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 $\mathit{FN}$ denote true positives, false positives, and false negatives, respectively. For $\beta=1$, $F_1$ is the harmonic mean of $P$ and $R$. $F_\beta$ encodes the relative importance of recall ($\beta > 1$) versus precision ($\beta < 1$).

Ramdhani (2022) decomposes $F_\beta$ into two independent random variables via: $F_\beta = X_1 X_2$ where $X_1 = r'(1+\beta^2)$ with $r' = pr$, and $X_2 = (r+\beta^{\prime\prime})^{-1}$, allowing statistical treatment and distributional sampling of $F_\beta$.

2. Distributional Assumptions and CDF Derivation

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

• Case 1: Uniform/Inverse-Uniform (U/IU)
  • $\beta',\beta'' \sim U(0,\beta^*)$ induce $X_1\sim U(r',\,r'+\beta^*)$, $X_2\sim IU(\frac1{r+\beta^*},\,\frac1r)$. This supports derivation of a closed-form, piecewise CDF $F_{U/IU}(z)$ quantifying $\Pr(F_\beta \leq z)$ as a function of $p$, $r$, $\beta^*$, and $z$.
• Case 2: Gaussian/Inverse-Exponential (G/IE)
  • $X_1\sim \mathcal{N}(r',\sigma^2)$, and $X_2\sim \mathrm{Inverse-Exp}(\lambda;r)$ where the PDF of $X_2$ is obtained by inverting and shifting an exponential. The resulting CDF $F_{G/IE}(z)$ exploits the standard Gaussian CDF $\Phi$ and an exponential term.

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

3. Identification of $\beta_{\rm opt}$ via Knee-Curve Detection

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

1. For each batch, compute observed $p$ and $r$.
2. For a grid of candidate $\beta$ values, compute $z_i = F_{\beta_i}(p,r)$ and corresponding CDF values $s_i$.
3. Construct the knee-curve $\{(\beta_i, s_i)\}$ and normalize to $[0,1]^2$.
4. The difference signal $d_i = s_i - \beta_i$ is analyzed for local maxima; $\beta_{\rm opt}$ is set as the mean of local-maxima $\{\beta_{\rm lmx}\}$, or defaults to $1$ in symmetric cases.

This knee detection locates the $\beta$ 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 $\beta_{\rm opt}$ is determined, it serves as a dynamic penalty parameter in the batch's loss function: $$\mathcal{L}_{\rm WBCE}(y,\hat y;\beta_{\rm opt}) = -\sum_{i=1}^m \left[y_i\,\log\hat y_i + (1-y_i)\,\log(1-\hat y_i)\,A_i \right]$$ where

$$A_i = \begin{cases} 1, & y_i = 1, \ 1+\beta_{\rm opt}^2, & y_i=0\, \land\, (1-\hat y_i\leq0.5), \ \frac{1}{1+\beta_{\rm opt}^2}, & y_i=0\, \land\, (1-\hat y_i>0.5). \end{cases}$$

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 $\beta_{\rm opt}$, while positive class predictions remain unweighted.

5. Precision–Recall Control via $\beta$ Dynamics

Classically, increasing $\beta$ in $F_\beta$ accentuates recall; $\beta<1$ stresses precision. Ramdhani’s WBCE framework preserves this semantics: a high $\beta_{\rm opt}$ increases penalties for false positives, thus incentivizing higher precision; low $\beta_{\rm opt}$ reduces those penalties, tolerating more false positives and seeking greater recall. The per-batch computed $\beta_{\rm opt}$ 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 $\beta$ (U/IU assumption) increases $F_1$ from $0.816$ to $0.826$.
• On IMDB sentiment data (7.4% positive), the Gaussian/IE approach achieves a $14\%$ $F_1$ uplift (from $0.675$ to $0.767$), attributed to both label noise mitigation and challenging feature space.
• UCI tabular and simulation datasets realize $12$–$28\%$ $F_1$ improvements in "easier," more separable regimes, with $\beta_{\rm opt}$ tracking domain-informed precision–recall trade-offs.

A pragmatic protocol: select $\beta_{\max}$ (e.g., $8$ or $16$) and grid size $n$ (e.g., $300$); for each batch, compute $(p, r)$, scan $\beta$, construct $(z_i, s_i)$, run knee detection to extract $\beta_{\rm opt}$, and insert into the loss. For quick deployment, U/IU with $\beta^*=8$ is robust; G/IE with $(\lambda,\sigma^2\approx5)$ allows calibration of recall–precision emphasis.

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