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I am highly confused on the terms mentioned above.

With posterior = (likelihood*prior) / marginal

MLE - You define a posterior, but during maximization or optimization, you maximize the likelihood only. MAP - This is when you maximize the posterior instead of Likelihood.. that will include the prior as well. Fully Bayesian - I don't know what is that. Is it to maximize the marginal likelihood?

Now, given all this. I have the following text:

This approach employs a fully Bayesian framework to achieve a model which is sparse in both sample and feature domains. We introduce a novel multi-step algorithm based on Variational Approximation to efficiently compute all model parameters in order to optimize the maximum a posteriori probability (MAP) measure.

Can somebody explain this? My second question is:

what is meant by giving a posterior distribution over both parameters and hyperparameters.

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