The principles underlying IB-MHT can be effectively adapted to optimize other information-theoretic metrics, such as Rényi divergences or f-divergences, by following a similar framework of hypothesis testing and Pareto optimization. Here’s how this can be achieved:
Metric Definition: First, define the target information-theoretic metric, such as Rényi divergence ( D_\\alpha(P || Q) ) for a given order ( \\alpha ), or f-divergences, which generalize various divergence measures. The optimization problem can then be framed to minimize the chosen divergence while satisfying certain constraints.
Estimation Techniques: Just as with mutual information, the estimation of Rényi divergences or f-divergences can be performed using empirical distributions derived from the data. Techniques such as kernel density estimation or plug-in estimators can be employed to compute these divergences from the observed data.
Hypothesis Testing Framework: Similar to the MHT approach in IB-MHT, a set of null hypotheses can be formulated for the chosen divergence metrics. For instance, one could test whether the divergence exceeds a specified threshold, analogous to testing mutual information constraints.
Pareto Front Estimation: The Pareto front can be constructed in the same manner as in IB-MHT, where the trade-offs between the divergence metric and other relevant quantities (e.g., model complexity or another divergence measure) are evaluated. This allows for the identification of hyperparameters that balance the trade-offs effectively.
Sequential Testing and FWER Control: The sequential testing approach can be adapted to control the family-wise error rate for the new divergence metrics, ensuring that the selected hyperparameters meet the desired statistical guarantees.
By leveraging these strategies, the IB-MHT framework can be generalized to optimize a wide range of information-theoretic metrics, enhancing its applicability in various domains such as machine learning, statistics, and information theory.