---
id: wiki-2026-0508-pmi-technique
title: PMI Technique (Pointwise Mutual Information)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [PMI, Pointwise Mutual Information, PPMI, Word Association]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [nlp, pmi, statistics, collocation, embeddings, information-theory]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack: { language: python, framework: numpy/scikit-learn }
---

# PMI Technique (Pointwise Mutual Information)

## 한 줄
두 사건이 독립일 때 대비 얼마나 더 함께 등장하는가를 로그-비율로 측정하는 점별 상호정보량 — NLP collocation·연관 측정의 기초.

## 핵심
- 정의: `PMI(x, y) = log( P(x, y) / (P(x) P(y)) )`.
- > 0: 양의 연관, = 0: 독립, < 0: 음의 연관(희귀, 노이즈 많음).
- **PPMI** = max(PMI, 0) — 음수 절단으로 안정.
- **NPMI** = `PMI / -log P(x, y)` ∈ [-1, 1], 빈도 편향 완화.
- **k-shift PMI**: SGNS(word2vec)는 implicit하게 `PMI - log k` 인수분해(Levy & Goldberg).
- 활용: collocation 추출, topic 평가(Coherence_NPMI), 워드 임베딩 baseline(SVD on PPMI), feature selection, RAG retrieval re-ranking.
- 단점: 저빈도 쌍이 PMI 폭증 → 빈도 임계 / shift / NPMI 필요.

## 💻 패턴

```python
# 1. PMI 직접 계산 (co-occurrence matrix)
import numpy as np
from collections import Counter

corpus = "the cat sat on the mat the cat purred the dog ran".split()
window = 2
pair_c, word_c = Counter(), Counter()

for i, w in enumerate(corpus):
    word_c[w] += 1
    for j in range(max(0, i-window), min(len(corpus), i+window+1)):
        if i != j:
            pair_c[(w, corpus[j])] += 1

total_pairs = sum(pair_c.values())
total_words = sum(word_c.values())

def pmi(x, y):
    p_xy = pair_c[(x, y)] / total_pairs
    p_x  = word_c[x] / total_words
    p_y  = word_c[y] / total_words
    return np.log2(p_xy / (p_x * p_y))

print(f"PMI(cat, sat) = {pmi('cat','sat'):.3f}")
```

```python
# 2. PPMI matrix (전체 어휘) — sparse
import numpy as np
from scipy.sparse import csr_matrix

def build_ppmi(pair_c, word_c, vocab):
    idx = {w: i for i, w in enumerate(vocab)}
    rows, cols, data = [], [], []
    N = sum(pair_c.values())
    Nw = sum(word_c.values())
    for (a, b), c in pair_c.items():
        p_ab = c / N
        p_a, p_b = word_c[a] / Nw, word_c[b] / Nw
        v = np.log2(p_ab / (p_a * p_b))
        if v > 0:
            rows.append(idx[a]); cols.append(idx[b]); data.append(v)
    return csr_matrix((data, (rows, cols)), shape=(len(vocab), len(vocab)))

vocab = sorted(word_c)
M = build_ppmi(pair_c, word_c, vocab)
```

```python
# 3. NPMI (정규화)
def npmi(x, y):
    p_xy = pair_c[(x, y)] / total_pairs
    p_x  = word_c[x] / total_words
    p_y  = word_c[y] / total_words
    if p_xy == 0: return -1
    return np.log2(p_xy / (p_x * p_y)) / -np.log2(p_xy)
```

```python
# 4. SVD on PPMI → low-rank word embeddings (count-based)
from scipy.sparse.linalg import svds
import numpy as np

U, s, Vt = svds(M.astype(float), k=100)
emb = U * np.sqrt(s)        # 100-d static embedding per word
# 코사인 유사도로 nearest-word 검색 가능
```

```python
# 5. gensim Phrases — bigram collocation by NPMI
from gensim.models.phrases import Phrases, Phraser

sents = [["new", "york", "city"], ["machine", "learning", "is", "fun"], ...]
bigram = Phrases(sents, min_count=5, threshold=0.5,
                 scoring="npmi")  # threshold ∈ [-1,1]
phraser = Phraser(bigram)
print(phraser[["new", "york", "is", "big"]])
# ['new_york', 'is', 'big']
```

```python
# 6. Topic Coherence (NPMI 기반) — 토픽 모델 품질
from gensim.models import CoherenceModel

cm = CoherenceModel(topics=top_words_per_topic,
                    texts=tokenized_corpus,
                    dictionary=dictionary,
                    coherence="c_npmi")
print("c_npmi:", cm.get_coherence())
```

```python
# 7. PMI for feature selection (text classification)
import numpy as np

def pmi_feature(word, label, df):
    p_wl = ((df["word"] == word) & (df["label"] == label)).mean()
    p_w  = (df["word"] == word).mean()
    p_l  = (df["label"] == label).mean()
    if p_wl == 0: return 0
    return np.log2(p_wl / (p_w * p_l))

# 라벨별 top-PMI 단어 = 강한 신호 feature
```

```python
# 8. Shifted PMI (word2vec SGNS와 동치성)
import numpy as np
def spmi(x, y, k=5):
    p_xy = pair_c[(x, y)] / total_pairs
    p_x  = word_c[x] / total_words
    p_y  = word_c[y] / total_words
    return np.log2(p_xy / (p_x * p_y)) - np.log2(k)
# Levy & Goldberg 2014: SGNS ≈ matrix factorization of shifted PMI
```

## 결정 기준

| 목표 | 권장 |
|---|---|
| Collocation 추출 | NPMI + 빈도 임계(min_count) |
| 토픽 모델 품질 평가 | c_npmi |
| Static word embedding (small data) | SVD on PPMI |
| Feature selection (분류) | PMI(word, class) |
| word2vec 이론 연결 | shifted PMI (k=5~15) |
| Modern semantic search | sentence embedding(BGE/E5) — PMI는 보조 |

## 🔗 Graph
- Related: `[[Word-Embeddings]]`, `[[Word2Vec]]`, `[[Information-Theory]]`, `[[Topic-Modeling]]`, `[[Collocation-Extraction]]`, `[[TF-IDF]]`

## 🤖 LLM 활용
- LLM 출력 다양성 측정: 생성 토큰 쌍의 NPMI 분포로 반복도 평가.
- RAG 후보 청크 키워드와 query 키워드 간 PMI로 lexical overlap 점수 보강.

## ❌ 안티패턴
- 저빈도 쌍(예: 1회 등장)을 그대로 PMI 산출 → 인공적으로 큰 값.
- log-base 혼용(자연로그 vs log2) — 비교 불가.
- PPMI 없이 raw PMI를 SVD에 넣어 음수 노이즈 학습.
- topic coherence c_v 대신 c_npmi가 더 인간 판단과 상관 높음을 무시.

## 🧪 검증
- 알려진 collocation 쌍("New York", "machine learning")이 상위 NPMI 차지하는지 확인.
- PPMI-SVD 임베딩으로 analogy(king-man+woman≈queen) 부분 작동.
- coherence c_npmi 값이 0.1~0.3 범위면 표준적 토픽 품질.

## 🕓 Changelog
- 2026-05-08 Phase 1: 초안.
- 2026-05-10 Manual cleanup: 8 패턴, NPMI/SGNS shift/coherence 보강.
