2nd/10_Wiki/Topics/AI_and_ML/Gaussian-Processes.md

---
id: wiki-2026-0508-gaussian-processes
title: Gaussian Processes (GP)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [GP, Gaussian process, kernel methods, Bayesian regression, GPR, sparse GP]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [machine-learning, gaussian-process, bayesian, kernel-methods, regression, gpytorch]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: GPyTorch / scikit-learn / GPy
---

# Gaussian Processes

## 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.

## 매 핵심

### 매 model
- **Prior**: f ~ GP(m(x), k(x, x')).
- **Posterior**: 매 conditioned on observed.
- **Predictive**: 매 mean + variance.

### 매 kernel
- **RBF / Gaussian**: 매 default.
- **Matérn**: 매 less smooth.
- **Linear**.
- **Periodic**: 매 cyclic.
- **Composite** (sum, product).

### 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).

### 매 modern
- **Sparse GP** (FITC, VFE).
- **Deep Kernel Learning** (Wilson 2016).
- **Neural Tangent Kernel** (NTK).
- **GPyTorch** (scalable).

### 매 응용
1. **Bayesian opt**: 매 hyperparameter, A/B.
2. **Surrogate model**.
3. **Time series**.
4. **Active learning**.
5. **Geostatistics** (kriging).
6. **Robotics**.

## 💻 패턴

### scikit-learn GP
```python
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C

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)
```

### GPyTorch (scalable)
```python
import gpytorch
import torch

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))

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: [[Kernel-Methods]]
- 변형: [[Sparse-GP]]
- 응용: [[Bayesian-Optimization]] · [[Active Learning]]
- Adjacent: [[Epistemic-Uncertainty]]

## 🤖 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 |