---
id: wiki-2026-0508-markov-random-fields
title: Markov Random Fields
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [MRF, Undirected Graphical Model, Markov Network, Gibbs Random Field]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [machine-learning, probability, graphical-model, vision, statistics]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: PyTorch/NetworkX
---

# Markov Random Fields

## 매 한 줄
> **"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.

## 매 핵심

### 매 정의
- **Graph** $G = (V, E)$, 매 node 의 random variable $X_v$.
- **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.
- **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.
- **Partition function**: $Z = \sum_x \prod_C \psi_C(x_C)$ — 매 intractable in general.

### 매 inference
- **Exact**: tree (sum-product / belief propagation).
- **Loopy BP**: 매 approximate, 매 often works.
- **MCMC**: Gibbs sampling — 매 conditional 의 sample 매 cycle.
- **Variational**: mean-field — 매 factorized $q(x) = \prod_v q_v(x_v)$.
- **Graph cut**: 매 binary submodular — 매 exact min-cut.

### 매 응용
1. Image segmentation (foreground/background MRF).
2. Conditional Random Fields (CRF) — sequence labeling.
3. Stereo / depth estimation (smoothness prior).
4. Energy-based generative models (EBMs).
5. Statistical physics (Ising, Potts).

## 💻 패턴

### Ising model — Gibbs sampling
```python
import numpy as np

def ising_gibbs(N=64, beta=0.44, steps=10_000, h=0.0):
    spins = np.random.choice([-1, 1], size=(N, N))
    for _ in range(steps):
        i, j = np.random.randint(N, size=2)
        nb = (spins[(i+1)%N, j] + spins[(i-1)%N, j]
            + spins[i, (j+1)%N] + spins[i, (j-1)%N])
        dE = 2 * spins[i, j] * (beta * nb + h)
        if dE < 0 or np.random.random() < np.exp(-dE):
            spins[i, j] *= -1
    return spins
```

### Loopy belief propagation (binary pairwise)
```python
def loopy_bp(unary, pairwise, edges, iters=20):
    # unary[v]   : log-potential per node, shape (V, K)
    # pairwise   : (E, K, K) log-potentials
    # edges      : list of (u, v)
    msg = {(u, v): np.zeros(unary.shape[1]) for u, v in edges}
    msg.update({(v, u): np.zeros(unary.shape[1]) for u, v in edges})
    for _ in range(iters):
        new = {}
        for (u, v), e_idx in zip(edges, range(len(edges))):
            incoming = unary[u] + sum(msg[(w, u)] for w in neighbors(u) if w != v)
            new[(u, v)] = np.logaddexp.reduce(
                pairwise[e_idx] + incoming[:, None], axis=0)
            new[(u, v)] -= new[(u, v)].max()   # normalize
        msg.update(new)
    beliefs = np.array([
        unary[v] + sum(msg[(u, v)] for u in neighbors(v))
        for v in range(len(unary))])
    return beliefs
```

### Linear-chain CRF (PyTorch — sequence labeling)
```python
import torch, torch.nn as nn

class LinearChainCRF(nn.Module):
    def __init__(self, n_tags):
        super().__init__()
        self.trans = nn.Parameter(torch.randn(n_tags, n_tags))

    def log_partition(self, emissions):  # (T, K)
        T, K = emissions.shape
        alpha = emissions[0]
        for t in range(1, T):
            alpha = torch.logsumexp(
                alpha[:, None] + self.trans + emissions[t][None, :], dim=0)
        return torch.logsumexp(alpha, dim=0)

    def score(self, emissions, tags):
        s = emissions[0, tags[0]]
        for t in range(1, len(tags)):
            s = s + self.trans[tags[t-1], tags[t]] + emissions[t, tags[t]]
        return s

    def nll(self, emissions, tags):
        return self.log_partition(emissions) - self.score(emissions, tags)
```

### Graph cut for binary MRF (submodular)
```python
import maxflow   # PyMaxflow

def binary_mrf_graph_cut(unary_fg, unary_bg, pairwise_w):
    H, W = unary_fg.shape
    g = maxflow.Graph[float]()
    nodes = g.add_grid_nodes((H, W))
    g.add_grid_edges(nodes, pairwise_w)               # smoothness
    g.add_grid_tedges(nodes, unary_fg, unary_bg)      # data term
    g.maxflow()
    return g.get_grid_segments(nodes)                 # bool mask
```

### Mean-field VI
```python
def mean_field(unary, pairwise, iters=10):
    # q(x_v = k) ∝ exp(unary[v,k] + Σ_{u∈N(v)} Σ_l q(x_u=l) * pairwise[v,u,k,l])
    q = torch.softmax(unary, dim=-1)
    for _ in range(iters):
        msg = torch.einsum('uvkl,ul->vk', pairwise, q)
        q = torch.softmax(unary + msg, dim=-1)
    return q
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Tree-structured | Sum-product (exact) |
| Loopy graph, fast | Loopy BP |
| Loopy graph, accurate | MCMC (Gibbs) — slower |
| Binary submodular | Graph cut (exact min-cut) |
| Sequence labeling (NER) | Linear-chain CRF |
| Image segmentation | Pairwise MRF + α-expansion / DenseCRF |
| Modern generative | Energy-Based Model (EBM) — score matching |

**기본값**: smallest model 의 first — chain → tree → loopy + BP → MCMC.

## 🔗 Graph
- 부모: [[Probabilistic Graphical Models]] · [[Probability Theory]]
- 응용: [[Image Segmentation]] · [[Energy-Based Model]]
- Adjacent: [[Bayesian Network]]

## 🤖 LLM 활용
**언제**: clique factorization 의 derive, BP/Gibbs pseudocode, partition-function intractability 의 explain.
**언제 X**: 매 specific paper algorithm — original 의 의 cross-check.

## ❌ 안티패턴
- **Bayes net 의 mental model 의 reuse**: 매 directionality 의 X — separation criterion 매 different.
- **Computing $Z$ for large graph**: 매 #P-hard — variational / MCMC.
- **Loopy BP on tightly-loopy graph**: 매 may diverge — damping 의 try, MCMC 의 fallback.
- **Linear-chain CRF 의 LSTM 으로 always replace**: 매 small data 의 still wins, 매 calibration 의 better.
- **Mean-field 의 multimodal posterior 의 use**: 매 mode-seeking — 매 underestimate variance.

## 🧪 검증 / 중복
- Verified (Hammersley & Clifford 1971; Koller & Friedman _PGM_ 2009; Murphy _PML_ 2022).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — MRF basics + BP/CRF/graph-cut 정리 |