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---
id: wiki-2026-0508-gaussian-processes
title: Gaussian Processes
title: Gaussian Processes (GP)
category: 10_Wiki/Topics
status: needs_review
status: verified
canonical_id: self
aliases: [MATH-GP-001]
aliases: [GP, Gaussian process, kernel methods, Bayesian regression, GPR, sparse GP]
duplicate_of: none
source_trust_level: A
confidence_score: 1.0
tags: ["Statistics|[Statistics", machine-learning, gaussian-processes, bayesian-inference, uncertainty]
confidence_score: 0.95
verification_status: applied
tags: [machine-learning, gaussian-process, bayesian, kernel-methods, regression, gpytorch]
raw_sources: []
last_reinforced: 2026-04-26
last_reinforced: 2026-05-10
github_commit: pending
inferred_by: Claude Opus 4.7 (auto-normalize 2026-05-08)
tech_stack:
language: Python
framework: GPyTorch / scikit-learn / GPy
---
# Gaussian Processes (가우시안 프로세스)
# Gaussian Processes
## 📌 한 줄 통찰 (The Karpathy Summary)
> "데이터를 지나는 무한히 많은 함수들의 분포를 정의하고, 가장 가능성 높은 길과 그 확신도를 계산하라" — 함수 자체를 무한 차원의 가우시안 분포로 취급하여, 어떤 입력에 대한 출력값을 평균과 표준편차를 가진 확률 분포로 제공하는 강력한 회귀 및 분류 도구.
## 매 한 줄
> **"매 distribution over functions"**. 매 mean function + kernel (covariance). 매 small data 의 의 의 SOTA, 매 uncertainty quantification 의 강함. 매 modern: 매 GPyTorch, 매 deep kernel, 매 sparse GP for large N. 매 Bayesian opt 의 backbone.
## 📖 구조화된 지식 (Synthesized Content)
- **추출된 패턴:** 입력 데이터 간의 유사성을 커널 함수(Kernel Function)로 정의하고, 가까운 지점은 비슷한 값을 가질 것이라는 믿음을 바탕으로 미지의 영역을 추론하는 베이지안 비모수 패턴.
- **핵심 요소:**
- **Mean Function:** 데이터가 없을 때의 기본 예측값 (보통 0).
- **Kernel (Covariance) Function:** 두 입력 사이의 관계(부드러움, 주기성 등)를 결정하는 핵심 파라미터.
- **[[Bayesian Inference|Bayesian Inference]]:** 새로운 관측 데이터가 유입될 때마다 사후 분포(Posterior)를 업데이트하여 불확실성 정제.
- **의의:** 딥러닝과 달리 모델의 '모름(Uncertainty)'을 정량화할 수 있어, 최적화(Bayesian [[Optimization|Optimization]])나 소량 데이터 학습에서 필수적으로 사용됨.
## 매 핵심
## ⚠️ 모순 및 업데이트 (Contradictions & Updates)
- **과거 데이터와의 충돌:** 연산 복잡도가 데이터 수의 세제곱($O(n^3)$)에 비례하여 대규모 데이터 처리가 불가능했으나, 최근에는 희소 근사(Sparse approximation) 기법을 통해 수백만 개의 데이터도 처리 가능하도록 진화.
- [[Epistemic-Uncertainty|Epistemic-Uncertainty]]를 정밀하게 측정하는 데 가장 적합한 도구로 재평가됨.
### 매 model
- **Prior**: f ~ GP(m(x), k(x, x')).
- **Posterior**: 매 conditioned on observed.
- **Predictive**: 매 mean + variance.
## 🔗 지식 연결 (Graph)
- [[Epistemic-Uncertainty|Epistemic-Uncertainty]], Bayesian-Inference, [[Black-Box-Optimization|Black-Box-Optimization]], Regression-Foundations
- **Raw Source:** 10_Wiki/Topics/AI/Gaussian-Processes.md
### 매 kernel
- **RBF / Gaussian**: 매 default.
- **Matérn**: 매 less smooth.
- **Linear**.
- **Periodic**: 매 cyclic.
- **Composite** (sum, product).
## 🤖 LLM 활용 힌트 (How to Use This Knowledge)
### 매 vs others
- **vs Linear regression**: 매 nonlinear.
- **vs NN**: 매 uncertainty native, 매 small-N.
- **vs Random Forest**: 매 smooth, 매 calibrated.
- **Limitation**: 매 O(N³).
**언제 이 지식을 쓰는가:**
- *(TODO)*
### 매 modern
- **Sparse GP** (FITC, VFE).
- **Deep Kernel Learning** (Wilson 2016).
- **Neural Tangent Kernel** (NTK).
- **GPyTorch** (scalable).
**언제 쓰면 안 되는가:**
- *(TODO)*
### 매 응용
1. **Bayesian opt**: 매 hyperparameter, A/B.
2. **Surrogate model**.
3. **Time series**.
4. **Active learning**.
5. **Geostatistics** (kriging).
6. **Robotics**.
## 🧪 검증 상태 (Validation)
## 💻 패턴
- **정보 상태:** needs_review
- **출처 신뢰도:** A
- **검토 이유:** *(P-Reinforce Phase 1 자동 정규화. 본문 검증 필요.)*
### scikit-learn GP
```python
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
## 🧬 중복 검사 (Duplicate Check)
kernel = C(1.0) * RBF(length_scale=1.0)
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=5)
gp.fit(X_train, y_train)
mean, std = gp.predict(X_test, return_std=True)
```
- **기존 유사 문서:** *(TODO: 인덱서 클러스터 리포트 참조)*
- **처리 방식:** UPDATE (자동 정규화)
- **처리 이유:** Phase 1 정규화 — 옛 템플릿/누락 필드 보강.
### GPyTorch (scalable)
```python
import gpytorch
import torch
## 🕓 변경 이력 (Changelog)
class ExactGP(gpytorch.models.ExactGP):
def __init__(self, X, y, likelihood):
super().__init__(X, y, likelihood)
self.mean = gpytorch.means.ConstantMean()
self.cov = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
def forward(self, x):
return gpytorch.distributions.MultivariateNormal(self.mean(x), self.cov(x))
| 날짜 | 변경 내용 | 처리 방식 | 신뢰도 |
|------|-----------|-----------|--------|
| 2026-05-08 | P-Reinforce Phase 1 정규화 (frontmatter + 헤더 표준화) | UPDATE | A |
likelihood = gpytorch.likelihoods.GaussianLikelihood()
model = ExactGP(X_train, y_train, likelihood).cuda()
optim = torch.optim.Adam(model.parameters(), lr=0.1)
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
model.train(); likelihood.train()
for _ in range(100):
optim.zero_grad()
out = model(X_train)
loss = -mll(out, y_train)
loss.backward(); optim.step()
model.eval(); likelihood.eval()
with torch.no_grad(), gpytorch.settings.fast_pred_var():
pred = likelihood(model(X_test))
mean, std = pred.mean, pred.stddev
```
### Bayesian opt (acquisition)
```python
from scipy.stats import norm
def expected_improvement(mean, std, best_y):
z = (mean - best_y) / std
return (mean - best_y) * norm.cdf(z) + std * norm.pdf(z)
def upper_confidence(mean, std, kappa=2.0):
return mean + kappa * std
def thompson_sample(gp, X_pool):
return gp.sample_y(X_pool, random_state=None).flatten().argmax()
```
### Sparse GP (large N)
```python
class SparseGP(gpytorch.models.ApproximateGP):
def __init__(self, inducing_points):
var_dist = gpytorch.variational.CholeskyVariationalDistribution(inducing_points.size(0))
var_strategy = gpytorch.variational.VariationalStrategy(self, inducing_points, var_dist)
super().__init__(var_strategy)
self.mean = gpytorch.means.ConstantMean()
self.cov = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
def forward(self, x):
return gpytorch.distributions.MultivariateNormal(self.mean(x), self.cov(x))
```
### Deep Kernel Learning
```python
class DeepKernelGP(gpytorch.models.ExactGP):
def __init__(self, X, y, likelihood):
super().__init__(X, y, likelihood)
self.feature_extractor = nn.Sequential(
nn.Linear(X.size(-1), 64), nn.ReLU(),
nn.Linear(64, 16),
)
self.mean = gpytorch.means.ConstantMean()
self.cov = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
def forward(self, x):
feat = self.feature_extractor(x)
return gpytorch.distributions.MultivariateNormal(self.mean(feat), self.cov(feat))
```
### Multi-output GP
```python
class MultiOutputGP(gpytorch.models.ExactGP):
def __init__(self, X, y, likelihood, n_outputs):
super().__init__(X, y, likelihood)
self.mean = gpytorch.means.MultitaskMean(gpytorch.means.ConstantMean(), num_tasks=n_outputs)
self.cov = gpytorch.kernels.MultitaskKernel(gpytorch.kernels.RBFKernel(), num_tasks=n_outputs)
def forward(self, x):
return gpytorch.distributions.MultitaskMultivariateNormal(self.mean(x), self.cov(x))
```
### Time series (with periodic kernel)
```python
periodic_kernel = gpytorch.kernels.PeriodicKernel()
trend_kernel = gpytorch.kernels.RBFKernel(length_scale=10)
self.cov = gpytorch.kernels.ScaleKernel(periodic_kernel + trend_kernel)
```
### Acquisition for BO loop
```python
def bo_loop(objective, bounds, n_iter=50, n_init=5):
X = sample_random(bounds, n_init)
y = np.array([objective(x) for x in X])
for _ in range(n_iter):
gp = fit_gp(X, y)
candidates = sample_random(bounds, 1000)
ei = expected_improvement(*gp.predict(candidates), best_y=y.max())
x_next = candidates[ei.argmax()]
y_next = objective(x_next)
X = np.vstack([X, x_next]); y = np.append(y, y_next)
return X[y.argmax()]
```
### Calibration check
```python
def calibration_plot(gp, X_test, y_test):
mean, std = gp.predict(X_test, return_std=True)
z_scores = (y_test - mean) / std
# 매 should be N(0, 1)
return np.histogram(z_scores)
```
## 매 결정 기준
| 상황 | Approach |
|---|---|
| Small N + uncertainty | Exact GP |
| Large N | Sparse GP |
| Deep features | Deep Kernel |
| Bayesian opt | Standard GP + EI |
| Time series | Periodic + RBF |
| Multi-output | Multi-task GP |
**기본값**: 매 GPyTorch + 매 RBF / Matérn kernel + 매 sparse for N > 1000 + 매 deep kernel for high-dim + 매 BO with EI.
## 🔗 Graph
- 부모: [[Bayesian-Methods]] · [[Kernel-Methods]]
- 변형: [[Sparse-GP]] · [[Deep-Kernel-Learning]]
- 응용: [[Bayesian-Optimization]] · [[Active-Learning]]
- Adjacent: [[Epistemic-Uncertainty]] · [[Neural-Tangent-Kernel]]
## 🤖 LLM 활용
**언제**: 매 small N. 매 uncertainty needed. 매 BO. 매 surrogate.
**언제 X**: 매 N > 100k (use sparse). 매 image / sequence.
## ❌ 안티패턴
- **Default kernel without thought**: 매 wrong assumption.
- **No length-scale optim**: 매 underfit.
- **N > 10k exact GP**: 매 OOM / slow.
- **GP for image**: 매 deep model better.
## 🧪 검증 / 중복
- Verified (Rasmussen & Williams GP for ML, GPyTorch).
- 신뢰도 A.
## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-04-26 | Auto |
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — exact + sparse + deep kernel + BO code |