[G1-Sync] Manual knowledge update
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Antigravity Agent
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id: wiki-2026-0508-markov-random-fields
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title: Markov Random Fields
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category: 10_Wiki/Topics
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status: needs_review
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status: verified
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canonical_id: self
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aliases: [P-REINFORCE-AUTO-B14FE1]
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aliases: [MRF, Undirected Graphical Model, Markov Network, Gibbs Random Field]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.9
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tags: [auto-reinforced]
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verification_status: applied
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tags: [machine-learning, probability, graphical-model, vision, statistics]
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raw_sources: []
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last_reinforced: 2026-04-20
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github_commit: "[P-Reinforce] Continuous Worker - Markov-Random-Fields"
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inferred_by: Claude Opus 4.7 (auto-normalize 2026-05-08)
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Python
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framework: PyTorch/NetworkX
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---
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# [[Markov-Random-Fields]]
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# Markov Random Fields
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## 📌 한 줄 통찰 (The Karpathy Summary)
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> 지식 요약 정보 추출 중...
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## 매 한 줄
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> **"매 undirected graph 의 joint distribution — local Markov property 의 satisfy"**. 매 Bayes net 의 directed counterpart — 매 conditional independence 의 graph separation 으로 read. Hammersley–Clifford (1971) 의 Gibbs distribution 의 equivalence — 2026 매 image segmentation, CRF, energy-based model (EBM) 의 underlie.
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## 📖 구조화된 지식 (Synthesized Content)
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본문 구조화 작업 중...
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## 매 핵심
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## ⚠️ 모순 및 업데이트 (Contradictions & Updates)
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- **과거 데이터와의 충돌:** 자동화 엔진에 의해 매핑된 지식으로, 추후 정밀 검증 필요.
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- **정책 변화:** General Knowledge 분야의 자동 자산화 수행.
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### 매 정의
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- **Graph** $G = (V, E)$, 매 node 의 random variable $X_v$.
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- **Local Markov**: $X_v \perp X_{V \setminus N[v]} \mid X_{N(v)}$ — 매 node 의 neighbors 의 condition 시 의 rest 의 independent.
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- **Hammersley–Clifford**: 매 strictly positive joint 의 매 Gibbs form $P(x) = \frac{1}{Z} \prod_{C \in \mathcal{C}} \psi_C(x_C)$ — 매 clique 의 product.
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- **Partition function**: $Z = \sum_x \prod_C \psi_C(x_C)$ — 매 intractable in general.
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## 🔗 지식 연결 (Graph)
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- Raw Source: [[00_Raw/2026-04-20/Markov-Random-Fields.md]]
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---
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### 매 inference
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- **Exact**: tree (sum-product / belief propagation).
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- **Loopy BP**: 매 approximate, 매 often works.
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- **MCMC**: Gibbs sampling — 매 conditional 의 sample 매 cycle.
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- **Variational**: mean-field — 매 factorized $q(x) = \prod_v q_v(x_v)$.
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- **Graph cut**: 매 binary submodular — 매 exact min-cut.
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## 🤖 LLM 활용 힌트 (How to Use This Knowledge)
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### 매 응용
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1. Image segmentation (foreground/background MRF).
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2. Conditional Random Fields (CRF) — sequence labeling.
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3. Stereo / depth estimation (smoothness prior).
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4. Energy-based generative models (EBMs).
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5. Statistical physics (Ising, Potts).
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**언제 이 지식을 쓰는가:**
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- *(TODO)*
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## 💻 패턴
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**언제 쓰면 안 되는가:**
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- *(TODO)*
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### Ising model — Gibbs sampling
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```python
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import numpy as np
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## 🧪 검증 상태 (Validation)
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def ising_gibbs(N=64, beta=0.44, steps=10_000, h=0.0):
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spins = np.random.choice([-1, 1], size=(N, N))
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for _ in range(steps):
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i, j = np.random.randint(N, size=2)
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nb = (spins[(i+1)%N, j] + spins[(i-1)%N, j]
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+ spins[i, (j+1)%N] + spins[i, (j-1)%N])
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dE = 2 * spins[i, j] * (beta * nb + h)
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if dE < 0 or np.random.random() < np.exp(-dE):
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spins[i, j] *= -1
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return spins
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```
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- **정보 상태:** needs_review
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- **출처 신뢰도:** A
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- **검토 이유:** *(P-Reinforce Phase 1 자동 정규화. 본문 검증 필요.)*
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### Loopy belief propagation (binary pairwise)
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```python
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def loopy_bp(unary, pairwise, edges, iters=20):
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# unary[v] : log-potential per node, shape (V, K)
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# pairwise : (E, K, K) log-potentials
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# edges : list of (u, v)
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msg = {(u, v): np.zeros(unary.shape[1]) for u, v in edges}
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msg.update({(v, u): np.zeros(unary.shape[1]) for u, v in edges})
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for _ in range(iters):
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new = {}
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for (u, v), e_idx in zip(edges, range(len(edges))):
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incoming = unary[u] + sum(msg[(w, u)] for w in neighbors(u) if w != v)
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new[(u, v)] = np.logaddexp.reduce(
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pairwise[e_idx] + incoming[:, None], axis=0)
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new[(u, v)] -= new[(u, v)].max() # normalize
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msg.update(new)
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beliefs = np.array([
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unary[v] + sum(msg[(u, v)] for u in neighbors(v))
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for v in range(len(unary))])
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return beliefs
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```
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## 🧬 중복 검사 (Duplicate Check)
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### Linear-chain CRF (PyTorch — sequence labeling)
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```python
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import torch, torch.nn as nn
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- **기존 유사 문서:** *(TODO: 인덱서 클러스터 리포트 참조)*
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- **처리 방식:** UPDATE (자동 정규화)
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- **처리 이유:** Phase 1 정규화 — 옛 템플릿/누락 필드 보강.
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class LinearChainCRF(nn.Module):
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def __init__(self, n_tags):
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super().__init__()
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self.trans = nn.Parameter(torch.randn(n_tags, n_tags))
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## 🕓 변경 이력 (Changelog)
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def log_partition(self, emissions): # (T, K)
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T, K = emissions.shape
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alpha = emissions[0]
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for t in range(1, T):
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alpha = torch.logsumexp(
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alpha[:, None] + self.trans + emissions[t][None, :], dim=0)
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return torch.logsumexp(alpha, dim=0)
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| 날짜 | 변경 내용 | 처리 방식 | 신뢰도 |
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|------|-----------|-----------|--------|
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| 2026-05-08 | P-Reinforce Phase 1 정규화 (frontmatter + 헤더 표준화) | UPDATE | A |
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def score(self, emissions, tags):
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s = emissions[0, tags[0]]
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for t in range(1, len(tags)):
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s = s + self.trans[tags[t-1], tags[t]] + emissions[t, tags[t]]
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return s
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def nll(self, emissions, tags):
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return self.log_partition(emissions) - self.score(emissions, tags)
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```
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### Graph cut for binary MRF (submodular)
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```python
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import maxflow # PyMaxflow
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def binary_mrf_graph_cut(unary_fg, unary_bg, pairwise_w):
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H, W = unary_fg.shape
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g = maxflow.Graph[float]()
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nodes = g.add_grid_nodes((H, W))
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g.add_grid_edges(nodes, pairwise_w) # smoothness
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g.add_grid_tedges(nodes, unary_fg, unary_bg) # data term
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g.maxflow()
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return g.get_grid_segments(nodes) # bool mask
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```
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### Mean-field VI
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```python
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def mean_field(unary, pairwise, iters=10):
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# q(x_v = k) ∝ exp(unary[v,k] + Σ_{u∈N(v)} Σ_l q(x_u=l) * pairwise[v,u,k,l])
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q = torch.softmax(unary, dim=-1)
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for _ in range(iters):
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msg = torch.einsum('uvkl,ul->vk', pairwise, q)
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q = torch.softmax(unary + msg, dim=-1)
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return q
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| Tree-structured | Sum-product (exact) |
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| Loopy graph, fast | Loopy BP |
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| Loopy graph, accurate | MCMC (Gibbs) — slower |
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| Binary submodular | Graph cut (exact min-cut) |
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| Sequence labeling (NER) | Linear-chain CRF |
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| Image segmentation | Pairwise MRF + α-expansion / DenseCRF |
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| Modern generative | Energy-Based Model (EBM) — score matching |
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**기본값**: smallest model 의 first — chain → tree → loopy + BP → MCMC.
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## 🔗 Graph
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- 부모: [[Probabilistic Graphical Models]] · [[Probability Theory]]
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- 변형: [[Conditional Random Fields]] · [[Ising Model]] · [[Boltzmann Machine]]
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- 응용: [[Image Segmentation]] · [[Sequence Labeling]] · [[Energy-Based Model]]
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- Adjacent: [[Bayesian Network]] · [[Belief Propagation]] · [[Hammersley-Clifford]]
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## 🤖 LLM 활용
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**언제**: clique factorization 의 derive, BP/Gibbs pseudocode, partition-function intractability 의 explain.
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**언제 X**: 매 specific paper algorithm — original 의 의 cross-check.
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## ❌ 안티패턴
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- **Bayes net 의 mental model 의 reuse**: 매 directionality 의 X — separation criterion 매 different.
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- **Computing $Z$ for large graph**: 매 #P-hard — variational / MCMC.
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- **Loopy BP on tightly-loopy graph**: 매 may diverge — damping 의 try, MCMC 의 fallback.
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- **Linear-chain CRF 의 LSTM 으로 always replace**: 매 small data 의 still wins, 매 calibration 의 better.
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- **Mean-field 의 multimodal posterior 의 use**: 매 mode-seeking — 매 underestimate variance.
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## 🧪 검증 / 중복
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- Verified (Hammersley & Clifford 1971; Koller & Friedman _PGM_ 2009; Murphy _PML_ 2022).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — MRF basics + BP/CRF/graph-cut 정리 |
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