更改

<math>

<math>

y=f(x)+\varepsilon, \varepsilon\sim\mathcal{N}(0,\sigma^2)

y=f(x)+\xi, \xi\sim\mathcal{N}(0,\sigma^2)

</math>

</math>

|{{EquationRef|4}}}}

|{{EquationRef|4}}}}

<nowiki>Here, [math]p(y|x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)[/math] is the conditional probability density of y given x. Since [math]\varepsilon[/math] follows a normal distribution with mean 0 and variance [math]\sigma^2[/math], [math]y=f(x)+\varepsilon[/math] follows a normal distribution with mean [math]f(x)[/math] and variance [math]\sigma^2[/math].</nowiki>

<nowiki>Here, [math]p(y|x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)[/math] is the conditional probability density of y given x. Since [math]\xi[/math] follows a normal distribution with mean 0 and variance [math]\sigma^2[/math], [math]y=f(x)+\xi[/math] follows a normal distribution with mean [math]f(x)[/math] and variance [math]\sigma^2[/math].</nowiki>

The integration range of y is [math]f([-\frac{L}{L},\frac{L}{2}])[/math], i.e., the range of y is formed by mapping the domain [math][-\frac{L}{2},\frac{L}{2}][/math] of x through the function f.

The integration range of y is [math]f([-\frac{L}{L},\frac{L}{2}])[/math], i.e., the range of y is formed by mapping the domain [math][-\frac{L}{2},\frac{L}{2}][/math] of x through the function f.

Among them,<math>\epsilon</math> and <math>\delta</math> represent the magnitude of observation noise and intervention noise, respectively.-->This kind of derivation was first seen in Hoel's 2013 paper [1] and was further discussed in detail in the "Neural Information Squeezer" paper [2].

Among them,<math>\epsilon</math> and <math>\delta</math> represent the magnitude of observation noise and intervention noise, respectively.-->This kind of derivation was first seen in Hoel's 2013 paper [1] and was further discussed in detail in the "Neural Information Squeezer" paper [2].

===High-Dimensional Case===

===High-Dimensional Case===

We can extend the EI (Effective Information) calculation for one-dimensional variables to a more general n-dimensional scenario. Specifically:{{NumBlk|:|

We can extend the EI (Effective Information) calculation for one-dimensional variables to a more general n-dimensional scenario. Specifically:{{NumBlk|:|