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→一维函数映射
<math>
<math>
\begin{aligned}
\begin{aligned}
\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_{-\infty}^{\infty}\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_{-\infty}^{\infty}\int_{\infty,\infty])}p(x)p(y|x)\ln p(y|x)dydx\\
&=\int_{-\infty}^{\infty}\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_{-\infty}^{\infty}\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}{\sqrt{2\pi e}})
&=\ln(\frac{L}{\sqrt{2\pi e}})
<math>
<math>
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y)dydx \approx \frac{1}{L}\cdot\frac{1}{f'(x_0)}
\begin{aligned}
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y)dydx \approx \frac{1}{L}\cdot\frac{1}{f'(x_0)}\\
&\approx \ln(\frac{L}{\sqrt{2\pi e}})+\frac{1}{2L}\int_{-L/2}^{L/2}\ln \left(\frac{f'(x)}{\epsilon}\right)^2dx
\end{aligned}
</math>
</math>
如果同时考虑两种噪声，并且如果干预空间大小为<math>L
如果同时考虑两种噪声，并且如果干预空间大小为<math>L