Assumption 1 (Existence of probability distribution)

In Problem 3, there exists a prob- ability distribution \(p(y|D)\) on the predictive space \(Y\) .

Definition 1 (Bayesian ML estimator)

For Problem 3 with Assumption 1, the estimator
\[
\newcommand{\argmax}{\mathop{\rm argmax}\limits}
\hat{y}^{(ML)} =\argmax_{y \in Y} p(y|D),
\]
which maximizes the Bayesian posterior probability \(p(y|D)\), is referred to as a Bayesian maximum likelihood (ML) estimator.

Example 1 (Pairwise alignment with maximum score)

In Problem 1 with a scoring model (e.g., gap costs and a substitution matrix), the distribution \(p(y|D)\) in Assumption 1 is derived from the Miyazawa model, and the Bayesian ML estimator is equivalent to the alignment that has the highest similarity score.

Example 2 (RNA structure with minimum free energy)

In Problem 2 with a McCaskill energy model, the distribution \(p(y|D)\) in Assumption 1 can be obtained as the Boltzmann distribution, and the Bayesian ML estimator is equivalent to the secondary structure that has the minimum free energy (MFE).