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大小无更改 、 2024年6月23日 (星期日)
第668行: 第668行:  
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y|x)dydx&\approx \int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{-\infty}^{\infty}p(x)p(y|x)\ln p(y|x)dydx\\
 
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y|x)dydx&\approx \int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{-\infty}^{\infty}p(x)p(y|x)\ln p(y|x)dydx\\
 
&=\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{-\infty}^{\infty}\frac{1}{L}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\ln\left[\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\right]dydx\\
 
&=\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{-\infty}^{\infty}\frac{1}{L}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\ln\left[\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\right]dydx\\
&=\ln(\frac{L}{\sigma\cdot\sqrt{2\pi e}})
+
&=\ln(\frac{1}{\sigma\cdot\sqrt{2\pi e}})
 
\end{aligned}
 
\end{aligned}
 
</math>
 
</math>
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