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The separation of performance metrics from gradient based loss functions maynot always give optimal results and may miss vital aggregate information. Thispaper investigates incorporating a performance metric alongside differentiableloss functions to inform training outcomes. The goal is to guide modelperformance and interpretation by assuming statistical distributions on thisperformance metric for dynamic weighting. The focus is on van Rijsbergens$F_{\beta}$ metric -- a popular choice for gauging classification performance.Through distributional assumptions on the $F_{\beta}$, an intermediary link canbe established to the standard binary cross-entropy via dynamic penaltyweights. First, the $F_{\beta}$ metric is reformulated to facilitate assumingstatistical distributions with accompanying proofs for the cumulative densityfunction. These probabilities are used within a knee curve algorithm to find anoptimal $\beta$ or $\beta_{opt}$. This $\beta_{opt}$ is used as a weight orpenalty in the proposed weighted binary cross-entropy. Experimentation onpublicly available data along with benchmark analysis mostly yields better andinterpretable results as compared to the baseline for both imbalanced andbalanced classes. For example, for the IMDB text data with known labelingerrors, a 14% boost in $F_1$ score is shown. The results also revealcommonalities between the penalty model families derived in this paper and thesuitability of recall-centric or precision-centric parameters used in theoptimization. The flexibility of this methodology can enhance interpretation.

Reformulating van Rijsbergen's $F_{\beta}$ metric for weighted binary cross-entropy