2nd/10_Wiki/Topics/Domain_Programming/Frontend/Probabilistic-Graphical-Models.md

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id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-probabilistic-graphical-models	Probabilistic Graphical Models	10_Wiki/Topics	verified	self	PGM Bayesian Network Markov Random Field MRF	none	A	0.9	applied	pgm bayesian-network mrf inference machine-learning		2026-05-10	pending	language framework Python pgmpy/pyro/numpyro

Probabilistic Graphical Models

매 한 줄

"매 graph 로 joint distribution 의 factorization 표현". 매 random variable = node, dependency = edge. Bayesian Network (DAG) 와 Markov Random Field (undirected) 의 두 family. 2026 의 매 deep learning 시대에도 medical diagnosis, fault detection, causal inference 에서 핵심.

매 핵심

매 Two Families

• Bayesian Network (Directed): P(X) = ∏ᵢ P(Xᵢ | Pa(Xᵢ)). 매 causal direction 명시.
• Markov Random Field (Undirected): P(X) = (1/Z) ∏ φc(Xc). 매 symmetric correlation.
• Factor Graph: 매 두 family 통합 representation — bipartite (variable + factor nodes).

매 핵심 Operation

• Marginal inference: P(Xᵢ) 매 sum out 다른 variable.
• MAP inference: argmax P(X) — most likely assignment.
• Conditional: P(Xᵢ | E=e) — evidence 주어진 belief update.
• Learning: 매 parameter (MLE, EM) + structure (score-based, constraint-based).

매 Inference Algorithm

• Exact: Variable Elimination, Belief Propagation (tree), Junction Tree.
• Approx: MCMC (Gibbs, Metropolis-Hastings), Variational Inference, Loopy BP.

매 응용

1. Medical diagnosis: 매 symptom → disease causal network.
2. Fault detection: 매 sensor reading → root cause.
3. Computer vision: 매 CRF for image segmentation (DeepLab v3 의 backbone).
4. Causal inference: 매 do-calculus, counterfactual.

💻 패턴

pgmpy: Bayesian Network 정의

from pgmpy.models import DiscreteBayesianNetwork
from pgmpy.factors.discrete import TabularCPD

model = DiscreteBayesianNetwork([('Rain', 'Sprinkler'), ('Rain', 'Wet'), ('Sprinkler', 'Wet')])

cpd_rain = TabularCPD('Rain', 2, [[0.8], [0.2]])
cpd_sprinkler = TabularCPD('Sprinkler', 2, [[0.9, 0.5], [0.1, 0.5]], evidence=['Rain'], evidence_card=[2])
cpd_wet = TabularCPD('Wet', 2,
    [[1.0, 0.2, 0.1, 0.01], [0.0, 0.8, 0.9, 0.99]],
    evidence=['Rain', 'Sprinkler'], evidence_card=[2, 2])

model.add_cpds(cpd_rain, cpd_sprinkler, cpd_wet)
assert model.check_model()

Variable Elimination inference

from pgmpy.inference import VariableElimination

infer = VariableElimination(model)
result = infer.query(variables=['Rain'], evidence={'Wet': 1})
print(result)

NumPyro: Bayesian regression

import numpyro
import numpyro.distributions as dist
from numpyro.infer import MCMC, NUTS
import jax.numpy as jnp

def model(X, y=None):
    beta = numpyro.sample('beta', dist.Normal(jnp.zeros(X.shape[1]), 1.0))
    sigma = numpyro.sample('sigma', dist.HalfNormal(1.0))
    mu = jnp.dot(X, beta)
    numpyro.sample('y', dist.Normal(mu, sigma), obs=y)

mcmc = MCMC(NUTS(model), num_warmup=500, num_samples=1000)
mcmc.run(jax.random.PRNGKey(0), X, y)

Markov Random Field (CRF for sequence labeling)

import torch
from torchcrf import CRF

num_tags = 5
crf = CRF(num_tags, batch_first=True)

emissions = torch.randn(2, 10, num_tags)  # (batch, seq, tags)
tags = torch.randint(0, num_tags, (2, 10))
loss = -crf(emissions, tags)  # neg log likelihood

best = crf.decode(emissions)  # Viterbi

Pyro: Variational Inference

import pyro
import pyro.distributions as dist
from pyro.infer import SVI, Trace_ELBO
from pyro.optim import Adam

def model(data):
    z = pyro.sample('z', dist.Normal(0, 1))
    pyro.sample('obs', dist.Normal(z, 1), obs=data)

def guide(data):
    mu = pyro.param('mu', torch.tensor(0.))
    sigma = pyro.param('sigma', torch.tensor(1.), constraint=dist.constraints.positive)
    pyro.sample('z', dist.Normal(mu, sigma))

svi = SVI(model, guide, Adam({'lr': 0.01}), Trace_ELBO())
for step in range(1000):
    svi.step(data)

매 결정 기준

상황	Approach
Discrete, small state space, exact inference	pgmpy + VE
Continuous, large model	NumPyro + NUTS
Sequence labeling	CRF
Image segmentation	CRF + CNN (DeepLab)
Causal inference	DoWhy + pgmpy
Real-time, approx OK	Loopy BP / VI

기본값: 매 small problem → pgmpy. 매 large continuous → NumPyro NUTS.

🔗 Graph

• 부모: Probability Theory · Graph_Theory
• 변형: Bayesian_Network · Markov_Random_Field
• 응용: Image Segmentation · Causal Inference
• Adjacent: Hidden_Markov_Model · Variational_Inference · MCMC

🤖 LLM 활용

언제: 매 explicit causal structure 필요, interpretability 중요, small-data domain (medicine). 언제 X: 매 large-scale perception (이미지/음성) — neural network 가 우수.

❌ 안티패턴

• 모든 변수 fully connected: 매 parameter explosion — sparsity 활용.
• Exact inference 의 강행 in dense graph: NP-hard — approximate 사용.
• Causal direction 의 임의 가정: 매 domain knowledge 없으면 PC algorithm 등으로 학습.

🧪 검증 / 중복

• Verified (Koller & Friedman 2009, Murphy 2012, pgmpy 0.1.26, NumPyro 0.16).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — pgmpy/NumPyro/Pyro modern stack