TP, TN, FP, FN Explained

              Actual Negative          Actual Positive
Predicted  ┌─────────────────────┬─────────────────────┐
Negative   │ TN (True Negative)  │ FN (False Negative) │
           │ Correctly cleared   │ Missed positive      │
Predicted  ├─────────────────────┼─────────────────────┤
Positive   │ FP (False Positive) │ TP (True Positive)  │
           │ False alarm         │ Correctly flagged   │
           └─────────────────────┴─────────────────────┘

# Patient HAS the condition AND the model flags them as positive
# The model is right — in the correct direction

# Example: Warfarin bleeding risk
# Reality:    patient has high bleeding risk (INR too high, renal impairment)
# Prediction: model flags as "high risk"
# Outcome:    dose reduction ordered — patient protected

# Good for:
# - Clinical alarm systems (alarm fired and was correct)
# - Drug screening (test detected a real interaction)
# - LLM safety (harmful content correctly blocked)

# Patient does NOT have the condition AND the model correctly says "negative"
# The model is right — in the safe direction

# Example:
# Reality:    patient has low bleeding risk
# Prediction: model says "low risk"
# Outcome:    dose maintained — no unnecessary intervention

# The "silent success" — models need lots of TNs to be useful in practice
# TN-heavy results are common in imbalanced datasets

# Patient does NOT have the condition BUT the model flags them as positive
# False alarm — the model raised an alert that wasn't needed

# Example:
# Reality:    patient has low bleeding risk
# Prediction: model says "high risk"
# Outcome:    unnecessary dose reduction — patient under-anticoagulated
#             risk of thromboembolic event (clot)

# FP is costly when:
#   - Treatment has significant side effects (under-dosing warfarin → clot risk)
#   - Alert fatigue is a risk (physician stops trusting the system)
#   - Downstream cost per false alarm is high (unnecessary biopsy, surgery)

# FP rate = FP / (FP + TN)  = 1 - Specificity

# Patient HAS the condition BUT the model says "negative"
# Missed case — the most dangerous error in clinical contexts

# Example:
# Reality:    patient has high bleeding risk
# Prediction: model says "low risk"
# Outcome:    dose not reduced — patient bleeds

# FN is costly when:
#   - Missing the positive case is dangerous (sepsis, PE, drug toxicity)
#   - No manual backup exists
#   - The whole point of the model is to catch things humans would miss

# FN rate = FN / (FN + TP) = 1 - Recall (also called miss rate)

from sklearn.metrics import confusion_matrix
import numpy as np

# Anticoagulant monitoring model — 200 patients, 30 high-risk
y_true = np.array([0]*170 + [1]*30)
y_pred = np.array([0]*158 + [1]*12 + [0]*7 + [1]*23)
#                  TN=158   FP=12   FN=7   TP=23

cm = confusion_matrix(y_true, y_pred)
tn, fp, fn, tp = cm.ravel()

print(f"TP = {tp:3d}  — Correctly flagged high-risk patients")
print(f"TN = {tn:3d}  — Correctly cleared low-risk patients")
print(f"FP = {fp:3d}  — Healthy patients incorrectly flagged (false alarm)")
print(f"FN = {fn:3d}  — High-risk patients missed (most dangerous)")

total = tn + fp + fn + tp

# From TP and FP: how trustworthy are the positive alerts?
precision = tp / (tp + fp)
print(f"Precision = {tp}/{tp+fp} = {precision:.3f}  — when we flag, we're right {precision:.0%}")

# From TP and FN: how many real positives did we catch?
recall    = tp / (tp + fn)
print(f"Recall    = {tp}/{tp+fn} = {recall:.3f}  — caught {recall:.0%} of high-risk patients")

# From TN and FP: how often do we clear safe patients correctly?
specificity = tn / (tn + fp)
print(f"Specificity = {tn}/{tn+fp} = {specificity:.3f}  — {specificity:.0%} of safe patients cleared correctly")

# From TN and FN: if we clear someone, how often are we right?
npv = tn / (tn + fn)
print(f"NPV = {tn}/{tn+fn} = {npv:.3f}  — cleared patients are truly safe {npv:.0%} of the time")

# Overall accuracy
accuracy = (tp + tn) / total
print(f"Accuracy  = {tp+tn}/{total} = {accuracy:.3f}  — correct {accuracy:.0%} overall")

import numpy as np
from sklearn.metrics import confusion_matrix

y_proba = model.predict_proba(X_test)[:, 1]

print(f"{'Thresh':>7}  {'TP':>5}  {'TN':>5}  {'FP':>5}  {'FN':>5}  {'Precision':>10}  {'Recall':>8}")
print("-" * 65)

for threshold in np.arange(0.2, 0.85, 0.1):
    y_pred_t = (y_proba >= threshold).astype(int)
    tn, fp, fn, tp = confusion_matrix(y_test, y_pred_t).ravel()
    prec = tp/(tp+fp) if (tp+fp) > 0 else 0
    rec  = tp/(tp+fn) if (tp+fn) > 0 else 0
    print(f"{threshold:>7.1f}  {tp:>5}  {tn:>5}  {fp:>5}  {fn:>5}  {prec:>10.3f}  {rec:>8.3f}")

# Lowering threshold:
# TP increases (more positives caught) → recall increases
# FP increases (more false alarms) → precision decreases
# FN decreases (fewer missed) → miss rate decreases
# TN decreases (more false alarms from true negatives)

# When errors have different costs, define a cost matrix
# Cost matrix: cost[i][j] = cost of predicting class j when actual is class i

cost_matrix = {
    "TP": 0,    # Correct flag — intervention ordered
    "TN": 0,    # Correct clear — no unnecessary action
    "FP": 50,   # False alarm — unnecessary dose reduction, risk of clot
    "FN": 500,  # Missed case — patient bleeds, potentially serious
}

tn, fp, fn, tp = confusion_matrix(y_true, y_pred).ravel()

total_cost = (
    tp * cost_matrix["TP"] +
    tn * cost_matrix["TN"] +
    fp * cost_matrix["FP"] +
    fn * cost_matrix["FN"]
)

print(f"Total cost: ${total_cost}")
print(f"FN contributes: ${fn * cost_matrix['FN']} ({fn * cost_matrix['FN'] / total_cost:.0%})")
print(f"FP contributes: ${fp * cost_matrix['FP']} ({fp * cost_matrix['FP'] / total_cost:.0%})")
# Missed cases are expensive — this argues for a lower threshold to catch more