36 lines
1.3 KiB
Markdown
36 lines
1.3 KiB
Markdown
---
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title: "Gibbs 后验"
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created: 2026-06-22
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updated: 2026-06-22
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type: concept
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tags: [robust-statistics, bayesian-inference, model-misspecification]
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sources: [nano-filter]
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---
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# Gibbs 后验
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Gibbs posterior 是标准 Bayesian 后验的推广，用于处理模型误设（model misspecification）场景。当真实数据生成过程与假定的似然模型不匹配时，Gibbs 后验用广义损失函数 $\ell_G(x_t, y_t)$ 替代负对数似然 $-\log p(y_t | x_t)$。
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## 定义
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Gibbs 后验是以下变分问题的解：
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$$
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p_G(x_t | y_{1:t}) = \arg\min_{q} \left\{ E_{q(x_t)}[\ell_G(x_t, y_t)] + D_{KL}(q(x_t) \| p(x_t | y_{1:t-1})) \right\}
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$$
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解析形式：
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$$
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p_G(x_t | y_{1:t}) = \frac{\exp\{-\ell_G(x_t, y_t)\} p(x_t | y_{1:t-1})}{\int \exp\{-\ell_G(x_t, y_t)\} p(x_t | y_{1:t-1}) dx_t}
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$$
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## NANO 的鲁棒扩展
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[[nano-filter|NANO filter]] 的推导仅依赖损失函数的一般形式，因此自然地支持 Gibbs 后验框架。论文提供两种损失函数选择：
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- **Huber 损失 / [[pseudo-huber-loss|Pseudo-Huber 损失]]**：对大残差以线性而非二次增长，抑制离群值影响
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- **加权对数似然**：通过数据依赖权重 $w(x_t, y_t)$ 缩放似然贡献
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## 参考
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- [[bayesian-filtering|Bayesian Filtering]]
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- [[pseudo-huber-loss|Pseudo-Huber Loss]]
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- [[nano-filter|NANO Filter]]
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