Math and science::INF ML AI
x times derivative of ln of x
The following expression simplifies:
\[
x \dv{x} \ln(x) = 1
\]
You see this in the rewrite of the weighted score function:
\[
p(x | \theta) \cdot \pdv{\theta} \ln(p(x | \theta)) \; = \;
\pdv{\theta} p(x | \theta)
\]
Which is used to prove that the expectation of the score function over \( x \) is zero:
\[
\begin{align*}
\mathbb{E}_{x \sim p(x | \theta)} [s(x, \theta)] &=
\int_{-\infty}^{\infty} \pdv{\theta} \ln(p(x | \theta)) \cdot p(x | \theta) \, \dd x \\
&= \int_{-\infty}^{\infty} \pdv{\theta} p(x | \theta) \, \dd x \\
&= \pdv{\theta} \int_{-\infty}^{\infty} p(x | \theta) \, \dd x \\
&= \pdv{\theta} 1 \\
&= 0
\end{align*}
\]
Which, in turn, is used to describe the Fisher information as being the variance of the score function.