observed choices
Published in Transportation Planning and Technology, 2018
Yuanying Zhao, Jacek Pawlak, John W. Polak
On the other hand, the MAP estimator is an estimate of an unknown quantity given both the actual observations and any prior knowledge the researcher may have about the estimated quantities. MAP assumes that a prior distribution over parameters is known (Sorenson 1980). Hence Bayes’ theorem gives the MAP estimate of model parameters: