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＃ bayesian network ＃＃ Laplace approximation

2022-07-18 19:01:00 【SyncStudy】

bayesian network

Laplace approximation

W M A P = a r g m a x W [ ] \bold{W}_{MAP}=arg max_\bold{W}[] WMAP​=argmaxW​[]

w r ∗ ∈ R w_r^* \in \mathbb{R} wr∗​∈R

[ p ( W ∣ y , X ) ∣ ∣ q ( W ) ] [p(\bold{W}|y,\bold{X})||q(\bold{W})] [p(W∣y,X)∣∣q(W)]

m i n W a s s [ N ( W ; W M A P , H − 1 ) ∣ ∣ N ( W s ; W M A P S , H s − 1 ) ] min Wass [\mathcal{N} (\bold{W};\bold{W}_{MAP}, H^{-1})|| \mathcal{N} (\bold{W}_s; \bold{W}_{MAP}^S, \bold{H}_s^{-1})] minWass[N(W;WMAP​,H−1)∣∣N(Ws​;WMAPS​,Hs−1​)]

f ( w ; w ) ≈ f ( w ; w M A P ) + ▽ f ( x ; w ) ∣ W M A P T f(w;w) \approx f(w;w_{MAP})+\triangledown f(x;w)|^T_{W_{MAP}} f(w;w)≈f(w;wMAP​)+▽f(x;w)∣WMAP​T​

y δ = A x + η y_\delta = Ax+\eta yδ​=Ax+η

A ∈ R d y × d x A \in \mathbb{R}^{d_y \times d_x} A∈Rdy​×dx​

η ∼ N ( 0 , σ y 2 I ) \eta \sim \mathcal{N}(0,\sigma^2_y I) η∼N(0,σy2​I)

d y < < d x d_y <<d_x dy​<<dx​

x ∈ R d x x \in \mathbb{R}^{d_x} x∈Rdx​

y δ ∈ R d y y_\delta \in \mathbb{R}^{d_y} yδ​∈Rdy​

w ∗ = a r g m i n w ∈ R d w ∣ ∣ A f ( w ) − y δ ∣ ∣ 2 + λ T V ( F ( w ) ) w^*= \mathop{arg min} \limits_{\bold{w} \in \mathbb{R}^{d_w}} ||\bold{A} f(w) - \bold{y}_\delta||^2+\lambda TV(F(w)) w∗=w∈Rdw​argmin​∣∣Af(w)−yδ​∣∣2+λTV(F(w))

T V ( f ( w ) ) = ∑ i , j TV(f(w))=\sum_{i,j} TV(f(w))=i,j∑​

p ( w ∣ l , σ 2 ) = N ( w ∣ 0 , ∑ ( l , σ 2 ) ) p(w|\mathcal{l}, \sigma^2)=\mathcal{N}(w|0,\sum(l, \sigma^2)) p(w∣l,σ2)=N(w∣0,∑(l,σ2))

∑ ( l , σ 2 ) \sum(l,\sigma^2) ∑(l,σ2)

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https://yzsam.com/2022/199/202207161511015371.html

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