My question is:

Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality?

Let me outline the classical proof and explain why I am interested in this question.

Choose some function $g(X,Y)$. Then, \begin{align} E[ (X-E[X|Y]) g(X,Y)] \le \left| E[ (X-E[X|Y]) g(X,Y)] \right| \le \sqrt{ E \left[ (X-E[X|Y])^2 \right] E[g(X,Y)^2] }. \end{align}

Therefore, \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{\left| E[ (X-E[X|Y]) g(X,Y)] \right|}{E[g(X,Y)^2]}. \end{align}

The proof is completed by choosing $g(x,y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ and noting that then $ E[ (X-E[X|Y]) g(X,Y)]=-1$. This gives us the Cramer-Rao lower bound \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}. \end{align}

The choice of $g(X,Y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ always seemed mysterious to me (but this is not the main reason for ask this question). That is why I am wondering whether there is a more "natural" proof where the quantity $\frac{d}{dx} \log (f_{XY}(x,y) )$ appearance is more obvious.

For example, it would be nice if we can derive an inequality by showing that

\begin{align} E \left[ (X-E[X|Y])^2 \right] = \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}+c, \end{align} where $c$ is non-negative.

My question is:

Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality?

Let me outline the classical proof and explain why I am interested in this question.

Choose some function $g(X,Y)$. Then, \begin{align} E[ (X-E[X\mid Y]) g(X,Y)] \le \left| E[ (X-E[X\mid Y]) g(X,Y)] \right| \le \sqrt{ E \left[ (X-E[X\mid Y])^2 \right] E[g(X,Y)^2] }. \end{align}

Therefore, \begin{align} E \left[ (X-E[X\mid Y])^2 \right] \ge \frac{\left| E[ (X-E[X\mid Y]) g(X,Y)] \right|}{E[g(X,Y)^2]}. \end{align}

The proof is completed by choosing $g(x,y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ and noting that then $ E[ (X-E[X\mid Y]) g(X,Y)]=-1$. This gives us the Cramer-Rao lower bound \begin{align} E \left[ (X-E[X\mid Y])^2 \right] \ge \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}. \end{align}

The choice of $g(X,Y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ always seemed mysterious to me (but this is not the main reason for ask this question). That is why I am wondering whether there is a more "natural" proof where the quantity $\frac{d}{dx} \log (f_{XY}(x,y) )$ appearance is more obvious.

For example, it would be nice if we can derive an inequality by showing that

\begin{align} E \left[ (X-E[X\mid Y])^2 \right] = \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}+c, \end{align} where $c$ is non-negative.

My question is:

Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality?

Let me outline the classical proof and explain why I am interested in this question.

Choose some function $g(X,Y)$. Then, \begin{align} E[ (X-E[X|Y]) g(X,Y)] \le \left| E[ (X-E[X|Y]) g(X,Y)] \right| \le \sqrt{ E \left[ (X-E[X|Y])^2 \right] E[g(X,Y)^2] }. \end{align}

Therefore, \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{\left| E[ (X-E[X|Y]) g(X,Y)] \right|}{E[g(X,Y)^2]}. \end{align}

The proof is completed by choosing $g(x,y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ and noting that then $ E[ (X-E[X|Y]) g(X,Y)]=-1$. This gives us the Cramer-Rao lower bound \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}. \end{align}

The choice of $g(X,Y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ always seemed mysterious to me. That is why I am wondering whether there is a more "natural" proof where the quantity $\frac{d}{dx} \log (f_{XY}(x,y) )$ appearance is more obvious.

For example, can we find an identity

\begin{align} E \left[ (X-E[X|Y])^2 \right] = \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}+c, \end{align} where $c$ is non-negative.

My question is:

Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality?

Let me outline the classical proof and explain why I am interested in this question.

Choose some function $g(X,Y)$. Then, \begin{align} E[ (X-E[X|Y]) g(X,Y)] \le \left| E[ (X-E[X|Y]) g(X,Y)] \right| \le \sqrt{ E \left[ (X-E[X|Y])^2 \right] E[g(X,Y)^2] }. \end{align}

Therefore, \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{\left| E[ (X-E[X|Y]) g(X,Y)] \right|}{E[g(X,Y)^2]}. \end{align}

The proof is completed by choosing $g(x,y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ and noting that then $ E[ (X-E[X|Y]) g(X,Y)]=-1$. This gives us the Cramer-Rao lower bound \begin{align} E \left[ (X-E[X|Y])^2 \right] \ge \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}. \end{align}

The choice of $g(X,Y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ always seemed mysterious to me (but this is not the main reason for ask this question). That is why I am wondering whether there is a more "natural" proof where the quantity $\frac{d}{dx} \log (f_{XY}(x,y) )$ appearance is more obvious.

For example, it would be nice if we can derive an inequality by showing that

\begin{align} E \left[ (X-E[X|Y])^2 \right] = \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) ) \right)^2 \right]}+c, \end{align} where $c$ is non-negative.