Bayesian Speech and Language Processing by Watanabe Shinji & Chien Jen-Tzung

Author:Watanabe, Shinji & Chien, Jen-Tzung
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2015-07-30T16:00:00+00:00

(6.8)

Here, does not depend on θ, and we can neglect it. Thus, is approximated by the following Gaussian distribution with mean vector and covariance matrix :

(6.9)

Note that since the dimensionality of the Hessian matrix H is the number of parameters J, and it is often large in our practical problems, we would have a numerical issue as to how to obtain . To summarize the Laplace approximation, if we have an arbitrary distribution with given continuous distribution , the Laplace approximation provides an analytical Gaussian distribution by using the mode and the Hessian matrix H. is usually obtained by some numerical computation or the MAP estimation.

Now, let us consider the relationship between MAP (Chapter 4) and Laplace approximations. As we discussed in Eq. (4.9), the MAP approximation of the posterior distribution is represented by a Dirac delta function as follows:

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