9 · Evaluation, Metrics & Diagnostics · Classification Metrics

Probability metrics

Log loss and Brier score evaluate probabilistic confidence, penalising confident wrong predictions. Calibration should be inspected when probabilities drive decisions. The important practical question is not only how the technique is defined, but what assumptions it introduces, which data are allowed to influence it, and how its effect should be validated on unseen evidence.

Reference lessonPython exampleVisual explanation

Intuition first

What this concept means in practice

The practical value of Probability metrics comes from understanding both the transformation and the boundary around it: what information is allowed to enter, what assumption is being made, and how you know the result is still valid after the transformation.

A beginner-friendly way to reason about it is to start with a tiny case where the correct result can be checked independently. Once the mechanism is clear, scale the exact same reasoning to larger tables, pipelines or models.

PurposeUse when probabilities feed risk, triage, ranking or expected-cost decisions.

MechanismEvaluate the full probability rather than only the final class. Log loss strongly penalises confident mistakes; Brier score is mean squared probability error.

EvidenceInspect intermediate and final output; compare with an independent expectation.

Main cautionGood discrimination does not guarantee calibrated probabilities.

Mechanism

Trace the operation from input to decision

Evaluate the full probability rather than only the final class. Log loss strongly penalises confident mistakes; Brier score is mean squared probability error.

1Input→

2Apply rule→

3Inspect state→

4Validate→

5Use result

Key rule

Brier = mean((p-y)^2)

Visual explanation

Make the structure visible

The interactive view uses a concept-specific plot when the topic maps naturally to one; otherwise it uses a workflow view instead of leaving a broken placeholder.

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Practical example

Where you would use it

Predicting 0.99 for the wrong class is punished much more by log loss than predicting 0.55.

Use when

Use when probabilities feed risk, triage, ranking or expected-cost decisions.

Pitfall

What can make the result misleading

Watch out

Good discrimination does not guarantee calibrated probabilities.

A useful diagnostic question is: Could the same code still run successfully if the analytical assumption were wrong? If yes, add an explicit validation check rather than relying on execution success.

Implementation

Miniature Python example

Keep the example small enough that you can inspect each stage manually.

Python

# Purpose: demonstrate Probability metrics with a small, inspectable example.
# Follow the comments and printed stages to connect each operation with its result.
# Import the library or helper used in this example.
# Step 1 — Import the module so its functions/classes are available to the rest of this example.
import numpy as np
# Import the library or helper used in this example.
# Step 2 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.metrics import confusion_matrix, accuracy_score, precision_score, recall_score, f1_score, roc_auc_score

# Create the numerical values used in the calculation.
# Step 3 — Construct `y_true` as an array so vectorised numerical operations can be applied consistently.
y_true = np.array([0,0,0,0,0,1,1,1,1,1,1,1])
# Step 4 — Construct `p` as an array so vectorised numerical operations can be applied consistently.
p = np.array([.08,.12,.22,.48,.56,.31,.58,.64,.70,.76,.82,.91])
# Store this intermediate value with a descriptive name for the next step.
# Step 5 — Execute this statement and inspect how it changes the current value, object or program state.
y_pred = (p >= .50).astype(int)
# Print this intermediate result so you can verify the workflow step by step.
# Step 6 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 1 · Threshold: 0.50")
# Print this intermediate result so you can verify the workflow step by step.
# Step 7 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · Confusion matrix:\n", confusion_matrix(y_true,y_pred))
# Print this intermediate result so you can verify the workflow step by step.
# Step 8 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · Accuracy:", round(accuracy_score(y_true,y_pred),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 9 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · Precision:", round(precision_score(y_true,y_pred),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 10 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · Recall:", round(recall_score(y_true,y_pred),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 11 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · F1:", round(f1_score(y_true,y_pred),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 12 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · ROC-AUC:", round(roc_auc_score(y_true,p),3))

Expected / illustrative output

STEP 1 · Threshold: 0.50
STEP 2 · Confusion matrix:
 [[4 1]
 [1 6]]
STEP 3 · Accuracy: 0.833
STEP 3 · Precision: 0.857
STEP 3 · Recall: 0.857
STEP 3 · F1: 0.857
STEP 3 · ROC-AUC: 0.943

Implementation checklist

Before you move on

Can you state what data or object enters the operation?
Can you explain what changes and what must remain invariant?
Have you checked the result on a tiny case you can verify independently?
Have you considered the main failure mode described above?
Can the operation be reproduced from code/formulas and documented assumptions?