46 lines
1.3 KiB
Markdown
46 lines
1.3 KiB
Markdown
---
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title: "高斯过程 (Gaussian Process)"
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created: 2026-06-17
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updated: 2026-06-17
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type: concept
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tags: [bayesian, stochastic-processes, kernel-methods, gaussian]
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sources: [raw/papers/ortega-phd-thesis-2026.md]
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confidence: high
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---
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# 高斯过程 (Gaussian Process)
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GP 是 Bayesian 机器学习的**核心非参数模型**——直接在函数空间上定义高斯分布先验。
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## 定义
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```
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f ~ GP(m(x), k(x, x'))
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```
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- `m(x) = E[f(x)]`：均值函数
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- `k(x, x') = Cov(f(x), f(x'))`：协方差/核函数
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有限点集上：`f(X) ~ N(m(X), K(X,X))`
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## 关键性质
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- **非参数**：模型容量随数据增长（无固定参数数量）
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- **解析后验**：观察到 (X,y) 后，f(x*) 的后验有封闭解
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- **校准不确定性**：预测方差 = 后验方差，天然校准
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- **核函数决定一切**：光滑性、周期性等由核编码
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## 在深度学习中的应用
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- **[[deep-gaussian-process|深度 GP]]**：层次化 GP 组合
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- **[[fixed-mean-gaussian-process|FMGP]]**：冻结 DNN 均值 + GP 协方差
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- **NTK 极限**：无限宽 NN ~ GP（[[neural-tangent-kernel|NTK]]）
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- **校准**：GP 后验提供原则性不确定性
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## 参考
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- [[deep-gaussian-process|深度 GP]]
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- [[fixed-mean-gaussian-process|FMGP]]
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- [[neural-tangent-kernel|NTK]]
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- [[ortega-phd-thesis|论文]]
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