Weighted Precision Metric

• Weighted precision metric is a performance measure that reshapes Fβ into a stochastic variable, adaptively balancing precision and recall based on batch-level statistics.
• The method employs distributional assumptions (Uniform/Inverse-Uniform and Gaussian/Inverse-Exponential) to derive closed-form CDFs and determine an optimal β via knee-curve detection.
• Integration into weighted binary cross-entropy loss enables dynamic adjustment of penalty terms, yielding significant F1 score improvements in imbalanced datasets like CIFAR-10 and IMDB.

A weighted precision metric is a class of performance measures and loss-shaping strategies that integrate traditional evaluation metrics—particularly van Rijsbergen’s $F_\beta$—directly into model training using data-driven, dynamically computed weights. Recent work by Ramdhani (2022) provides a formalism to convert $F_\beta$ into a stochastic variable suitable for tight integration with a weighted binary cross-entropy (WBCE) objective, dynamically emphasizing precision or recall in response to batch-level statistics and their statistical distributions (Ramdhani, 2022).

1. $F_\beta$ and Its Reformulation

$F_\beta$ is a parametric metric combining precision $P$ and recall $R$: $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$

where

$$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$

Here, $\mathit{TP}$, $\mathit{FP}$, and $F_\beta$0 denote true positives, false positives, and false negatives, respectively. For $F_\beta$1, $F_\beta$2 is the harmonic mean of $F_\beta$3 and $F_\beta$4. $F_\beta$5 encodes the relative importance of recall ($F_\beta$6) versus precision ($F_\beta$7).

Ramdhani (2022) decomposes $F_\beta$8 into two independent random variables via: $F_\beta$9 where $F_\beta$0 with $F_\beta$1, and $F_\beta$2, allowing statistical treatment and distributional sampling of $F_\beta$3.

2. Distributional Assumptions and CDF Derivation

To enable statistical reasoning over $F_\beta$4, two main distributional case studies are formulated:

• Case 1: Uniform/Inverse-Uniform (U/IU)
  • $F_\beta$5 induce $F_\beta$6, $F_\beta$7. This supports derivation of a closed-form, piecewise CDF $F_\beta$8 quantifying $F_\beta$9 as a function of $F_\beta$0, $F_\beta$1, $F_\beta$2, and $F_\beta$3.
• Case 2: Gaussian/Inverse-Exponential (G/IE)
  • $F_\beta$4, and $F_\beta$5 where the PDF of $F_\beta$6 is obtained by inverting and shifting an exponential. The resulting CDF $F_\beta$7 exploits the standard Gaussian CDF $F_\beta$8 and an exponential term.

Both constructions enable one to model batch-level $F_\beta$9 statistics under specific assumptions, producing interpretable CDF surfaces as a function of $P$0.

3. Identification of $P$1 via Knee-Curve Detection

The core methodology applies these CDFs to algorithmically select an "optimal" $P$2 per batch:

1. For each batch, compute observed $P$3 and $P$4.
2. For a grid of candidate $P$5 values, compute $P$6 and corresponding CDF values $P$7.
3. Construct the knee-curve $P$8 and normalize to $P$9.
4. The difference signal $R$0 is analyzed for local maxima; $R$1 is set as the mean of local-maxima $R$2, or defaults to $R$3 in symmetric cases.

This knee detection locates the $R$4 value at which further increases yield diminishing returns to the precision/recall trade-off, operationalizing "turning points" on the CDF surface.

4. Integration into Weighted Binary Cross-Entropy Loss

Once $R$5 is determined, it serves as a dynamic penalty parameter in the batch's loss function: $R$6 where

$R$7

This reweighting penalizes or incentivizes certain types of errors depending on the current batch’s precision–recall profile: negative examples (majority class) receive increased penalty or are rewarded according to current mispredictions and $R$8, while positive class predictions remain unweighted.

5. Precision–Recall Control via $R$9 Dynamics

Classically, increasing $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$0 in $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$1 accentuates recall; $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$2 stresses precision. Ramdhani’s WBCE framework preserves this semantics: a high $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$3 increases penalties for false positives, thus incentivizing higher precision; low $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$4 reduces those penalties, tolerating more false positives and seeking greater recall. The per-batch computed $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$5 allows real-time shift of the model’s operational focus along the precision–recall spectrum, matching data distribution or downstream task desiderata.

6. Empirical Performance and Practical Guidance

Empirical evaluation demonstrates:

• For CIFAR-10 (10% positive imbalance), WBCE with fixed $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$6 (U/IU assumption) increases $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$7 from $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$8 to $F_\beta = \frac{(1+\beta^2) P R}{\beta^2 P + R}$9.
• On IMDB sentiment data (7.4% positive), the Gaussian/IE approach achieves a $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$0 $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$1 uplift (from $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$2 to $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$3), attributed to both label noise mitigation and challenging feature space.
• UCI tabular and simulation datasets realize $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$4–$$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$5 $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$6 improvements in "easier," more separable regimes, with $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$7 tracking domain-informed precision–recall trade-offs.

A pragmatic protocol: select $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$8 (e.g., $$P = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FP}}, \quad R = \frac{\mathit{TP}}{\mathit{TP} + \mathit{FN}}$$9 or $\mathit{TP}$0) and grid size $\mathit{TP}$1 (e.g., $\mathit{TP}$2); for each batch, compute $\mathit{TP}$3, scan $\mathit{TP}$4, construct $\mathit{TP}$5, run knee detection to extract $\mathit{TP}$6, and insert into the loss. For quick deployment, U/IU with $\mathit{TP}$7 is robust; G/IE with $\mathit{TP}$8 allows calibration of recall–precision emphasis.

In summary, the weighted precision metric—here instantiated as a data-driven, dynamically weighted $\mathit{TP}$9–BCE hybrid—provides a methodology for transitioning $\mathit{FP}$0 from a post-hoc evaluator to an actively loss-shaping oracle during training, adaptively steering optimization toward evolving class trade-offs (Ramdhani, 2022).