Machine LearningModel Evaluation Metrics

Ranking Metrics

LevelAdvanced

Duration120 mins

TopicModel Evaluation Metrics

1 / 5

Precision@k and Recall@k

Beyond Classification: The Ranking Paradigm

When you type a query into a search engine, the system doesn't present you with a simple yes/no answer. Instead, it returns a ranked list of results—documents ordered by predicted relevance. When a streaming service recommends movies, it doesn't merely classify each film as 'relevant' or 'not relevant'; it produces an ordered sequence where position reflects confidence and importance.

This shift from classification to ranking fundamentally changes how we must evaluate model performance. In classification, a prediction is either correct or incorrect. In ranking, correctness is position-dependent: a relevant document at position 1 is more valuable than the same document at position 100. The user who finds their answer in the first result has a qualitatively different experience from one who must scroll through pages of irrelevant content.

Precision@k and Recall@k are the foundational metrics that bridge classification thinking with ranking reality. They answer the questions every search and recommendation system must address: Of the top-k items we show the user, how many are actually relevant? And of all the relevant items that exist, how many appear in our top-k?

What You Will Learn

By the end of this page, you will understand Precision@k and Recall@k from first principles—their definitions, computational procedures, statistical properties, and practical applications. You will be able to implement these metrics, interpret their values correctly, and understand their relationships to both classification metrics and more sophisticated ranking measures.

The Ranking Problem Formulation

Before defining metrics, we must precisely formulate the ranking problem. Consider a query $q$ (which may be explicit, like a search string, or implicit, like a user's context) and a corpus of $N$ items $\mathcal{D} = {d_1, d_2, \ldots, d_N}$. Each item has a true relevance to the query, denoted $\text{rel}(d_i, q) \in {0, 1}$ for binary relevance (relevant or not) or $\text{rel}(d_i, q) \in {0, 1, 2, \ldots, L}$ for graded relevance.

The Ranking Task:

A ranking model produces a permutation $\pi$ of items, where $\pi(i)$ denotes the item placed at rank $i$. Equivalently, we can describe this as a ranked list $[d_{\pi(1)}, d_{\pi(2)}, \ldots, d_{\pi(N)}]$ where the item at position 1 is ranked highest.

Key Notation:

Symbol	Meaning
$k$	The cut-off rank (we evaluate only the top-$k$ items)
$R$	The set of all truly relevant items for query $q$
$\text{rel}_i$	Relevance of the item at rank $i$ (shorthand for $\text{rel}(\pi(i), q)$)
$R_k$	Relevant items in the top-$k$: ${\pi(i) : i \leq k \text{ and } \text{rel}_i = 1}$

The ranking problem differs fundamentally from classification in that:

Order matters: The same set of predictions in different orders yields different metric values
Truncation is natural: Users typically examine only the first few results
Context is query-dependent: The same item may be relevant to one query and irrelevant to another

Why Position Matters

User studies consistently show that click probability drops dramatically with position. In web search, position 1 receives 30-40% of clicks, position 2 receives about 15%, and positions beyond 10 receive negligible attention. A metric that ignores position fails to capture the user's actual experience.

Precision@k: Definition and Intuition

Precision@k measures the proportion of relevant items among the top-$k$ ranked items. It answers the question: Of the items we showed the user, how many were actually relevant?

Formal Definition:

$$\text{Precision@}k = \frac{|R_k|}{k} = \frac{\text{Number of relevant items in top-}k}{k}$$

Equivalently, using indicator notation:

$$\text{Precision@}k = \frac{1}{k} \sum_{i=1}^{k} \mathbb{1}[\text{rel}_i = 1]$$

where $\mathbb{1}[\cdot]$ is the indicator function that equals 1 if the condition is true and 0 otherwise.

Properties:

Range: $[0, 1]$ where 1 means all top-$k$ items are relevant
Monotonicity in k: Not guaranteed—P@k can increase or decrease as k grows
Upper bound: $\min(1, |R|/k)$ when $|R| < k$ (can't achieve P@k = 1 if fewer than k relevant items exist)
Best possible: Achieved when all relevant items occupy the top positions

Precision@k Example: Search Results for Query 'machine learning'
Rank	Document	Relevant?	Cumulative Relevant	P@k
1	ML Tutorial	✓ Yes	1	1/1 = 1.000
2	Python Basics	✗ No	1	1/2 = 0.500
3	Deep Learning Guide	✓ Yes	2	2/3 = 0.667
4	Neural Networks	✓ Yes	3	3/4 = 0.750
5	Cooking Recipes	✗ No	3	3/5 = 0.600
6	Data Science Intro	✓ Yes	4	4/6 = 0.667
7	Random Blog Post	✗ No	4	4/7 = 0.571
8	ML Research Paper	✓ Yes	5	5/8 = 0.625
9	Weather Forecast	✗ No	5	5/9 = 0.556
10	Algorithm Book	✓ Yes	6	6/10 = 0.600

Interpreting the Example:

Notice that P@k fluctuates as k increases:

P@1 = 1.0 (perfect—the first result is relevant)
P@2 = 0.5 (drops because position 2 is irrelevant)
P@4 = 0.75 (recovers as more relevant items appear)
P@5 = 0.6 (drops again with an irrelevant result)

This non-monotonic behavior is characteristic of P@k. Unlike cumulative metrics, inserting an irrelevant item at any position reduces the precision at that cutoff and beyond (until enough relevant items accumulate).

The Denominator is Fixed

A crucial property of P@k is that the denominator is always k—the cutoff you chose. This means P@k is independent of how many relevant items exist in the corpus. Whether there are 5 or 5000 relevant documents, P@10 still divides by 10. This property has important implications for comparing systems across queries with different numbers of relevant items.

Recall@k: Definition and Intuition

Recall@k measures the proportion of all relevant items that appear in the top-$k$ ranked items. It answers the question: Of all the relevant items that exist, how many did we surface in our top-k?

Formal Definition:

$$\text{Recall@}k = \frac{|R_k|}{|R|} = \frac{\text{Number of relevant items in top-}k}{\text{Total number of relevant items}}$$

Equivalently:

$$\text{Recall@}k = \frac{\sum_{i=1}^{k} \mathbb{1}[\text{rel}_i = 1]}{|R|}$$

Properties:

Range: $[0, 1]$ where 1 means all relevant items appear in the top-$k$
Monotonicity in k: Guaranteed non-decreasing (more positions can only add relevant items, never remove them)
Upper bound: $\min(1, k/|R|)$ when $k < |R|$ (can't achieve R@k = 1 if k is smaller than the number of relevant items)
Dependency on |R|: Unlike P@k, Recall@k explicitly depends on the total number of relevant items

Recall@k Example: Same Search Results (Assume 8 Relevant Documents Exist)
Rank	Document	Relevant?	Cumulative Relevant	R@k (\|R\|=8)
1	ML Tutorial	✓ Yes	1	1/8 = 0.125
2	Python Basics	✗ No	1	1/8 = 0.125
3	Deep Learning Guide	✓ Yes	2	2/8 = 0.250
4	Neural Networks	✓ Yes	3	3/8 = 0.375
5	Cooking Recipes	✗ No	3	3/8 = 0.375
6	Data Science Intro	✓ Yes	4	4/8 = 0.500
7	Random Blog Post	✗ No	4	4/8 = 0.500
8	ML Research Paper	✓ Yes	5	5/8 = 0.625
9	Weather Forecast	✗ No	5	5/8 = 0.625
10	Algorithm Book	✓ Yes	6	6/8 = 0.750

Interpreting the Example:

Unlike P@k, R@k is monotonically non-decreasing:

R@1 = 0.125 (found 1 of 8 relevant items)
R@4 = 0.375 (found 3 of 8)
R@10 = 0.750 (found 6 of 8)

Recall can never decrease as k increases because we can only accumulate more relevant items (or stay flat if the new position is irrelevant).

The Coverage Interpretation:

Recall@k captures coverage—how much of the relevant content have we exposed to the user? In applications where users need comprehensive results (e.g., legal document discovery, prior art search in patents), high recall is critical. Missing even one relevant document could have serious consequences.

The R = 0 Edge Case

When a query has no relevant items (|R| = 0), Recall@k is undefined (division by zero). This edge case requires explicit handling: some systems report NaN, others report 0, and others exclude such queries from evaluation. Always document your handling of this case.

The Precision-Recall Relationship in Ranking

Precision@k and Recall@k are intimately related through their shared numerator—the count of relevant items in the top-k. This creates a fundamental relationship:

$$\text{Precision@}k \times k = \text{Recall@}k \times |R|$$

Both sides equal $|R_k|$, the number of relevant items found.

The Precision-Recall Tradeoff:

As $k$ increases:

Recall@k can only increase (or stay flat)—we might find more relevant items
Precision@k typically decreases—the denominator grows, and we're forced to include less-relevant items

This creates the classic precision-recall tradeoff:

Small k (conservative): High precision, low recall—we show only items we're very confident about, but miss many relevant items
Large k (aggressive): Low precision, high recall—we cast a wide net and find most relevant items, but include many irrelevant ones

Mathematical Relationship:

We can derive one from the other:

$$\text{Recall@}k = \frac{k \cdot \text{Precision@}k}{|R|}$$

$$\text{Precision@}k = \frac{|R| \cdot \text{Recall@}k}{k}$$

precision_recall_relationship.py
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import numpy as np
import matplotlib.pyplot as plt
 
def compute_precision_recall_at_k(relevance_list: list[int], total_relevant: int):
    """
    Compute P@k and R@k for all values of k.
    
    Args:
        relevance_list: Binary list where 1 = relevant item at that rank
        total_relevant: Total number of relevant items in corpus |R|
    
    Returns:
        Dict with precision@k and recall@k arrays
    """
    relevance = np.array(relevance_list)
    n = len(relevance)
    
    # Cumulative relevant count at each position
    cumulative_relevant = np.cumsum(relevance)
    
    # P@k = cumulative_relevant / k
    k_values = np.arange(1, n + 1)
    precision_at_k = cumulative_relevant / k_values
    
    # R@k = cumulative_relevant / |R|
    recall_at_k = cumulative_relevant / total_relevant if total_relevant > 0 else np.zeros(n)
    
    return {
        'k': k_values,
        'precision': precision_at_k,
        'recall': recall_at_k,
        'cumulative_relevant': cumulative_relevant
    }
 
# Example: Ranked list with relevance judgments
# 1 = relevant, 0 = not relevant
ranked_relevance = [1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1]
total_relevant = 10  # Total relevant items in corpus
 
results = compute_precision_recall_at_k(ranked_relevance, total_relevant)
 
# Display results
print("Precision@k and Recall@k Analysis")
print("=" * 50)
print(f"Total items ranked: {len(ranked_relevance)}")
print(f"Total relevant in corpus: {total_relevant}")
print(f"Relevant in ranked list: {sum(ranked_relevance)}")
print()
 
print(f"{'k':>3} | {'Rel@k':>5} | {'P@k':>6} | {'R@k':>6}")
print("-" * 30)
for k in [1, 3, 5, 10, 15, 20]:
    if k <= len(ranked_relevance):
        print(f"{k:>3} | {int(results['cumulative_relevant'][k-1]):>5} | "
              f"{results['precision'][k-1]:>6.3f} | {results['recall'][k-1]:>6.3f}")
 
# Verify the relationship: P@k * k = R@k * |R|
print("\nVerifying P@k × k = R@k × |R|:")
for k in [5, 10]:
    lhs = results['precision'][k-1] * k
    rhs = results['recall'][k-1] * total_relevant
    print(f"k={k}: P@k × k = {lhs:.2f}, R@k × |R| = {rhs:.2f}")

Visualizing the Tradeoff:

Plotting P@k against R@k as k varies produces a precision-recall curve for the ranking. Unlike classification PR curves (which vary with threshold), ranking PR curves vary with cutoff $k$. The curve typically shows:

High precision at low recall: At small k, we're selective
Declining precision as recall grows: As k increases, we include weaker candidates
Plateaus: When consecutive ranks are irrelevant, recall stays flat while precision drops
Steps: When a relevant item is found, recall jumps and precision may recover slightly

Computational Implementation

Implementing P@k and R@k correctly requires attention to several practical considerations. Let's build a robust implementation from first principles.

Input Representations:

Ranking metrics can be computed from various input formats:

Explicit ranking: A list of item IDs in ranked order + a set of relevant item IDs
Score-based: A list of (item_id, score, relevance) tuples, sorted by score
Binary relevance vector: A vector where position indicates rank and value indicates relevance

Algorithm Complexity:

Operation	Time Complexity	Space Complexity
P@k from relevance list	O(k)	O(1)
R@k from relevance list	O(k)	O(1)
All P@k, R@k for k=1..n	O(n)	O(n)
Building from item IDs	O(k) with hash set for relevant	O(

precision_recall_implementation.py
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import numpy as np
from typing import List, Set, Optional, Dict, Any
 
class RankingMetrics:
    """
    Production-grade implementation of ranking metrics.
    Handles edge cases, supports multiple input formats, and provides
    comprehensive metric computation.
    """
    
    def __init__(self, relevant_items: Set[Any], ranked_items: List[Any]):
        """
        Initialize with ground truth and ranked predictions.
        
        Args:
            relevant_items: Set of item IDs that are relevant
            ranked_items: List of item IDs in ranked order (position 0 = rank 1)
        """
        self.relevant_items = set(relevant_items)
        self.ranked_items = ranked_items
        self.n_relevant = len(self.relevant_items)
        self.n_ranked = len(ranked_items)
        
        # Precompute binary relevance array for efficiency
        self._relevance_array = np.array([
            1 if item in self.relevant_items else 0 
            for item in ranked_items
        ])
        
        # Precompute cumulative relevance
        self._cumulative_relevant = np.cumsum(self._relevance_array)
    
    def precision_at_k(self, k: int) -> float:
        """
        Compute Precision@k.
        
        Args:
            k: Cutoff rank (1-indexed conceptually, but we handle internally)
        
        Returns:
            Precision@k value in [0, 1]
        
        Raises:
            ValueError: If k < 1
        """
        if k < 1:
            raise ValueError(f"k must be >= 1, got {k}")
        
        # Handle k larger than our ranked list
        effective_k = min(k, self.n_ranked)
        
        if effective_k == 0:
            return 0.0
        
        relevant_in_top_k = self._cumulative_relevant[effective_k - 1]
        return float(relevant_in_top_k) / k  # Note: divide by k, not effective_k
    
    def recall_at_k(self, k: int) -> Optional[float]:
        """
        Compute Recall@k.
        
        Args:
            k: Cutoff rank
        
        Returns:
            Recall@k value in [0, 1], or None if no relevant items exist
        """
        if k < 1:
            raise ValueError(f"k must be >= 1, got {k}")
        
        if self.n_relevant == 0:
            return None  # Undefined when no relevant items
        
        effective_k = min(k, self.n_ranked)
        
        if effective_k == 0:
            return 0.0
        
        relevant_in_top_k = self._cumulative_relevant[effective_k - 1]
        return float(relevant_in_top_k) / self.n_relevant
    
    def precision_recall_at_all_k(self) -> Dict[str, np.ndarray]:
        """
        Compute P@k and R@k for all k from 1 to n_ranked.
        
        Returns:
            Dictionary with 'k', 'precision', 'recall', 'f1' arrays
        """
        if self.n_ranked == 0:
            return {
                'k': np.array([]),
                'precision': np.array([]),
                'recall': np.array([]),
                'f1': np.array([])
            }
        
        k_values = np.arange(1, self.n_ranked + 1)
        precision = self._cumulative_relevant / k_values
        
        if self.n_relevant > 0:
            recall = self._cumulative_relevant / self.n_relevant
        else:
            recall = np.zeros(self.n_ranked)
        
        # F1@k = harmonic mean of P@k and R@k
        with np.errstate(divide='ignore', invalid='ignore'):
            f1 = 2 * precision * recall / (precision + recall)
            f1 = np.nan_to_num(f1)  # Handle 0/0 cases
        
        return {
            'k': k_values,
            'precision': precision,
            'recall': recall,
            'f1': f1
        }
    
    def r_precision(self) -> float:
        """
        Compute R-Precision: Precision at k=|R|.
        A single-number summary that balances precision and recall.
        """
        if self.n_relevant == 0:
            return 0.0
        return self.precision_at_k(self.n_relevant)
 
 
# Example usage with realistic data
if __name__ == "__main__":
    # Scenario: Product search results
    # User searched for "wireless headphones"
    
    relevant_products = {
        'sony_wh1000xm4', 'bose_qc45', 'apple_airpods_max',
        'sennheiser_momentum', 'jabra_elite_85h', 'audio_technica_m50x'
    }  # 6 relevant products
    
    # Model's ranked output (top 15 results)
    search_results = [
        'sony_wh1000xm4',      # Rank 1: Relevant ✓
        'cheap_wired_earbuds', # Rank 2: Not relevant
        'bose_qc45',           # Rank 3: Relevant ✓
        'phone_case',          # Rank 4: Not relevant
        'apple_airpods_max',   # Rank 5: Relevant ✓
        'usb_cable',           # Rank 6: Not relevant
        'jabra_elite_85h',     # Rank 7: Relevant ✓
        'random_speaker',      # Rank 8: Not relevant
        'bluetooth_adapter',   # Rank 9: Not relevant
        'sennheiser_momentum', # Rank 10: Relevant ✓
        'gaming_headset',      # Rank 11: Not relevant
        'microphone',          # Rank 12: Not relevant
        'audio_technica_m50x', # Rank 13: Relevant ✓
        'laptop_stand',        # Rank 14: Not relevant
        'mouse_pad',           # Rank 15: Not relevant
    ]
    
    metrics = RankingMetrics(relevant_products, search_results)
    
    print("Product Search Ranking Evaluation")
    print("=" * 50)
    print(f"Total relevant products: {len(relevant_products)}")
    print(f"Products in ranked list: {len(search_results)}")
    print()
    
    # Report key cutoffs
    for k in [1, 3, 5, 10]:
        p_at_k = metrics.precision_at_k(k)
        r_at_k = metrics.recall_at_k(k)
        print(f"P@{k:2d} = {p_at_k:.3f}, R@{k:2d} = {r_at_k:.3f}")
    
    print(f"\nR-Precision (P@{len(relevant_products)}): {metrics.r_precision():.3f}")

Statistical Properties and Interpretation

Understanding the statistical properties of P@k and R@k is essential for proper interpretation and comparison.

Expected Values Under Random Ranking:

If items are ranked uniformly at random, what are the expected values of P@k and R@k?

Let $\rho = |R|/N$ be the base rate—the proportion of relevant items in the corpus.

$$\mathbb{E}[\text{P@}k] = \rho$$ $$\mathbb{E}[\text{R@}k] = \frac{k}{N}$$ (approximately, for large N)

A good ranker should significantly outperform these baselines. If your P@10 equals the base rate ρ, your model is no better than random.

Variance and Confidence:

The variance of P@k (under random ranking) follows a binomial distribution:

$$\text{Var}[\text{P@}k] = \frac{\rho(1-\rho)}{k}$$

This has important implications:

Larger k reduces variance: P@100 is more stable than P@5
Extreme base rates increase variance: Very rare or very common relevant items lead to higher variance
Small k requires caution: P@1 can swing dramatically between queries

Interpretation Guide for Precision@k Values
P@k Range	Interpretation	Typical Action
0.9 - 1.0	Excellent: Almost all shown items are relevant	Possibly too conservative; check recall
0.7 - 0.9	Good: Strong majority of results are relevant	Solid performance for most applications
0.5 - 0.7	Moderate: About half the results are useful	Room for improvement; check feature quality
0.3 - 0.5	Weak: More noise than signal	Significant model improvements needed
< 0.3	Poor: Mostly irrelevant results	Fundamental issues with ranking model

Context Matters

These interpretations assume balanced relevance. If only 1% of items are relevant (|R|/N = 0.01), then P@10 = 0.3 means your model is 30× better than random—which is excellent! Always contextualize P@k against the base rate and the specific application requirements.

Averaging Across Queries:

In practice, we evaluate a ranking system across many queries and report average metrics:

$$\text{Mean P@}k = \frac{1}{|Q|} \sum_{q \in Q} \text{P@}k(q)$$

$$\text{Mean R@}k = \frac{1}{|Q|} \sum_{q \in Q} \text{R@}k(q)$$

Statistical Significance:

When comparing two ranking systems, use appropriate statistical tests:

Paired t-test: For comparing Mean P@k across systems
Bootstrap confidence intervals: More robust for non-normal distributions
Two-sided binomial test: For P@1 comparisons (is one system better at the top position?)

A typical threshold is p < 0.05 for claiming one system is significantly better.

R-Precision: Bridging Precision and Recall

R-Precision is a special case of Precision@k where $k$ is set to $|R|$, the total number of relevant items for the query:

$$\text{R-Precision} = \text{Precision@}|R| = \frac{|R_{|R|}|}{|R|}$$

Why R-Precision is Elegant:

At $k = |R|$, something mathematically beautiful happens:

$$\text{R-Precision} = \text{Precision@}|R| = \text{Recall@}|R|$$

Proof: At k = |R|:

Precision@|R| = (relevant in top-|R|) / |R|
Recall@|R| = (relevant in top-|R|) / |R|

Both reduce to the same quantity!

Interpretation:

R-Precision asks: "If you could only show |R| items (the exact number that are relevant), how many of those would be correct?" This is a natural performance measure because:

It adapts to query difficulty (queries with more relevant items use larger k)
It equals both precision and recall at this special cutoff
It ranges from 0 (no relevant items in top-|R|) to 1 (all relevant items in top-|R|, perfectly ranked)

r_precision_analysis.py
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import numpy as np
 
def r_precision(ranked_relevance: list[int], total_relevant: int) -> float:
    """
    Compute R-Precision: Precision at k = |R|.
    
    Args:
        ranked_relevance: Binary relevance of items in ranked order
        total_relevant: Total number of relevant items |R|
    
    Returns:
        R-Precision value
    """
    if total_relevant == 0:
        return 0.0
    
    # Look at top |R| positions
    k = total_relevant
    top_k = ranked_relevance[:k] if k <= len(ranked_relevance) else ranked_relevance
    
    # Count relevant in top k
    relevant_found = sum(top_k)
    
    return relevant_found / total_relevant
 
def compare_with_precision_recall(ranked_relevance: list[int], total_relevant: int):
    """
    Demonstrate that R-Precision = P@|R| = R@|R|
    """
    k = total_relevant
    relevant_in_top_k = sum(ranked_relevance[:k]) if k <= len(ranked_relevance) else sum(ranked_relevance)
    
    precision_at_R = relevant_in_top_k / k if k > 0 else 0
    recall_at_R = relevant_in_top_k / total_relevant if total_relevant > 0 else 0
    r_prec = r_precision(ranked_relevance, total_relevant)
    
    print(f"k = |R| = {k}")
    print(f"Relevant items in top-{k}: {relevant_in_top_k}")
    print(f"P@{k} = {relevant_in_top_k}/{k} = {precision_at_R:.4f}")
    print(f"R@{k} = {relevant_in_top_k}/{total_relevant} = {recall_at_R:.4f}")
    print(f"R-Precision = {r_prec:.4f}")
    print(f"P@|R| == R@|R|: {abs(precision_at_R - recall_at_R) < 1e-10}")
 
# Example 1: Perfect ranking (all relevant items at top)
print("Example 1: Perfect Ranking")
print("-" * 40)
# 5 relevant items, ranked at positions 1-5
ranked = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
compare_with_precision_recall(ranked, total_relevant=5)
 
print()
 
# Example 2: Imperfect ranking
print("Example 2: Imperfect Ranking")
print("-" * 40)
# 5 relevant items scattered in top 10
ranked = [1, 0, 1, 0, 1, 0, 0, 1, 0, 1]
compare_with_precision_recall(ranked, total_relevant=5)
 
print()
 
# Example 3: Worst case (all relevant after position |R|)
print("Example 3: Worst Ranking")
print("-" * 40)
# 3 relevant items, all ranked after position 3
ranked = [0, 0, 0, 1, 1, 1, 0, 0, 0, 0]
compare_with_precision_recall(ranked, total_relevant=3)

When to Use R-Precision

R-Precision is particularly useful when you want a single-number summary that doesn't require choosing an arbitrary k. It's commonly used in TREC evaluations and when comparing systems that must adapt to queries with varying numbers of relevant documents.

Practical Applications and Domain Considerations

P@k and R@k manifest differently across application domains. Understanding these differences is crucial for selecting appropriate values of k and interpreting results correctly.

Domain-Specific Considerations for P@k and R@k
Domain	Typical k Values	Primary Focus	Key Considerations
Web Search	k=1,3,5,10	P@k (user sees few results)	Position 1 dominates; P@1 often most important
E-commerce Recommendations	k=5,10,20	Balance P@k and R@k	Revenue depends on clicks; explore-exploit tradeoff
Document Retrieval (Legal)	k=100,1000	R@k (completeness vital)	Missing a document could lose a case
Medical Literature	k=20,50,100	High R@k with acceptable P@k	Must not miss relevant studies; false positives cost time
Social Media Feeds	k=10,20,50	P@k (engagement focus)	User attention is limited; stale content penalized
Question Answering	k=1,3,5	P@k, especially P@1	User expects answer in first result

Web Search: The Position 1 Obsession

In web search, P@1 (also called Hit Rate@1) is often the most critical metric. User behavior studies show:

~30-40% of clicks go to position 1
~50% of users never scroll below the fold
The "I'm Feeling Lucky" mentality: users expect the first result to answer their query

For Google and Bing, improving P@1 by even 0.1% translates to millions of better experiences daily.

Recommendation Systems: The Coverage Challenge

In recommendations, there's tension between:

Precision focus: Show only items you're confident the user will like
Recall focus: Expose users to the breadth of relevant content

Overly precision-focused systems create "filter bubbles" where users never discover new content. Systems often track R@50 or R@100 to ensure diversity.

Legal/Medical: The "Can't Miss" Requirement

In high-stakes domains, recall often trumps precision:

In legal discovery, missing a relevant document risks sanctions
In systematic medical reviews, missing a study undermines the entire review

These domains often require R@k ≥ 0.95 for large k, accepting low precision as the cost of completeness.

Choosing k: Practical Guidelines

•Match UI constraints: If your UI shows 10 items per page, evaluate P@10 and R@10
•Consider user behavior: If analytics show users examine ~5 items, focus on P@5
•Include standard benchmarks: Report P@1,3,5,10 for comparability with literature
•Recall requirements drive k: If you need 90% recall, find the minimum k that achieves R@k ≥ 0.9
•Report multiple k values: Single k rarely tells the full story; use P@k curves

Limitations and Extensions

While P@k and R@k are foundational, they have important limitations that motivate more sophisticated metrics.

Limitation 1: Position Blindness Within Top-k

P@k treats all positions within the top-k equally. Whether the 3 relevant items are at positions {1,2,3} or {8,9,10}, P@10 = 0.3 in both cases. But the user experience differs dramatically! This limitation motivates position-weighted metrics like NDCG and MRR.

Limitation 2: Binary Relevance Only

P@k and R@k assume binary relevance (relevant or not). In reality, relevance is often graded:

Highly relevant (perfect match)
Somewhat relevant (partial match)
Marginally relevant (tangentially related)
Not relevant

Graded relevance requires metrics like NDCG (covered in a later page).

Limitation 3: Arbitrary k Selection

The choice of k is somewhat arbitrary. Different k values can lead to different conclusions about which system is better. This motivates:

R-Precision (adaptive k = |R|)
Mean Average Precision (integrates over all k)
Expected Reciprocal Rank (models user stopping behavior)

Limitation 4: No Penalty for Specific Error Types

P@k doesn't distinguish between:

Showing a marginally relevant item vs. a completely irrelevant item
Ranking a highly relevant item at position 5 vs. position 500 (if k < 5, neither counts)

These distinctions require more expressive metrics.

The Metric Zoo

The limitations of P@k and R@k have spawned dozens of alternative ranking metrics. Subsequent pages will cover Mean Average Precision (MAP), NDCG, MRR, and partial AUC—each addressing specific limitations. P@k and R@k remain foundational because they're intuitive, easy to compute, and often sufficiently informative for many applications.

limitation_examples.py
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# Demonstrating limitations of P@k
 
# Example: Same P@k, very different user experience
ranking_a = [1, 1, 1, 0, 0, 0, 0, 0, 0, 0]  # Relevant at positions 1,2,3
ranking_b = [0, 0, 0, 0, 0, 0, 0, 1, 1, 1]  # Relevant at positions 8,9,10
 
# Both have identical P@10
p_at_10_a = sum(ranking_a) / 10  # = 0.3
p_at_10_b = sum(ranking_b) / 10  # = 0.3
 
print("Limitation 1: Position Blindness")
print(f"Ranking A (relevant at top): P@10 = {p_at_10_a}")
print(f"Ranking B (relevant at bottom): P@10 = {p_at_10_b}")
print("Same P@10, but Ranking A is clearly better for users!")
print()
 
# Example: Inconsistent winner at different k
ranking_x = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0]
ranking_y = [0, 0, 1, 1, 1, 0, 0, 0, 0, 0]
 
print("Limitation 3: k-Dependent Conclusions")
for k in [2, 3, 5, 10]:
    p_x = sum(ranking_x[:k]) / k
    p_y = sum(ranking_y[:k]) / k
    winner = "X" if p_x > p_y else ("Y" if p_y > p_x else "Tie")
    print(f"P@{k}: X={p_x:.2f}, Y={p_y:.2f} → Winner: {winner}")
print("Different k values give different winners!")

Summary: Precision@k and Recall@k

We have established Precision@k and Recall@k as the foundational metrics for evaluating ranked retrieval systems. Let's consolidate our understanding:

Key Takeaways

•Precision@k = (relevant in top-k) / k — measures the proportion of shown items that are relevant; answers 'how pure are my results?'
•Recall@k = (relevant in top-k) / |R| — measures the proportion of all relevant items that appear in top-k; answers 'how complete is my coverage?'
•The precision-recall tradeoff applies: larger k typically increases recall while decreasing precision
•R-Precision sets k = |R|, creating a special point where P@|R| = R@|R|
•Domain determines emphasis: search engines prioritize P@k; legal discovery prioritizes R@k
•Limitations exist: P@k ignores position within top-k, assumes binary relevance, and requires arbitrary k selection
•Statistical context matters: always compare against random baseline and report confidence intervals

Mathematical Summary:

Metric	Formula	Range	Monotonic in k?
P@k	$\frac{\sum_{i=1}^{k} \text{rel}_i}{k}$	[0, 1]	No
R@k	$\frac{\sum_{i=1}^{k} \text{rel}_i}{	R	}$
R-Precision	$\frac{\sum_{i=1}^{	R	} \text{rel}_i}{

What's Next:

With P@k and R@k established, we're ready to examine Mean Average Precision (MAP)—a metric that elegantly combines precision at all relevant positions into a single summary statistic, addressing the arbitrary k-selection limitation.

Page Complete

You now understand Precision@k and Recall@k as the foundational metrics for ranking evaluation. This knowledge is prerequisite to understanding more sophisticated metrics like MAP, NDCG, and MRR—each of which builds upon and addresses limitations of these fundamental measures.

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Machine LearningModel Evaluation Metrics

Ranking Metrics

LevelAdvanced

Duration120 mins

TopicModel Evaluation Metrics

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Precision@k and Recall@k

Beyond Classification: The Ranking Paradigm

What You Will Learn

The Ranking Problem Formulation

The Ranking Task:

Key Notation:

Symbol	Meaning
$k$	The cut-off rank (we evaluate only the top-$k$ items)
$R$	The set of all truly relevant items for query $q$
$\text{rel}_i$	Relevance of the item at rank $i$ (shorthand for $\text{rel}(\pi(i), q)$)
$R_k$	Relevant items in the top-$k$: ${\pi(i) : i \leq k \text{ and } \text{rel}_i = 1}$

The ranking problem differs fundamentally from classification in that:

Order matters: The same set of predictions in different orders yields different metric values
Truncation is natural: Users typically examine only the first few results
Context is query-dependent: The same item may be relevant to one query and irrelevant to another

Why Position Matters

Precision@k: Definition and Intuition

Precision@k measures the proportion of relevant items among the top-$k$ ranked items. It answers the question: Of the items we showed the user, how many were actually relevant?

Formal Definition:

$$\text{Precision@}k = \frac{|R_k|}{k} = \frac{\text{Number of relevant items in top-}k}{k}$$

Equivalently, using indicator notation:

$$\text{Precision@}k = \frac{1}{k} \sum_{i=1}^{k} \mathbb{1}[\text{rel}_i = 1]$$

where $\mathbb{1}[\cdot]$ is the indicator function that equals 1 if the condition is true and 0 otherwise.

Properties:

Range: $[0, 1]$ where 1 means all top-$k$ items are relevant
Monotonicity in k: Not guaranteed—P@k can increase or decrease as k grows
Upper bound: $\min(1, |R|/k)$ when $|R| < k$ (can't achieve P@k = 1 if fewer than k relevant items exist)
Best possible: Achieved when all relevant items occupy the top positions

Precision@k Example: Search Results for Query 'machine learning'
Rank	Document	Relevant?	Cumulative Relevant	P@k
1	ML Tutorial	✓ Yes	1	1/1 = 1.000
2	Python Basics	✗ No	1	1/2 = 0.500
3	Deep Learning Guide	✓ Yes	2	2/3 = 0.667
4	Neural Networks	✓ Yes	3	3/4 = 0.750
5	Cooking Recipes	✗ No	3	3/5 = 0.600
6	Data Science Intro	✓ Yes	4	4/6 = 0.667
7	Random Blog Post	✗ No	4	4/7 = 0.571
8	ML Research Paper	✓ Yes	5	5/8 = 0.625
9	Weather Forecast	✗ No	5	5/9 = 0.556
10	Algorithm Book	✓ Yes	6	6/10 = 0.600

Interpreting the Example:

Notice that P@k fluctuates as k increases:

P@1 = 1.0 (perfect—the first result is relevant)
P@2 = 0.5 (drops because position 2 is irrelevant)
P@4 = 0.75 (recovers as more relevant items appear)
P@5 = 0.6 (drops again with an irrelevant result)

The Denominator is Fixed

Recall@k: Definition and Intuition

Formal Definition:

$$\text{Recall@}k = \frac{|R_k|}{|R|} = \frac{\text{Number of relevant items in top-}k}{\text{Total number of relevant items}}$$

Equivalently:

$$\text{Recall@}k = \frac{\sum_{i=1}^{k} \mathbb{1}[\text{rel}_i = 1]}{|R|}$$

Properties:

Range: $[0, 1]$ where 1 means all relevant items appear in the top-$k$
Monotonicity in k: Guaranteed non-decreasing (more positions can only add relevant items, never remove them)
Upper bound: $\min(1, k/|R|)$ when $k < |R|$ (can't achieve R@k = 1 if k is smaller than the number of relevant items)
Dependency on |R|: Unlike P@k, Recall@k explicitly depends on the total number of relevant items

Recall@k Example: Same Search Results (Assume 8 Relevant Documents Exist)
Rank	Document	Relevant?	Cumulative Relevant	R@k (\|R\|=8)
1	ML Tutorial	✓ Yes	1	1/8 = 0.125
2	Python Basics	✗ No	1	1/8 = 0.125
3	Deep Learning Guide	✓ Yes	2	2/8 = 0.250
4	Neural Networks	✓ Yes	3	3/8 = 0.375
5	Cooking Recipes	✗ No	3	3/8 = 0.375
6	Data Science Intro	✓ Yes	4	4/8 = 0.500
7	Random Blog Post	✗ No	4	4/8 = 0.500
8	ML Research Paper	✓ Yes	5	5/8 = 0.625
9	Weather Forecast	✗ No	5	5/8 = 0.625
10	Algorithm Book	✓ Yes	6	6/8 = 0.750

Interpreting the Example:

Unlike P@k, R@k is monotonically non-decreasing:

R@1 = 0.125 (found 1 of 8 relevant items)
R@4 = 0.375 (found 3 of 8)
R@10 = 0.750 (found 6 of 8)

Recall can never decrease as k increases because we can only accumulate more relevant items (or stay flat if the new position is irrelevant).

The Coverage Interpretation:

The R = 0 Edge Case

The Precision-Recall Relationship in Ranking

Precision@k and Recall@k are intimately related through their shared numerator—the count of relevant items in the top-k. This creates a fundamental relationship:

$$\text{Precision@}k \times k = \text{Recall@}k \times |R|$$

Both sides equal $|R_k|$, the number of relevant items found.

The Precision-Recall Tradeoff:

As $k$ increases:

Recall@k can only increase (or stay flat)—we might find more relevant items
Precision@k typically decreases—the denominator grows, and we're forced to include less-relevant items

This creates the classic precision-recall tradeoff:

Small k (conservative): High precision, low recall—we show only items we're very confident about, but miss many relevant items
Large k (aggressive): Low precision, high recall—we cast a wide net and find most relevant items, but include many irrelevant ones

Mathematical Relationship:

We can derive one from the other:

$$\text{Recall@}k = \frac{k \cdot \text{Precision@}k}{|R|}$$

$$\text{Precision@}k = \frac{|R| \cdot \text{Recall@}k}{k}$$

precision_recall_relationship.py
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import numpy as np
import matplotlib.pyplot as plt
 
def compute_precision_recall_at_k(relevance_list: list[int], total_relevant: int):
    """
    Compute P@k and R@k for all values of k.
    
    Args:
        relevance_list: Binary list where 1 = relevant item at that rank
        total_relevant: Total number of relevant items in corpus |R|
    
    Returns:
        Dict with precision@k and recall@k arrays
    """
    relevance = np.array(relevance_list)
    n = len(relevance)
    
    # Cumulative relevant count at each position
    cumulative_relevant = np.cumsum(relevance)
    
    # P@k = cumulative_relevant / k
    k_values = np.arange(1, n + 1)
    precision_at_k = cumulative_relevant / k_values
    
    # R@k = cumulative_relevant / |R|
    recall_at_k = cumulative_relevant / total_relevant if total_relevant > 0 else np.zeros(n)
    
    return {
        'k': k_values,
        'precision': precision_at_k,
        'recall': recall_at_k,
        'cumulative_relevant': cumulative_relevant
    }
 
# Example: Ranked list with relevance judgments
# 1 = relevant, 0 = not relevant
ranked_relevance = [1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1]
total_relevant = 10  # Total relevant items in corpus
 
results = compute_precision_recall_at_k(ranked_relevance, total_relevant)
 
# Display results
print("Precision@k and Recall@k Analysis")
print("=" * 50)
print(f"Total items ranked: {len(ranked_relevance)}")
print(f"Total relevant in corpus: {total_relevant}")
print(f"Relevant in ranked list: {sum(ranked_relevance)}")
print()
 
print(f"{'k':>3} | {'Rel@k':>5} | {'P@k':>6} | {'R@k':>6}")
print("-" * 30)
for k in [1, 3, 5, 10, 15, 20]:
    if k <= len(ranked_relevance):
        print(f"{k:>3} | {int(results['cumulative_relevant'][k-1]):>5} | "
              f"{results['precision'][k-1]:>6.3f} | {results['recall'][k-1]:>6.3f}")
 
# Verify the relationship: P@k * k = R@k * |R|
print("\nVerifying P@k × k = R@k × |R|:")
for k in [5, 10]:
    lhs = results['precision'][k-1] * k
    rhs = results['recall'][k-1] * total_relevant
    print(f"k={k}: P@k × k = {lhs:.2f}, R@k × |R| = {rhs:.2f}")

Visualizing the Tradeoff:

High precision at low recall: At small k, we're selective
Declining precision as recall grows: As k increases, we include weaker candidates
Plateaus: When consecutive ranks are irrelevant, recall stays flat while precision drops
Steps: When a relevant item is found, recall jumps and precision may recover slightly

Computational Implementation

Implementing P@k and R@k correctly requires attention to several practical considerations. Let's build a robust implementation from first principles.

Input Representations:

Ranking metrics can be computed from various input formats:

Explicit ranking: A list of item IDs in ranked order + a set of relevant item IDs
Score-based: A list of (item_id, score, relevance) tuples, sorted by score
Binary relevance vector: A vector where position indicates rank and value indicates relevance

Algorithm Complexity:

Operation	Time Complexity	Space Complexity
P@k from relevance list	O(k)	O(1)
R@k from relevance list	O(k)	O(1)
All P@k, R@k for k=1..n	O(n)	O(n)
Building from item IDs	O(k) with hash set for relevant	O(

precision_recall_implementation.py
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import numpy as np
from typing import List, Set, Optional, Dict, Any
 
class RankingMetrics:
    """
    Production-grade implementation of ranking metrics.
    Handles edge cases, supports multiple input formats, and provides
    comprehensive metric computation.
    """
    
    def __init__(self, relevant_items: Set[Any], ranked_items: List[Any]):
        """
        Initialize with ground truth and ranked predictions.
        
        Args:
            relevant_items: Set of item IDs that are relevant
            ranked_items: List of item IDs in ranked order (position 0 = rank 1)
        """
        self.relevant_items = set(relevant_items)
        self.ranked_items = ranked_items
        self.n_relevant = len(self.relevant_items)
        self.n_ranked = len(ranked_items)
        
        # Precompute binary relevance array for efficiency
        self._relevance_array = np.array([
            1 if item in self.relevant_items else 0 
            for item in ranked_items
        ])
        
        # Precompute cumulative relevance
        self._cumulative_relevant = np.cumsum(self._relevance_array)
    
    def precision_at_k(self, k: int) -> float:
        """
        Compute Precision@k.
        
        Args:
            k: Cutoff rank (1-indexed conceptually, but we handle internally)
        
        Returns:
            Precision@k value in [0, 1]
        
        Raises:
            ValueError: If k < 1
        """
        if k < 1:
            raise ValueError(f"k must be >= 1, got {k}")
        
        # Handle k larger than our ranked list
        effective_k = min(k, self.n_ranked)
        
        if effective_k == 0:
            return 0.0
        
        relevant_in_top_k = self._cumulative_relevant[effective_k - 1]
        return float(relevant_in_top_k) / k  # Note: divide by k, not effective_k
    
    def recall_at_k(self, k: int) -> Optional[float]:
        """
        Compute Recall@k.
        
        Args:
            k: Cutoff rank
        
        Returns:
            Recall@k value in [0, 1], or None if no relevant items exist
        """
        if k < 1:
            raise ValueError(f"k must be >= 1, got {k}")
        
        if self.n_relevant == 0:
            return None  # Undefined when no relevant items
        
        effective_k = min(k, self.n_ranked)
        
        if effective_k == 0:
            return 0.0
        
        relevant_in_top_k = self._cumulative_relevant[effective_k - 1]
        return float(relevant_in_top_k) / self.n_relevant
    
    def precision_recall_at_all_k(self) -> Dict[str, np.ndarray]:
        """
        Compute P@k and R@k for all k from 1 to n_ranked.
        
        Returns:
            Dictionary with 'k', 'precision', 'recall', 'f1' arrays
        """
        if self.n_ranked == 0:
            return {
                'k': np.array([]),
                'precision': np.array([]),
                'recall': np.array([]),
                'f1': np.array([])
            }
        
        k_values = np.arange(1, self.n_ranked + 1)
        precision = self._cumulative_relevant / k_values
        
        if self.n_relevant > 0:
            recall = self._cumulative_relevant / self.n_relevant
        else:
            recall = np.zeros(self.n_ranked)
        
        # F1@k = harmonic mean of P@k and R@k
        with np.errstate(divide='ignore', invalid='ignore'):
            f1 = 2 * precision * recall / (precision + recall)
            f1 = np.nan_to_num(f1)  # Handle 0/0 cases
        
        return {
            'k': k_values,
            'precision': precision,
            'recall': recall,
            'f1': f1
        }
    
    def r_precision(self) -> float:
        """
        Compute R-Precision: Precision at k=|R|.
        A single-number summary that balances precision and recall.
        """
        if self.n_relevant == 0:
            return 0.0
        return self.precision_at_k(self.n_relevant)
 
 
# Example usage with realistic data
if __name__ == "__main__":
    # Scenario: Product search results
    # User searched for "wireless headphones"
    
    relevant_products = {
        'sony_wh1000xm4', 'bose_qc45', 'apple_airpods_max',
        'sennheiser_momentum', 'jabra_elite_85h', 'audio_technica_m50x'
    }  # 6 relevant products
    
    # Model's ranked output (top 15 results)
    search_results = [
        'sony_wh1000xm4',      # Rank 1: Relevant ✓
        'cheap_wired_earbuds', # Rank 2: Not relevant
        'bose_qc45',           # Rank 3: Relevant ✓
        'phone_case',          # Rank 4: Not relevant
        'apple_airpods_max',   # Rank 5: Relevant ✓
        'usb_cable',           # Rank 6: Not relevant
        'jabra_elite_85h',     # Rank 7: Relevant ✓
        'random_speaker',      # Rank 8: Not relevant
        'bluetooth_adapter',   # Rank 9: Not relevant
        'sennheiser_momentum', # Rank 10: Relevant ✓
        'gaming_headset',      # Rank 11: Not relevant
        'microphone',          # Rank 12: Not relevant
        'audio_technica_m50x', # Rank 13: Relevant ✓
        'laptop_stand',        # Rank 14: Not relevant
        'mouse_pad',           # Rank 15: Not relevant
    ]
    
    metrics = RankingMetrics(relevant_products, search_results)
    
    print("Product Search Ranking Evaluation")
    print("=" * 50)
    print(f"Total relevant products: {len(relevant_products)}")
    print(f"Products in ranked list: {len(search_results)}")
    print()
    
    # Report key cutoffs
    for k in [1, 3, 5, 10]:
        p_at_k = metrics.precision_at_k(k)
        r_at_k = metrics.recall_at_k(k)
        print(f"P@{k:2d} = {p_at_k:.3f}, R@{k:2d} = {r_at_k:.3f}")
    
    print(f"\nR-Precision (P@{len(relevant_products)}): {metrics.r_precision():.3f}")

Statistical Properties and Interpretation

Understanding the statistical properties of P@k and R@k is essential for proper interpretation and comparison.

Expected Values Under Random Ranking:

If items are ranked uniformly at random, what are the expected values of P@k and R@k?

Let $\rho = |R|/N$ be the base rate—the proportion of relevant items in the corpus.

$$\mathbb{E}[\text{P@}k] = \rho$$ $$\mathbb{E}[\text{R@}k] = \frac{k}{N}$$ (approximately, for large N)

A good ranker should significantly outperform these baselines. If your P@10 equals the base rate ρ, your model is no better than random.

Variance and Confidence:

The variance of P@k (under random ranking) follows a binomial distribution:

$$\text{Var}[\text{P@}k] = \frac{\rho(1-\rho)}{k}$$

This has important implications:

Larger k reduces variance: P@100 is more stable than P@5
Extreme base rates increase variance: Very rare or very common relevant items lead to higher variance
Small k requires caution: P@1 can swing dramatically between queries

Interpretation Guide for Precision@k Values
P@k Range	Interpretation	Typical Action
0.9 - 1.0	Excellent: Almost all shown items are relevant	Possibly too conservative; check recall
0.7 - 0.9	Good: Strong majority of results are relevant	Solid performance for most applications
0.5 - 0.7	Moderate: About half the results are useful	Room for improvement; check feature quality
0.3 - 0.5	Weak: More noise than signal	Significant model improvements needed
< 0.3	Poor: Mostly irrelevant results	Fundamental issues with ranking model

Context Matters

Averaging Across Queries:

In practice, we evaluate a ranking system across many queries and report average metrics:

$$\text{Mean P@}k = \frac{1}{|Q|} \sum_{q \in Q} \text{P@}k(q)$$

$$\text{Mean R@}k = \frac{1}{|Q|} \sum_{q \in Q} \text{R@}k(q)$$

Statistical Significance:

When comparing two ranking systems, use appropriate statistical tests:

Paired t-test: For comparing Mean P@k across systems
Bootstrap confidence intervals: More robust for non-normal distributions
Two-sided binomial test: For P@1 comparisons (is one system better at the top position?)

A typical threshold is p < 0.05 for claiming one system is significantly better.

R-Precision: Bridging Precision and Recall

R-Precision is a special case of Precision@k where $k$ is set to $|R|$, the total number of relevant items for the query:

$$\text{R-Precision} = \text{Precision@}|R| = \frac{|R_{|R|}|}{|R|}$$

Why R-Precision is Elegant:

At $k = |R|$, something mathematically beautiful happens:

$$\text{R-Precision} = \text{Precision@}|R| = \text{Recall@}|R|$$

Proof: At k = |R|:

Precision@|R| = (relevant in top-|R|) / |R|
Recall@|R| = (relevant in top-|R|) / |R|

Both reduce to the same quantity!

Interpretation:

R-Precision asks: "If you could only show |R| items (the exact number that are relevant), how many of those would be correct?" This is a natural performance measure because:

It adapts to query difficulty (queries with more relevant items use larger k)
It equals both precision and recall at this special cutoff
It ranges from 0 (no relevant items in top-|R|) to 1 (all relevant items in top-|R|, perfectly ranked)

r_precision_analysis.py
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import numpy as np
 
def r_precision(ranked_relevance: list[int], total_relevant: int) -> float:
    """
    Compute R-Precision: Precision at k = |R|.
    
    Args:
        ranked_relevance: Binary relevance of items in ranked order
        total_relevant: Total number of relevant items |R|
    
    Returns:
        R-Precision value
    """
    if total_relevant == 0:
        return 0.0
    
    # Look at top |R| positions
    k = total_relevant
    top_k = ranked_relevance[:k] if k <= len(ranked_relevance) else ranked_relevance
    
    # Count relevant in top k
    relevant_found = sum(top_k)
    
    return relevant_found / total_relevant
 
def compare_with_precision_recall(ranked_relevance: list[int], total_relevant: int):
    """
    Demonstrate that R-Precision = P@|R| = R@|R|
    """
    k = total_relevant
    relevant_in_top_k = sum(ranked_relevance[:k]) if k <= len(ranked_relevance) else sum(ranked_relevance)
    
    precision_at_R = relevant_in_top_k / k if k > 0 else 0
    recall_at_R = relevant_in_top_k / total_relevant if total_relevant > 0 else 0
    r_prec = r_precision(ranked_relevance, total_relevant)
    
    print(f"k = |R| = {k}")
    print(f"Relevant items in top-{k}: {relevant_in_top_k}")
    print(f"P@{k} = {relevant_in_top_k}/{k} = {precision_at_R:.4f}")
    print(f"R@{k} = {relevant_in_top_k}/{total_relevant} = {recall_at_R:.4f}")
    print(f"R-Precision = {r_prec:.4f}")
    print(f"P@|R| == R@|R|: {abs(precision_at_R - recall_at_R) < 1e-10}")
 
# Example 1: Perfect ranking (all relevant items at top)
print("Example 1: Perfect Ranking")
print("-" * 40)
# 5 relevant items, ranked at positions 1-5
ranked = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]
compare_with_precision_recall(ranked, total_relevant=5)
 
print()
 
# Example 2: Imperfect ranking
print("Example 2: Imperfect Ranking")
print("-" * 40)
# 5 relevant items scattered in top 10
ranked = [1, 0, 1, 0, 1, 0, 0, 1, 0, 1]
compare_with_precision_recall(ranked, total_relevant=5)
 
print()
 
# Example 3: Worst case (all relevant after position |R|)
print("Example 3: Worst Ranking")
print("-" * 40)
# 3 relevant items, all ranked after position 3
ranked = [0, 0, 0, 1, 1, 1, 0, 0, 0, 0]
compare_with_precision_recall(ranked, total_relevant=3)

When to Use R-Precision

Practical Applications and Domain Considerations

P@k and R@k manifest differently across application domains. Understanding these differences is crucial for selecting appropriate values of k and interpreting results correctly.

Domain-Specific Considerations for P@k and R@k
Domain	Typical k Values	Primary Focus	Key Considerations
Web Search	k=1,3,5,10	P@k (user sees few results)	Position 1 dominates; P@1 often most important
E-commerce Recommendations	k=5,10,20	Balance P@k and R@k	Revenue depends on clicks; explore-exploit tradeoff
Document Retrieval (Legal)	k=100,1000	R@k (completeness vital)	Missing a document could lose a case
Medical Literature	k=20,50,100	High R@k with acceptable P@k	Must not miss relevant studies; false positives cost time
Social Media Feeds	k=10,20,50	P@k (engagement focus)	User attention is limited; stale content penalized
Question Answering	k=1,3,5	P@k, especially P@1	User expects answer in first result

Web Search: The Position 1 Obsession

In web search, P@1 (also called Hit Rate@1) is often the most critical metric. User behavior studies show:

~30-40% of clicks go to position 1
~50% of users never scroll below the fold
The "I'm Feeling Lucky" mentality: users expect the first result to answer their query

For Google and Bing, improving P@1 by even 0.1% translates to millions of better experiences daily.

Recommendation Systems: The Coverage Challenge

In recommendations, there's tension between:

Precision focus: Show only items you're confident the user will like
Recall focus: Expose users to the breadth of relevant content

Overly precision-focused systems create "filter bubbles" where users never discover new content. Systems often track R@50 or R@100 to ensure diversity.

Legal/Medical: The "Can't Miss" Requirement

In high-stakes domains, recall often trumps precision:

In legal discovery, missing a relevant document risks sanctions
In systematic medical reviews, missing a study undermines the entire review

These domains often require R@k ≥ 0.95 for large k, accepting low precision as the cost of completeness.

Choosing k: Practical Guidelines

•Match UI constraints: If your UI shows 10 items per page, evaluate P@10 and R@10
•Consider user behavior: If analytics show users examine ~5 items, focus on P@5
•Include standard benchmarks: Report P@1,3,5,10 for comparability with literature
•Recall requirements drive k: If you need 90% recall, find the minimum k that achieves R@k ≥ 0.9
•Report multiple k values: Single k rarely tells the full story; use P@k curves

Limitations and Extensions

While P@k and R@k are foundational, they have important limitations that motivate more sophisticated metrics.

Limitation 1: Position Blindness Within Top-k

Limitation 2: Binary Relevance Only

P@k and R@k assume binary relevance (relevant or not). In reality, relevance is often graded:

Highly relevant (perfect match)
Somewhat relevant (partial match)
Marginally relevant (tangentially related)
Not relevant

Graded relevance requires metrics like NDCG (covered in a later page).

Limitation 3: Arbitrary k Selection

The choice of k is somewhat arbitrary. Different k values can lead to different conclusions about which system is better. This motivates:

R-Precision (adaptive k = |R|)
Mean Average Precision (integrates over all k)
Expected Reciprocal Rank (models user stopping behavior)

Limitation 4: No Penalty for Specific Error Types

P@k doesn't distinguish between:

Showing a marginally relevant item vs. a completely irrelevant item
Ranking a highly relevant item at position 5 vs. position 500 (if k < 5, neither counts)

These distinctions require more expressive metrics.

The Metric Zoo

limitation_examples.py
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# Demonstrating limitations of P@k
 
# Example: Same P@k, very different user experience
ranking_a = [1, 1, 1, 0, 0, 0, 0, 0, 0, 0]  # Relevant at positions 1,2,3
ranking_b = [0, 0, 0, 0, 0, 0, 0, 1, 1, 1]  # Relevant at positions 8,9,10
 
# Both have identical P@10
p_at_10_a = sum(ranking_a) / 10  # = 0.3
p_at_10_b = sum(ranking_b) / 10  # = 0.3
 
print("Limitation 1: Position Blindness")
print(f"Ranking A (relevant at top): P@10 = {p_at_10_a}")
print(f"Ranking B (relevant at bottom): P@10 = {p_at_10_b}")
print("Same P@10, but Ranking A is clearly better for users!")
print()
 
# Example: Inconsistent winner at different k
ranking_x = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0]
ranking_y = [0, 0, 1, 1, 1, 0, 0, 0, 0, 0]
 
print("Limitation 3: k-Dependent Conclusions")
for k in [2, 3, 5, 10]:
    p_x = sum(ranking_x[:k]) / k
    p_y = sum(ranking_y[:k]) / k
    winner = "X" if p_x > p_y else ("Y" if p_y > p_x else "Tie")
    print(f"P@{k}: X={p_x:.2f}, Y={p_y:.2f} → Winner: {winner}")
print("Different k values give different winners!")

Summary: Precision@k and Recall@k

We have established Precision@k and Recall@k as the foundational metrics for evaluating ranked retrieval systems. Let's consolidate our understanding:

Key Takeaways

•Precision@k = (relevant in top-k) / k — measures the proportion of shown items that are relevant; answers 'how pure are my results?'
•Recall@k = (relevant in top-k) / |R| — measures the proportion of all relevant items that appear in top-k; answers 'how complete is my coverage?'
•The precision-recall tradeoff applies: larger k typically increases recall while decreasing precision
•R-Precision sets k = |R|, creating a special point where P@|R| = R@|R|
•Domain determines emphasis: search engines prioritize P@k; legal discovery prioritizes R@k
•Limitations exist: P@k ignores position within top-k, assumes binary relevance, and requires arbitrary k selection
•Statistical context matters: always compare against random baseline and report confidence intervals

Mathematical Summary:

Metric	Formula	Range	Monotonic in k?
P@k	$\frac{\sum_{i=1}^{k} \text{rel}_i}{k}$	[0, 1]	No
R@k	$\frac{\sum_{i=1}^{k} \text{rel}_i}{	R	}$
R-Precision	$\frac{\sum_{i=1}^{	R	} \text{rel}_i}{

What's Next:

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