Hit Rate (HR)

1. Definition

Hit Rate measures whether the set of recommended items contains at least one of the items that the user actually interacted with (e.g., purchased, clicked, watched).

For a single user:

$HR@K = \begin{cases} 1 & \text{if at least one relevant item is in the top K recommendations} \\ 0 & \text{otherwise} \end{cases}$

For multiple users:

$HR@K = \frac{1}{N} \sum_{i=1}^N HR_i@K$

where $N$ = number of users.

2. Intuition

• It checks “Did the recommender system hit the target at least once within the top K results?”
• Binary per user: either a hit (1) or a miss (0).

3. Example

Suppose a recommender system shows Top-3 recommendations for a user:

• Predicted = [Movie A, Movie B, Movie C]
• Actual watched = Movie C

Since Movie C is in the top 3, HR@3 = 1.

If Actual watched = Movie D (not in top 3), then HR@3 = 0.

For multiple users, average across them.

4. Interpretation

• Range: 0 → 1
• 1 = perfect (every user’s relevant item is included in top K).
• Higher Hit Rate means the system is good at surfacing at least one relevant recommendation.

5. Use Cases

• Recommender Systems (Netflix, Amazon):
  • HR@10 → checks if the true item is somewhere in the top 10 recommendations.
• Information Retrieval:
  • Similar to Recall@K, but more lenient since it doesn’t care how many relevant items are retrieved, just whether at least one is present.

6. Comparison with Other Metrics

• Recall@K: Counts all relevant items in top K (not just one).
• Precision@K: Focuses on proportion of recommended items that are relevant.
• Hit Rate@K: Only checks if at least one relevant item is present.

Example:

• Top-5 results contain 3 relevant items → Recall@5 = 3 / Total relevant.
• Precision@5 = 3/5.
• Hit Rate@5 = 1.

7. Python Example

import numpy as np

def hit_rate_at_k(y_true, y_pred, k=5):
    """
    y_true: list of true relevant items (e.g., items the user actually interacted with)
    y_pred: list of predicted ranked items
    k: cutoff
    """
    top_k = y_pred[:k]
    return int(any(item in y_true for item in top_k))

# Example for 3 users
y_true_list = [[3], [2], [5]]  # actual items
y_pred_list = [[1,3,4], [2,6,7], [1,4,8]]  # predicted top-K lists

hits = [hit_rate_at_k(y_true, y_pred, k=3) 
        for y_true, y_pred in zip(y_true_list, y_pred_list)]
hit_rate = np.mean(hits)

print("Hit Rate@3:", hit_rate)

import numpy as np

def hit_rate_at_k(y_true, y_pred, k=5):
    """
    y_true: list of true relevant items (e.g., items the user actually interacted with)
    y_pred: list of predicted ranked items
    k: cutoff
    """
    top_k = y_pred[:k]
    return int(any(item in y_true for item in top_k))

# Example for 3 users
y_true_list = [[3], [2], [5]]  # actual items
y_pred_list = [[1,3,4], [2,6,7], [1,4,8]]  # predicted top-K lists

hits = [hit_rate_at_k(y_true, y_pred, k=3) 
        for y_true, y_pred in zip(y_true_list, y_pred_list)]
hit_rate = np.mean(hits)

print("Hit Rate@3:", hit_rate)

Output:

Hit Rate@3: 0.67

Hit Rate@3: 0.67

→ The system got 2 out of 3 users correct.

Summary

• Hit Rate@K = did the top K recommendations include at least one relevant item?
• Range = [0,1], higher is better.
• Simple and intuitive but does not measure how many relevant items were recommended.