optimal-classification-cutoffs 0.6.1

Utilities for computing optimal classification cutoffs for binary and multiclass classification

Optimal Classification Cut-Offs

Select optimal probability thresholds for binary and multiclass classification.
Maximize F1, precision, recall, accuracy, or custom cost-sensitive utilities with algorithms designed for piecewise‑constant classification metrics.

---

Why thresholds—and what are we optimizing?

Most probabilistic classifiers output scores or probabilities p = P(y=1|x) (binary) or a probability vector over classes (multiclass). Turning those into decisions requires thresholds:

• Binary: predict 1 if p > τ, else 0.
• Multiclass: predict class argmax_k p_k or use per‑class thresholds τ_k.

The default τ = 0.5 is rarely optimal for your objective (e.g., F1 under imbalance, cost asymmetry, etc.). Because metrics like F1/precision/recall/accuracy only change when thresholds cross unique probability values, they are piecewise‑constant. That structure lets us compute globally optimal thresholds quickly and exactly.

---

Methods at a glance (from basic → advanced)

Intuition: we want the cut(s) over sorted probabilities that maximize your objective.

• Unique scan (unique cuts) — baseline / safe: evaluate the metric at all unique predicted probabilities and pick the best. Competitive when n_unique is moderate.
  Method: "unique_scan".

• Sort & scan (exact, fast) — recommended for piecewise metrics: sort probabilities once and compute all candidate scores with vectorized cumulative counts. O(n log n), exact optimum for F1/precision/recall/accuracy.
  Method: "sort_scan".

• Expected Fβ (Dinkelbach; calibrated) — analytical, fastest when valid: solves a fractional program for expected Fβ under perfect calibration. Currently supports F1. Use when you trust calibration and want the expected‑metric optimum.
  Mode: "expected".

• Continuous optimizers — for non‑piecewise targets or micro‑averaged multiclass joint objectives: fallback to scipy.optimize or simple gradient heuristics. Not guaranteed optimal for stepwise metrics.
  Methods: "minimize", "gradient".

Multiclass strategies:

• One‑vs‑Rest (OvR) — optimize each class's threshold independently (macro/weighted/none averaging). Simple and effective; by default we predict the highest‑probability class above its threshold, falling back to argmax if none pass.
  Method: "auto", "unique_scan", "sort_scan", "minimize", "gradient".

• Coordinate Ascent (coupled, single‑label consistent) — optimizes F1 for the single‑label rule argmax_k (p_k − τ_k). Typically better for imbalanced problems; currently F1 only, comparison ">" only, and no sample weights.
  Method: "coord_ascent".

---

Practical validation: holdout & cross‑validation

Thresholds are hyperparameters. To estimate a threshold you can trust:

1. Split: Train your model; reserve validation data (or use cross‑validation) to choose τ.
2. (Optional) Calibrate probabilities (CalibratedClassifierCV) for better transportability.
3. Select thresholds on validation/CV using this library.
4. Freeze the threshold and evaluate on a held‑out test set.

This repository includes cross‑validation utilities to estimate thresholds and quantify uncertainty.

---

🚀 Quick start

Install

pip install optimal-classification-cutoffs

Optional dependencies for enhanced performance and testing:

# For performance optimization (recommended)
pip install optimal-classification-cutoffs[performance]

# For running examples
pip install optimal-classification-cutoffs[examples]  

# For development and testing
pip install optimal-classification-cutoffs[dev]

# All optional dependencies
pip install optimal-classification-cutoffs[all]

Binary

from optimal_cutoffs import get_optimal_threshold

y_true = [0, 1, 1, 0, 1]
y_prob = [0.2, 0.8, 0.7, 0.3, 0.9]

# Optimize F1 threshold
result = get_optimal_threshold(y_true, y_prob, metric="f1", method="auto")
print(result.threshold)          # e.g. 0.7...
y_pred = result.predict(y_prob)  # boolean labels

Multiclass (OvR thresholds)

import numpy as np
from optimal_cutoffs import get_optimal_threshold

y_true = [0, 1, 2, 0, 1]
y_prob = np.array([
    [0.7, 0.2, 0.1],
    [0.1, 0.8, 0.1],
    [0.1, 0.1, 0.8],
    [0.6, 0.3, 0.1],
    [0.2, 0.7, 0.1],
])

result = get_optimal_threshold(y_true, y_prob, metric="f1")  # auto-detects multiclass
print(result.thresholds)                   # per-class τ_k
y_pred = result.predict(y_prob)            # integer class labels

Cost-Sensitive Binary

from optimal_cutoffs import get_optimal_threshold, bayes_thresholds_from_costs

# Empirical (finite-sample) optimum from labeled data
result = get_optimal_threshold(
    y_true, y_prob,
    utility={"tp": 50.0, "tn": 0.0, "fp": -1.0, "fn": -10.0},  # benefits/costs
)
tau = result.threshold

# Closed-form Bayes threshold (calibrated probabilities)
result_bayes = bayes_thresholds_from_costs(
    fp_costs=[1.0], fn_costs=[10.0]  # costs per class
)
tau_bayes = result_bayes.thresholds[0]

API Decision Stack

1. Problem: binary or multiclass (auto‑detected).

2. Objective: metric ("f1", "precision", "recall", "accuracy") or utility/cost (binary‑only).

3. Estimation regime (choose one):     • Empirical (finite sample) — optimize on labeled data.     • Expected under calibration —       – Bayes (utility, closed‑form; binary‑only), or       – Dinkelbach (expected F1; no weights).

4. Method (empirical only): "auto", "sort_scan", "unique_scan", "minimize", "gradient"; multiclass adds "coord_ascent". For expected F1, use mode="expected".

5. Tolerance: control numerical precision for floating-point comparisons (default: 1e-10).

6. Validation: holdout or cross‑validation (cv_threshold_optimization, nested_cv_threshold_optimization).

Examples

• Empirical metric (binary):

get_optimal_threshold(y, p, metric="f1", method="auto")

• Empirical utility (binary):

get_optimal_threshold(y, p, utility={"fp":-1, "fn":-5}, method="sort_scan")

• Bayes utility (calibrated, binary):

bayes_thresholds_from_costs(fp_costs=[1], fn_costs=[5]) # or
get_optimal_threshold(None, p, utility={"fp":-1,"fn":-5}, mode="bayes")

• Expected F1 via Dinkelbach (calibrated, binary):

get_optimal_threshold(y, p, metric="f1", mode="expected")

• Custom tolerance for numerical precision:

get_optimal_threshold(y, p, metric="f1", tolerance=1e-6)