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To be precise, I'm checking this presentation https://kaybrodersen.github.io/talks/Brodersen_2013_03_22.pdf, but I don't understand what is the connection between Laplace method and variational bayes? I mean, I know perfectly what Laplace approximation does and its limitations, but I don't follow how it is connected to variational bayes. Maybe the last one it's like an extension of laplace approximation?

Thanks in advance for your help.

Renzo Mauricio Guzmán Anaya
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    The only connection is that they're both methods of approximating a density. The slides say that variational Bayes generalizes the Laplace method but it doesn't really; it's just _more general_ than the Laplace method because the form of a Laplace approximation is always normal whereas you get to choose a parametric family of densities in variational Bayes. – Cyan Oct 06 '17 at 00:58

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As already stated in the comment section, both the Laplace Method and a certain class of Variational Inference Methods (convex-type representations) are based on locally approximating a (non-Gaussian) density.

Chris Bishop's book 'Pattern recognition and machine learning' has a chapter on this (Chapter 10.5. Local Variational Methods).

I hope that helps.

Effesian
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