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The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior density over the quantity one wants to estimate.
where ^ is the location of a mode of the joint target density, also known as the maximum a posteriori or MAP point and is the positive definite matrix of second derivatives of the negative log joint target density at the mode = ^. Thus, the Gaussian approximation matches the value and the log-curvature of the un-normalised target density at the ...
The EM method was modified to compute maximum a posteriori (MAP) estimates for Bayesian inference in the original paper by Dempster, Laird, and Rubin. Other methods exist to find maximum likelihood estimates, such as gradient descent, conjugate gradient, or variants of the Gauss–Newton algorithm. Unlike EM, such methods typically require the ...
From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPDI). [4] But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated. [5]
In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).
The maximum a posteriori, which is the mode of the posterior and is often computed in Bayesian statistics using mathematical optimization methods, remains the same. The posterior can be approximated even without computing the exact value of P ( B ) {\displaystyle P(B)} with methods such as Markov chain Monte Carlo or variational Bayesian methods .
The approximation was then improved by applying smoothing techniques to the outcomes of the simulations. While the idea of using simulation for hypothesis testing was not new, [5] [6] Diggle and Gratton seemingly introduced the first procedure using simulation to do statistical inference under a circumstance where the likelihood is intractable.
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events.