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Naive Bayes Log-Likelihood (Bernoulli)

Classic ML

Hard

Train a Bernoulli Naive Bayes model on binary features and return the unnormalized log posterior for every test sample and class.

\theta_{jc}=\frac{N_{jc}+1}{N_c+2}

\log \widetilde{P}(c\mid\mathbf{x})=\log P(c)+\sum_{j=1}^{D}\left[x_j\log\theta_{jc}+(1-x_j)\log(1-\theta_{jc})\right]

Here, $N_c$ is the number of training samples in class $c$ , $N_{jc}$ counts class- $c$ samples whose feature $j$ is one, and $D$ is the feature count. The added one and two implement Laplace smoothing. Order output columns by ascending class label, round values to four decimals, and return a NumPy array of shape (n_test, n_classes).

Loading visualization...

Examples

Input: X_train = [[1, 0], [0, 1]], y_train = [1, 0], X_test = [[1, 0]]

Output: [[-2.8904, -1.5041]]

Explanation: The columns correspond to classes 0 and 1, and the observed feature pattern is more likely under class 1.

Input: X_train = [[1, 0], [1, 1], [0, 0], [0, 1]], y_train = [0, 0, 1, 1], X_test = [[1, 0], [0, 1]]

Output: [[-1.674, -2.7726], [-2.7726, -1.674]]

Hint 1

Use np.unique(y_train, return_counts=True) to obtain sorted classes and priors.

Hint 2

For one class, X_train[y_train == label].sum(axis=0) gives all feature-one counts.

Hint 3

Evaluate the Bernoulli terms with matrix multiplication against np.log(theta) and np.log1p(-theta).

Requirements

Estimate class priors from training frequencies
Apply Laplace smoothing to every Bernoulli feature probability
Include contributions from both present and absent features in log space
Return a NumPy array rounded to four decimals

Constraints

Training and test features contain only 0 and 1
Training labels contain at least one sample per represented class
Use NumPy only

Try Similar Problems

Gaussian Naive Bayes Bernoulli Pmf Bigram Probabilities Logistic Regression Training Gini Impurity

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Naive Bayes Log-Likelihood (Bernoulli)

Classic ML

Hard

Train a Bernoulli Naive Bayes model on binary features and return the unnormalized log posterior for every test sample and class.

\theta_{jc}=\frac{N_{jc}+1}{N_c+2}

\log \widetilde{P}(c\mid\mathbf{x})=\log P(c)+\sum_{j=1}^{D}\left[x_j\log\theta_{jc}+(1-x_j)\log(1-\theta_{jc})\right]

Loading visualization...

Examples

Input: X_train = [[1, 0], [0, 1]], y_train = [1, 0], X_test = [[1, 0]]

Output: [[-2.8904, -1.5041]]

Explanation: The columns correspond to classes 0 and 1, and the observed feature pattern is more likely under class 1.

Input: X_train = [[1, 0], [1, 1], [0, 0], [0, 1]], y_train = [0, 0, 1, 1], X_test = [[1, 0], [0, 1]]

Output: [[-1.674, -2.7726], [-2.7726, -1.674]]

Hint 1

Use np.unique(y_train, return_counts=True) to obtain sorted classes and priors.

Hint 2

For one class, X_train[y_train == label].sum(axis=0) gives all feature-one counts.

Hint 3

Evaluate the Bernoulli terms with matrix multiplication against np.log(theta) and np.log1p(-theta).

Requirements

Estimate class priors from training frequencies
Apply Laplace smoothing to every Bernoulli feature probability
Include contributions from both present and absent features in log space
Return a NumPy array rounded to four decimals

Constraints

Training and test features contain only 0 and 1
Training labels contain at least one sample per represented class
Use NumPy only

Try Similar Problems

Gaussian Naive Bayes Bernoulli Pmf Bigram Probabilities Logistic Regression Training Gini Impurity

Case 1

Case 2

X_train =

Accepts: array

y_train =

Accepts: array

X_test =

Accepts: array