Correction: Zhang, J.; Liu, K. Neural Information Squeezer for Causal Emergence. Entropy 2023, 25, 26

There was an error in the original publication. We found that due to previous negligence, some important content was missing in the previous manuscript [1]. It is a lemma in the appendix that supports the theorem.

A correction has been made to Appendix B:

Lemma A1.

(Bijection mapping does not affect mutual information): For any given continuous random variables $X$ and $Z$, if there is a bijection (one to one) mapping $f$ and another random variable $Y$ such that for any $x\in Dom \left(X\right)$  there is a $y=f \left(x\right)\in Dom \left(Y\right)$, and vice versa, where $Dom \left(X\right)$ denotes the domain of the variable $X$, then the mutual information between $X$ and $Z$ is equal to the information between $Y$ and $Z$, that is:

$$I \left(X;Z\right)=I \left(Y;Z\right).$$

(A15)

Proof. 

Because there is a one to one mapping $f:X\to Y$, we have:

$$p_X\left(x\right)=p_Y\left(y\right)\left|det J\right|,$$

(A16)

where $p_Y$ and $p_X$ are the density functions of $X,Y$, $J={\left.\frac{\partial f}{\partial X}\right|}_x$ is the Jacobian matrix of $f$, and if we insert Equation (A16) into the expression of the mutual information of $I (X;Z)$, and replace the integration for $x$ with the one for $y$, we have:

$$\begin{array}{cc}\hfill I \left(X;Z\right)& ={\int }_X{\int }_Z^{ }{p}_{XZ}\left(x,z\right)·\mathrm{l}\mathrm{n}\frac{p_{XZ}\left(x,z\right)}{p_X\left(x\right)p_Z\left(z\right)}·dz·dx\hfill \\ & ={\int }_X{\int }_Z^{ }{p}_X\left(x\right)p_{Z|X}\left(z|x\right)·\mathrm{l}\mathrm{n}\frac{p_X\left(x\right)p_{Z|X}\left(z|x\right)}{p_X\left(x\right)p_Z\left(z\right)}·dz·dx\hfill \\ & ={\int }_Y{\int }_Z^{ }|\mathrm{d}\mathrm{e}\mathrm{t}(J\left)\right|·p_Y\left(y\right){·p}_{Z|Y}\left(z|y\right)·\mathrm{l}\mathrm{n}\frac{p_{Z|Y}\left(z|y\right)}{p_Z\left(z\right)}·\left|det\left(J^{-1}\right)\right|·dz·dy\hfill \\ & =I \left(Y;Z\right).\hfill \end{array}$$

(A17)

And Equation (A17) can also be proved because of the commutativeness of the mutual information. □

Due to the insertion of Lemma A1, the number of following Lemma are changed.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.