00:00:00

Description

Editorial

Probability-Weighted Ensemble Classification

MEDIUM20 pts

In ensemble machine learning, combining the predictions of multiple models often yields better performance than any single model alone. Soft voting is a powerful ensemble technique that leverages the full probability distributions from each classifier rather than just their hard class predictions.

Unlike hard voting, which simply counts class votes from each classifier and picks the majority, soft voting computes a weighted average of predicted class probabilities across all classifiers. This approach captures the confidence level of each model's predictions, allowing highly certain predictions to carry more influence than uncertain ones.

The Soft Voting Algorithm

Given:

K classifiers, each producing probability distributions over C classes for N samples
Optional weights (w_1, w_2, ..., w_K) representing the relative importance of each classifier

For each sample i and class j, the weighted average probability is:

$$\bar{p}{ij} = \frac{\sum{k=1}^{K} w_k \cdot p_{k,i,j}}{\sum_{k=1}^{K} w_k}$$

The final predicted class for sample i is:

$$\hat{y}i = \arg\max{j} \bar{p}_{ij}$$

Weight Normalization

When weights are provided, they must be normalized to sum to 1 before computing the weighted average. If no weights are provided, use uniform weights where each classifier contributes equally ((w_k = 1/K)).

Your Task

Implement a function that performs soft voting ensemble classification. Given:

A 3D array of shape (n_classifiers, n_samples, n_classes) containing predicted probabilities
An optional list of classifier weights

Return a list of predicted class labels for each sample.

Example

Input

probabilities = [[[0.7, 0.2, 0.1], [0.3, 0.5, 0.2]], [[0.3, 0.4, 0.3], [0.2, 0.3, 0.5]]]
weights = None

Output

[0, 1]

Explanation

We have 2 classifiers predicting probabilities for 2 samples across 3 classes.

With uniform weights (0.5 each):

Sample 0: • Classifier 1 probabilities: [0.7, 0.2, 0.1] • Classifier 2 probabilities: [0.3, 0.4, 0.3] • Weighted average: [(0.7+0.3)/2, (0.2+0.4)/2, (0.1+0.3)/2] = [0.5, 0.3, 0.2] • Class 0 has highest probability (0.5), so predicted class = 0

Sample 1: • Classifier 1 probabilities: [0.3, 0.5, 0.2] • Classifier 2 probabilities: [0.2, 0.3, 0.5] • Weighted average: [(0.3+0.2)/2, (0.5+0.3)/2, (0.2+0.5)/2] = [0.25, 0.4, 0.35] • Class 1 has highest probability (0.4), so predicted class = 1

Final output: [0, 1]

Example

Input

probabilities = [[[0.6, 0.3, 0.1], [0.2, 0.6, 0.2]], [[0.4, 0.4, 0.2], [0.3, 0.3, 0.4]]]
weights = [0.7, 0.3]

Output

[0, 1]

Explanation

We have 2 classifiers with custom weights [0.7, 0.3] (already normalized to sum to 1).

Sample 0: • Classifier 1 (weight 0.7): [0.6, 0.3, 0.1] • Classifier 2 (weight 0.3): [0.4, 0.4, 0.2] • Weighted average:

Class 0: (0.7×0.6 + 0.3×0.4) = 0.42 + 0.12 = 0.54
Class 1: (0.7×0.3 + 0.3×0.4) = 0.21 + 0.12 = 0.33
Class 2: (0.7×0.1 + 0.3×0.2) = 0.07 + 0.06 = 0.13 • Class 0 wins with 0.54, so predicted class = 0

Sample 1: • Weighted average:

Class 0: (0.7×0.2 + 0.3×0.3) = 0.14 + 0.09 = 0.23
Class 1: (0.7×0.6 + 0.3×0.3) = 0.42 + 0.09 = 0.51
Class 2: (0.7×0.2 + 0.3×0.4) = 0.14 + 0.12 = 0.26 • Class 1 wins with 0.51, so predicted class = 1

Final output: [0, 1]

Example

Input

probabilities = [[[0.8, 0.1, 0.1], [0.1, 0.8, 0.1]], [[0.7, 0.2, 0.1], [0.2, 0.6, 0.2]], [[0.6, 0.3, 0.1], [0.3, 0.5, 0.2]]]
weights = [0.5, 0.3, 0.2]

Output

[0, 1]

Explanation

We have 3 classifiers with custom weights [0.5, 0.3, 0.2] for 2 samples over 3 classes.

Sample 0: • Classifier 1 (w=0.5): [0.8, 0.1, 0.1] • Classifier 2 (w=0.3): [0.7, 0.2, 0.1] • Classifier 3 (w=0.2): [0.6, 0.3, 0.1] • Weighted average:

Class 0: (0.5×0.8) + (0.3×0.7) + (0.2×0.6) = 0.40 + 0.21 + 0.12 = 0.73
Class 1: (0.5×0.1) + (0.3×0.2) + (0.2×0.3) = 0.05 + 0.06 + 0.06 = 0.17
Class 2: (0.5×0.1) + (0.3×0.1) + (0.2×0.1) = 0.05 + 0.03 + 0.02 = 0.10 • Class 0 has highest weighted average (0.73), so predicted class = 0

Sample 1: • Weighted average:

Class 0: (0.5×0.1) + (0.3×0.2) + (0.2×0.3) = 0.05 + 0.06 + 0.06 = 0.17
Class 1: (0.5×0.8) + (0.3×0.6) + (0.2×0.5) = 0.40 + 0.18 + 0.10 = 0.68
Class 2: (0.5×0.1) + (0.3×0.2) + (0.2×0.2) = 0.05 + 0.06 + 0.04 = 0.15 • Class 1 has highest weighted average (0.68), so predicted class = 1

Final output: [0, 1]

Accepted0/0·0% Acceptance

Constraints

1 ≤ n_classifiers ≤ 50 (number of models in the ensemble)
1 ≤ n_samples ≤ 10,000 (number of samples to classify)
2 ≤ n_classes ≤ 100 (number of classification categories)
0.0 ≤ probabilities[k][i][j] ≤ 1.0 for all values
Each classifier's probabilities for a sample sum to 1.0 (valid probability distribution)
If weights is not None, len(weights) == n_classifiers
If weights is provided, all weights are positive (weights[k] > 0)
Weights will be normalized to sum to 1 during computation

Code

Visualizer

Solutions

14px

Test Cases3

Results

Submissions

weights =

null

probabilities =

[[[0.7,0.2,0.1],[0.3,0.5,0.2]],[[0.3,0.4,0.3],[0.2,0.3,0.5]]]

Loading problem...

101

00:00:00

Description

Editorial

Probability-Weighted Ensemble Classification

MEDIUM20 pts

The Soft Voting Algorithm

Given:

K classifiers, each producing probability distributions over C classes for N samples
Optional weights (w_1, w_2, ..., w_K) representing the relative importance of each classifier

For each sample i and class j, the weighted average probability is:

$$\bar{p}{ij} = \frac{\sum{k=1}^{K} w_k \cdot p_{k,i,j}}{\sum_{k=1}^{K} w_k}$$

The final predicted class for sample i is:

$$\hat{y}i = \arg\max{j} \bar{p}_{ij}$$

Weight Normalization

Your Task

Implement a function that performs soft voting ensemble classification. Given:

A 3D array of shape (n_classifiers, n_samples, n_classes) containing predicted probabilities
An optional list of classifier weights

Return a list of predicted class labels for each sample.

Example

Input

probabilities = [[[0.7, 0.2, 0.1], [0.3, 0.5, 0.2]], [[0.3, 0.4, 0.3], [0.2, 0.3, 0.5]]]
weights = None

Output

[0, 1]

Explanation

We have 2 classifiers predicting probabilities for 2 samples across 3 classes.

With uniform weights (0.5 each):

Final output: [0, 1]

Example

Input

probabilities = [[[0.6, 0.3, 0.1], [0.2, 0.6, 0.2]], [[0.4, 0.4, 0.2], [0.3, 0.3, 0.4]]]
weights = [0.7, 0.3]

Output

[0, 1]

Explanation

We have 2 classifiers with custom weights [0.7, 0.3] (already normalized to sum to 1).

Sample 0: • Classifier 1 (weight 0.7): [0.6, 0.3, 0.1] • Classifier 2 (weight 0.3): [0.4, 0.4, 0.2] • Weighted average:

Class 0: (0.7×0.6 + 0.3×0.4) = 0.42 + 0.12 = 0.54
Class 1: (0.7×0.3 + 0.3×0.4) = 0.21 + 0.12 = 0.33
Class 2: (0.7×0.1 + 0.3×0.2) = 0.07 + 0.06 = 0.13 • Class 0 wins with 0.54, so predicted class = 0

Sample 1: • Weighted average:

Class 0: (0.7×0.2 + 0.3×0.3) = 0.14 + 0.09 = 0.23
Class 1: (0.7×0.6 + 0.3×0.3) = 0.42 + 0.09 = 0.51
Class 2: (0.7×0.2 + 0.3×0.4) = 0.14 + 0.12 = 0.26 • Class 1 wins with 0.51, so predicted class = 1

Final output: [0, 1]

Example

Input

probabilities = [[[0.8, 0.1, 0.1], [0.1, 0.8, 0.1]], [[0.7, 0.2, 0.1], [0.2, 0.6, 0.2]], [[0.6, 0.3, 0.1], [0.3, 0.5, 0.2]]]
weights = [0.5, 0.3, 0.2]

Output

[0, 1]

Explanation

We have 3 classifiers with custom weights [0.5, 0.3, 0.2] for 2 samples over 3 classes.

Sample 0: • Classifier 1 (w=0.5): [0.8, 0.1, 0.1] • Classifier 2 (w=0.3): [0.7, 0.2, 0.1] • Classifier 3 (w=0.2): [0.6, 0.3, 0.1] • Weighted average:

Class 0: (0.5×0.8) + (0.3×0.7) + (0.2×0.6) = 0.40 + 0.21 + 0.12 = 0.73
Class 1: (0.5×0.1) + (0.3×0.2) + (0.2×0.3) = 0.05 + 0.06 + 0.06 = 0.17
Class 2: (0.5×0.1) + (0.3×0.1) + (0.2×0.1) = 0.05 + 0.03 + 0.02 = 0.10 • Class 0 has highest weighted average (0.73), so predicted class = 0

Sample 1: • Weighted average:

Class 0: (0.5×0.1) + (0.3×0.2) + (0.2×0.3) = 0.05 + 0.06 + 0.06 = 0.17
Class 1: (0.5×0.8) + (0.3×0.6) + (0.2×0.5) = 0.40 + 0.18 + 0.10 = 0.68
Class 2: (0.5×0.1) + (0.3×0.2) + (0.2×0.2) = 0.05 + 0.06 + 0.04 = 0.15 • Class 1 has highest weighted average (0.68), so predicted class = 1

Final output: [0, 1]

Accepted0/0·0% Acceptance

Constraints

1 ≤ n_classifiers ≤ 50 (number of models in the ensemble)
1 ≤ n_samples ≤ 10,000 (number of samples to classify)
2 ≤ n_classes ≤ 100 (number of classification categories)
0.0 ≤ probabilities[k][i][j] ≤ 1.0 for all values
Each classifier's probabilities for a sample sum to 1.0 (valid probability distribution)
If weights is not None, len(weights) == n_classifiers
If weights is provided, all weights are positive (weights[k] > 0)
Weights will be normalized to sum to 1 during computation

Code

Visualizer

Solutions

14px

Test Cases3

Results

Submissions

weights =

null

probabilities =

[[[0.7,0.2,0.1],[0.3,0.5,0.2]],[[0.3,0.4,0.3],[0.2,0.3,0.5]]]

Probability-Weighted Ensemble Classification

The Soft Voting Algorithm

Weight Normalization

Your Task

Hints

Probability-Weighted Ensemble Classification

The Soft Voting Algorithm

Weight Normalization

Your Task

Hints